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Centripetal force

A centripetal force (from Latin centrum, "center" and petere, "to seek" ) is a force that makes a body follow a curved path. The direction of the centripetal force is always orthogonal to the motion of the body and towards the fixed point of the instantaneous center of curvature of the path. Isaac Newton described it as "a force by which bodies are drawn or impelled, or in any way tend, towards a point as to a centre". In Newtonian mechanics, gravity provides the centripetal force causing astronomical orbits.

One common example involving centripetal force is the case in which a body moves with uniform speed along a circular path. The centripetal force is directed at right angles to the motion and also along the radius towards the centre of the circular path. The mathematical description was derived in 1659 by the Dutch physicist Christiaan Huygens.

Formula
From the kinematics of curved motion it is known that an object moving at tangential speed v along a path with radius of curvature r accelerates toward the center of curvature at a rate

Here, $$a_c$$ is the centripetal acceleration and $$\Delta \textbf{v}$$ is the difference between the velocity vectors at $$t+\Delta{t}$$ and $$t$$.

By Newton's second law, the cause of acceleration is a net force acting on the object, which is proportional to its mass m and its acceleration. The force, usually referred to as a centripetal force, has a magnitude

and is, like centripetal acceleration, directed toward the center of curvature of the object's trajectory.

Derivation

The centripetal acceleration can be inferred from the diagram of the velocity vectors at two instances. In the case of uniform circular motion the velocities have constant magnitude. Because each one is perpendicular to its respective position vector, simple vector subtraction implies two similar isosceles triangles with congruent angles – one comprising a base of $$\Delta \textbf{v}$$ and a leg length of $$v$$, and the other a base of $$\Delta \textbf{r}$$ (position vector difference) and a leg length of $$r$$: $$\frac{|\Delta \textbf{v}|}{v} = \frac{|\Delta \textbf{r}|}{r}$$ $$|\Delta \textbf{v}| = \frac{v}{r}|\Delta \textbf{r}|$$ Therefore, $$|\Delta\textbf{v}|$$ can be substituted with $$\frac{v}{r} |\Delta \textbf{r}|$$: $$a_c = \lim_{\Delta t \to 0} \frac{|\Delta \textbf{v}|}{\Delta t} = \frac{v}{r} \lim_{\Delta t \to 0} \frac{|\Delta \textbf{r}|}{\Delta t} = \omega\lim_{\Delta t \to 0} \frac{|\Delta \textbf{r}|}{\Delta t} = v\omega = \frac{v^2}{r}$$ The direction of the force is toward the center of the circle in which the object is moving, or the osculating circle (the circle that best fits the local path of the object, if the path is not circular). The speed in the formula is squared, so twice the speed needs four times the force, at a given radius.

This force is also sometimes written in terms of the angular velocity ω of the object about the center of the circle, related to the tangential velocity by the formula $$v = \omega r$$ so that $$F_c = m r \omega^2 \,.$$

Expressed using the orbital period T for one revolution of the circle,

the equation becomes $$F_c = m r \left(\frac{2\pi}{T}\right)^2.$$

In particle accelerators, velocity can be very high (close to the speed of light in vacuum) so the same rest mass now exerts greater inertia (relativistic mass) thereby requiring greater force for the same centripetal acceleration, so the equation becomes: $$F_c = \frac{\gamma m v^2}{r}$$ where

is the Lorentz factor.

Thus the centripetal force is given by: $$F_c = \gamma m v \omega$$ which is the rate of change of relativistic momentum $$\gamma m v$$.

Analysis of several cases
Below are three examples of increasing complexity, with derivations of the formulas governing velocity and acceleration.

Uniform circular motion
Uniform circular motion refers to the case of constant rate of rotation. Here are two approaches to describing this case.

Calculus derivation
In two dimensions, the position vector $$\textbf{r}$$, which has magnitude (length) $$r$$ and directed at an angle $$\theta$$ above the x-axis, can be expressed in Cartesian coordinates using the unit vectors and : $$ \textbf{r} = r \cos(\theta) \hat\mathbf x + r \sin(\theta) \hat\mathbf y. $$

The assumption of uniform circular motion requires three things:
 * 1) The object moves only on a circle.
 * 2) The radius of the circle $$r$$ does not change in time.
 * 3) The object moves with constant angular velocity $$\omega$$ around the circle. Therefore, $$\theta = \omega t$$ where $$t$$ is time.

The velocity $$\textbf{v}$$ and acceleration $$\textbf{a}$$ of the motion are the first and second derivatives of position with respect to time:

$$ \textbf{r} = r \cos(\omega t) \hat\mathbf x + r \sin(\omega t) \hat\mathbf y, $$

The term in parentheses is the original expression of $$\textbf{r}$$ in Cartesian coordinates. Consequently, $$ \textbf{a} = - \omega^2 \textbf{r}. $$ negative shows that the acceleration is pointed towards the center of the circle (opposite the radius), hence it is called "centripetal" (i.e. "center-seeking"). While objects naturally follow a straight path (due to inertia), this centripetal acceleration describes the circular motion path caused by a centripetal force.

Derivation using vectors
The image at right shows the vector relationships for uniform circular motion. The rotation itself is represented by the angular velocity vector Ω, which is normal to the plane of the orbit (using the right-hand rule) and has magnitude given by:
 * $$ |\mathbf{\Omega}| = \frac {\mathrm{d} \theta } {\mathrm{d}t} = \omega \, $$

with θ the angular position at time t. In this subsection, dθ/dt is assumed constant, independent of time. The distance traveled dℓ of the particle in time dt along the circular path is
 * $$ \mathrm{d}\boldsymbol{\ell} = \mathbf {\Omega} \times \mathbf{r}(t) \mathrm{d}t \, $$

which, by properties of the vector cross product, has magnitude rdθ and is in the direction tangent to the circular path.

Consequently,
 * $$\frac {\mathrm{d} \mathbf{r}}{\mathrm{d}t} = \lim_{{\Delta}t \to 0} \frac {\mathbf{r}(t + {\Delta}t)-\mathbf{r}(t)}{{\Delta}t} = \frac{\mathrm{d} \boldsymbol{\ell}}{\mathrm{d}t} \ .$$

In other words,
 * $$ \mathbf{v}\ \stackrel{\mathrm{def}}{ = }\ \frac {\mathrm{d} \mathbf{r}}{\mathrm{d}t} = \frac {\mathrm{d}\mathbf{\boldsymbol{\ell}}}{\mathrm{d}t} = \mathbf {\Omega} \times \mathbf{r}(t)\ . $$

Differentiating with respect to time, $$ \mathbf{a}\ \stackrel{\mathrm{def}}{ = }\ \frac {\mathrm{d} \mathbf{v}} {d\mathrm{t}} = \mathbf {\Omega} \times \frac{\mathrm{d} \mathbf{r}(t)}{\mathrm{d}t} = \mathbf{\Omega} \times \left[ \mathbf {\Omega} \times \mathbf{r}(t)\right] \ .$$

Lagrange's formula states: $$ \mathbf{a} \times \left ( \mathbf{b} \times \mathbf{c} \right ) = \mathbf{b} \left ( \mathbf{a} \cdot \mathbf{c} \right ) - \mathbf{c} \left ( \mathbf{a} \cdot \mathbf{b} \right ) \ .$$

Applying Lagrange's formula with the observation that Ω • r(t) = 0 at all times, $$ \mathbf{a} = - {|\mathbf{\Omega|}}^2 \mathbf{r}(t) \ .$$

In words, the acceleration is pointing directly opposite to the radial displacement r at all times, and has a magnitude: $$ |\mathbf{a}| = |\mathbf{r}(t)| \left ( \frac {\mathrm{d} \theta}{\mathrm{d}t} \right) ^2 = r {\omega}^2 $$ where vertical bars |...| denote the vector magnitude, which in the case of r(t) is simply the radius r of the path. This result agrees with the previous section, though the notation is slightly different.

When the rate of rotation is made constant in the analysis of nonuniform circular motion, that analysis agrees with this one.

A merit of the vector approach is that it is manifestly independent of any coordinate system.

Example: The banked turn

The upper panel in the image at right shows a ball in circular motion on a banked curve. The curve is banked at an angle θ from the horizontal, and the surface of the road is considered to be slippery. The objective is to find what angle the bank must have so the ball does not slide off the road. Intuition tells us that, on a flat curve with no banking at all, the ball will simply slide off the road; while with a very steep banking, the ball will slide to the center unless it travels the curve rapidly.

Apart from any acceleration that might occur in the direction of the path, the lower panel of the image above indicates the forces on the ball. There are two forces; one is the force of gravity vertically downward through the center of mass of the ball mg, where m is the mass of the ball and g is the gravitational acceleration; the second is the upward normal force exerted by the road at a right angle to the road surface man. The centripetal force demanded by the curved motion is also shown above. This centripetal force is not a third force applied to the ball, but rather must be provided by the net force on the ball resulting from vector addition of the normal force and the force of gravity. The resultant or net force on the ball found by vector addition of the normal force exerted by the road and vertical force due to gravity must equal the centripetal force dictated by the need to travel a circular path. The curved motion is maintained so long as this net force provides the centripetal force requisite to the motion.

The horizontal net force on the ball is the horizontal component of the force from the road, which has magnitude $|F_{h}| = m|a_{n}| sin θ$. The vertical component of the force from the road must counteract the gravitational force: $|F_{v}| = m|a_{n}| cos θ = m|g|$, which implies $|a_{n}| = |g| / cos θ$. Substituting into the above formula for $|F_{h}|$ yields a horizontal force to be:

On the other hand, at velocity |v| on a circular path of radius r, kinematics says that the force needed to turn the ball continuously into the turn is the radially inward centripetal force Fc of magnitude: $$|\mathbf{F}_\mathrm{c}| = m |\mathbf{a}_\mathrm{c}| = \frac{m|\mathbf{v}|^2}{r} \,. $$

Consequently, the ball is in a stable path when the angle of the road is set to satisfy the condition: $$m |\mathbf{g}| \tan \theta = \frac{m|\mathbf{v}|^2}{r} \, ,$$ or, $$ \tan \theta = \frac {|\mathbf{v}|^2} {|\mathbf{g}|r} \, .$$

As the angle of bank θ approaches 90°, the tangent function approaches infinity, allowing larger values for |v|2/r. In words, this equation states that for greater speeds (bigger |v|) the road must be banked more steeply (a larger value for θ), and for sharper turns (smaller r) the road also must be banked more steeply, which accords with intuition. When the angle θ does not satisfy the above condition, the horizontal component of force exerted by the road does not provide the correct centripetal force, and an additional frictional force tangential to the road surface is called upon to provide the difference. If friction cannot do this (that is, the coefficient of friction is exceeded), the ball slides to a different radius where the balance can be realized.

These ideas apply to air flight as well. See the FAA pilot's manual.

Nonuniform circular motion

As a generalization of the uniform circular motion case, suppose the angular rate of rotation is not constant. The acceleration now has a tangential component, as shown the image at right. This case is used to demonstrate a derivation strategy based on a polar coordinate system.

Let r(t) be a vector that describes the position of a point mass as a function of time. Since we are assuming circular motion, let $r(t) = R·u_{r}$, where R is a constant (the radius of the circle) and ur is the unit vector pointing from the origin to the point mass. The direction of ur is described by θ, the angle between the x-axis and the unit vector, measured counterclockwise from the x-axis. The other unit vector for polar coordinates, uθ is perpendicular to ur and points in the direction of increasing θ. These polar unit vectors can be expressed in terms of Cartesian unit vectors in the x and y directions, denoted $$\hat\mathbf i$$ and $$\hat\mathbf j$$ respectively: $$\mathbf u_r = \cos \theta \ \hat\mathbf i + \sin \theta \ \hat\mathbf j$$ and $$\mathbf u_\theta = - \sin \theta \ \hat\mathbf i + \cos \theta \ \hat\mathbf j.$$

One can differentiate to find velocity: $$\begin{align} \mathbf{v} &= r \frac {d \mathbf{u}_r}{dt} \\ &= r \frac {d}{dt} \left( \cos \theta \ \hat\mathbf{i} + \sin \theta \ \hat\mathbf{j}\right) \\ &= r \frac {d \theta}{dt} \frac{d}{d \theta} \left( \cos \theta \ \hat\mathbf{i} + \sin \theta \ \hat\mathbf{j}\right) \\ & = r \frac {d \theta} {dt} \left( -\sin \theta \ \hat\mathbf{i} + \cos \theta \ \hat\mathbf{j}\right)\\ & = r \frac{d\theta}{dt} \mathbf{u}_\theta \\ & = \omega r \mathbf{u}_\theta \end{align}$$ where $ω$ is the angular velocity $dθ/dt$.

This result for the velocity matches expectations that the velocity should be directed tangentially to the circle, and that the magnitude of the velocity should be $rω$. Differentiating again, and noting that $$\frac {d\mathbf{u}_\theta}{dt} = -\frac{d\theta}{dt} \mathbf{u}_r = - \omega \mathbf{u}_r \, $$ we find that the acceleration, a is: $$\mathbf{a} = r \left( \frac {d\omega}{dt} \mathbf{u}_\theta - \omega^2 \mathbf{u}_r \right) \. $$

Thus, the radial and tangential components of the acceleration are: $$\mathbf{a}_{r} = - \omega^{2} r \ \mathbf{u}_r = - \frac{|\mathbf{v}|^2}{r} \ \mathbf{u}_r $$ and $$\mathbf{a}_\theta = r \ \frac {d\omega}{dt} \ \mathbf{u}_\theta = \frac {d | \mathbf{v} | }{dt} \ \mathbf{u}_\theta \, $$ where $|v| = r ω$ is the magnitude of the velocity (the speed).

These equations express mathematically that, in the case of an object that moves along a circular path with a changing speed, the acceleration of the body may be decomposed into a perpendicular component that changes the direction of motion (the centripetal acceleration), and a parallel, or tangential component, that changes the speed.

Polar coordinates
The above results can be derived perhaps more simply in polar coordinates, and at the same time extended to general motion within a plane, as shown next. Polar coordinates in the plane employ a radial unit vector uρ and an angular unit vector uθ, as shown above. A particle at position r is described by:

$$\mathbf{r} = \rho \mathbf{u}_{\rho} \, $$

where the notation ρ is used to describe the distance of the path from the origin instead of R to emphasize that this distance is not fixed, but varies with time. The unit vector uρ travels with the particle and always points in the same direction as r(t). Unit vector uθ also travels with the particle and stays orthogonal to uρ. Thus, uρ and uθ form a local Cartesian coordinate system attached to the particle, and tied to the path travelled by the particle. By moving the unit vectors so their tails coincide, as seen in the circle at the left of the image above, it is seen that uρ and uθ form a right-angled pair with tips on the unit circle that trace back and forth on the perimeter of this circle with the same angle θ(t) as r(t).

When the particle moves, its velocity is

 * $$ \mathbf{v} = \frac {\mathrm{d} \rho }{\mathrm{d}t} \mathbf{u}_{\rho} + \rho \frac {\mathrm{d} \mathbf{u}_{\rho}}{\mathrm{d}t} \, . $$

To evaluate the velocity, the derivative of the unit vector uρ is needed. Because uρ is a unit vector, its magnitude is fixed, and it can change only in direction, that is, its change duρ has a component only perpendicular to uρ. When the trajectory r(t) rotates an amount dθ, uρ, which points in the same direction as r(t), also rotates by dθ. See image above. Therefore, the change in uρ is

 * $$ \mathrm{d} \mathbf{u}_{\rho} = \mathbf{u}_{\theta} \mathrm{d}\theta \,, $$

or

 * $$ \frac {\mathrm{d} \mathbf{u}_{\rho}}{\mathrm{d}t} = \mathbf{u}_{\theta} \frac {\mathrm{d}\theta}{\mathrm{d}t} \, . $$

In a similar fashion, the rate of change of uθ is found. As with uρ, uθ is a unit vector and can only rotate without changing size. To remain orthogonal to uρ while the trajectory r(t) rotates an amount dθ, uθ, which is orthogonal to r(t), also rotates by dθ. See image above. Therefore, the change duθ is orthogonal to uθ and proportional to dθ (see image above):

 * $$ \frac{\mathrm{d} \mathbf{u}_{\theta}}{\mathrm{d}t} = -\frac {\mathrm{d} \theta} {\mathrm{d}t} \mathbf{u}_{\rho} \, . $$

The equation above shows the sign to be negative: to maintain orthogonality, if duρ is positive with dθ, then duθ must decrease.

Substituting the derivative of uρ into the expression for velocity:

 * $$ \mathbf{v} = \frac {\mathrm{d} \rho }{\mathrm{d}t} \mathbf{u}_{\rho} + \rho \mathbf{u}_{\theta} \frac {\mathrm{d} \theta} {\mathrm{d}t} = v_{\rho} \mathbf{u}_{\rho} + v_{\theta} \mathbf{u}_{\theta} = \mathbf{v}_{\rho} + \mathbf{v}_{\theta} \, . $$

To obtain the acceleration, another time differentiation is done:
 * $$ \mathbf{a} = \frac {\mathrm{d}^2 \rho }{\mathrm{d}t^2} \mathbf{u}_{\rho} + \frac {\mathrm{d} \rho }{\mathrm{d}t} \frac{\mathrm{d} \mathbf{u}_{\rho}}{\mathrm{d}t} + \frac {\mathrm{d} \rho}{\mathrm{d}t} \mathbf{u}_{\theta} \frac {\mathrm{d} \theta} {\mathrm{d}t} + \rho \frac{\mathrm{d} \mathbf{u}_{\theta}}{\mathrm{d}t} \frac {\mathrm{d} \theta} {\mathrm{d}t} + \rho \mathbf{u}_{\theta} \frac {\mathrm{d}^2 \theta} {\mathrm{d}t^2} \, . $$

Substituting the derivatives of uρ and uθ, the acceleration of the particle is:
 * $$\begin{align}

\mathbf{a} & = \frac {\mathrm{d}^2 \rho }{\mathrm{d}t^2} \mathbf{u}_{\rho} + 2\frac {\mathrm{d} \rho}{\mathrm{d}t} \mathbf{u}_{\theta} \frac {\mathrm{d} \theta} {\mathrm{d}t} - \rho \mathbf{u}_{\rho} \left( \frac {\mathrm{d} \theta} {\mathrm{d}t}\right)^2 + \rho \mathbf{u}_{\theta} \frac {\mathrm{d}^2 \theta} {\mathrm{d}t^2} \, \\ & = \mathbf{u}_{\rho} \left[ \frac {\mathrm{d}^2 \rho }{\mathrm{d}t^2}-\rho\left( \frac {\mathrm{d} \theta} {\mathrm{d}t}\right)^2 \right] + \mathbf{u}_{\theta}\left[ 2\frac {\mathrm{d} \rho}{\mathrm{d}t} \frac {\mathrm{d} \theta} {\mathrm{d}t} + \rho \frac {\mathrm{d}^2 \theta} {\mathrm{d}t^2}\right] \\ & = \mathbf{u}_{\rho} \left[ \frac {\mathrm{d}v_{\rho}}{\mathrm{d}t}-\frac{v_{\theta}^2}{\rho}\right] + \mathbf{u}_{\theta}\left[ \frac{2}{\rho}v_{\rho} v_{\theta} + \rho\frac{\mathrm{d}}{\mathrm{d}t}\frac{v_{\theta}}{\rho}\right] \,. \end{align}$$

As a particular example, if the particle moves in a circle of constant radius R, then dρ/dt = 0, v = vθ, and:

$$\mathbf{a} = \mathbf{u}_{\rho} \left[ -\rho\left( \frac {\mathrm{d} \theta} {\mathrm{d}t}\right)^2 \right] + \mathbf{u}_{\theta}\left[ \rho \frac {\mathrm{d}^2 \theta} {\mathrm{d}t^2}\right] = \mathbf{u}_{\rho} \left[ -\frac{v^2}{r}\right] + \mathbf{u}_{\theta}\left[ \frac {\mathrm{d} v} {\mathrm{d}t}\right] \ $$

where $$ v = v_{\theta}. $$

These results agree with those above for nonuniform circular motion. See also the article on non-uniform circular motion. If this acceleration is multiplied by the particle mass, the leading term is the centripetal force and the negative of the second term related to angular acceleration is sometimes called the Euler force.

For trajectories other than circular motion, for example, the more general trajectory envisioned in the image above, the instantaneous center of rotation and radius of curvature of the trajectory are related only indirectly to the coordinate system defined by uρ and uθ and to the length |r(t)| = ρ. Consequently, in the general case, it is not straightforward to disentangle the centripetal and Euler terms from the above general acceleration equation. To deal directly with this issue, local coordinates are preferable, as discussed next.

Local coordinates

Local coordinates mean a set of coordinates that travel with the particle, and have orientation determined by the path of the particle. Unit vectors are formed as shown in the image at right, both tangential and normal to the path. This coordinate system sometimes is referred to as intrinsic or path coordinates or nt-coordinates, for normal-tangential, referring to these unit vectors. These coordinates are a very special example of a more general concept of local coordinates from the theory of differential forms.

Distance along the path of the particle is the arc length s, considered to be a known function of time.

 * $$ s = s(t) \ . $$

A center of curvature is defined at each position s located a distance ρ (the radius of curvature) from the curve on a line along the normal un (s). The required distance ρ(s) at arc length s is defined in terms of the rate of rotation of the tangent to the curve, which in turn is determined by the path itself. If the orientation of the tangent relative to some starting position is θ(s), then ρ(s) is defined by the derivative dθ/ds:

 * $$\frac{1} {\rho (s)} = \kappa (s) = \frac {\mathrm{d}\theta}{\mathrm{d}s}\ . $$

The radius of curvature usually is taken as positive (that is, as an absolute value), while the curvature κ is a signed quantity.

A geometric approach to finding the center of curvature and the radius of curvature uses a limiting process leading to the osculating circle. See image above.

Using these coordinates, the motion along the path is viewed as a succession of circular paths of ever-changing center, and at each position s constitutes non-uniform circular motion at that position with radius ρ. The local value of the angular rate of rotation then is given by:

 * $$ \omega(s) = \frac{\mathrm{d}\theta}{\mathrm{d}t} = \frac{\mathrm{d}\theta}{\mathrm{d}s} \frac {\mathrm{d}s}{\mathrm{d}t} = \frac{1}{\rho(s)}\ \frac {\mathrm{d}s}{\mathrm{d}t} = \frac{v(s)}{\rho(s)}\ ,$$

with the local speed v given by:

 * $$ v(s) = \frac {\mathrm{d}s}{\mathrm{d}t}\ . $$

As for the other examples above, because unit vectors cannot change magnitude, their rate of change is always perpendicular to their direction (see the left-hand insert in the image above):

 * $$\frac{d\mathbf{u}_\mathrm{n}(s)}{ds} = \mathbf{u}_\mathrm{t}(s)\frac{d\theta}{ds} = \mathbf{u}_\mathrm{t}(s)\frac{1}{\rho} \ ; $$ $$\frac{d\mathbf{u}_\mathrm{t}(s)}{\mathrm{d}s} = -\mathbf{u}_\mathrm{n}(s)\frac{\mathrm{d}\theta}{\mathrm{d}s} = - \mathbf{u}_\mathrm{n}(s)\frac{1}{\rho} \ . $$

Consequently, the velocity and acceleration are:
 * $$ \mathbf{v}(t) = v \mathbf{u}_\mathrm{t}(s)\ ; $$

and using the chain-rule of differentiation:

 * $$ \mathbf{a}(t) = \frac{\mathrm{d}v}{\mathrm{d}t} \mathbf{u}_\mathrm{t}(s) - \frac{v^2}{\rho}\mathbf{u}_\mathrm{n}(s) \ ; $$ with the tangential acceleration $$\frac{\mathrm{\mathrm{d}}v}{\mathrm{\mathrm{d}}t} = \frac{\mathrm{d}v}{\mathrm{d}s}\ \frac{\mathrm{d}s}{\mathrm{d}t} = \frac{\mathrm{d}v}{\mathrm{d}s}\ v \ . $$

In this local coordinate system, the acceleration resembles the expression for nonuniform circular motion with the local radius ρ(s), and the centripetal acceleration is identified as the second term.

Extending this approach to three dimensional space curves leads to the Frenet–Serret formulas.

Alternative approach
Looking at the image above, one might wonder whether adequate account has been taken of the difference in curvature between ρ(s) and ρ(s + ds) in computing the arc length as ds = ρ(s)dθ. Reassurance on this point can be found using a more formal approach outlined below. This approach also makes connection with the article on curvature.

To introduce the unit vectors of the local coordinate system, one approach is to begin in Cartesian coordinates and describe the local coordinates in terms of these Cartesian coordinates. In terms of arc length s, let the path be described as: $$\mathbf{r}(s) = \left[ x(s),\ y(s) \right]. $$

Then an incremental displacement along the path ds is described by: $$\mathrm{d}\mathbf{r}(s) = \left[ \mathrm{d}x(s),\ \mathrm{d}y(s) \right] = \left[ x'(s),\ y'(s) \right] \mathrm{d}s \, $$

where primes are introduced to denote derivatives with respect to s. The magnitude of this displacement is ds, showing that:
 * $$\left[ x'(s)^2 + y'(s)^2 \right] = 1 \ . $$ (Eq. 1)

This displacement is necessarily a tangent to the curve at s, showing that the unit vector tangent to the curve is: $$\mathbf{u}_\mathrm{t}(s) = \left[ x'(s), \ y'(s) \right], $$ while the outward unit vector normal to the curve is $$\mathbf{u}_\mathrm{n}(s) = \left[ y'(s),\ -x'(s) \right], $$

Orthogonality can be verified by showing that the vector dot product is zero. The unit magnitude of these vectors is a consequence of Eq. 1. Using the tangent vector, the angle θ of the tangent to the curve is given by: $$\sin \theta = \frac{y'(s)}{\sqrt{x'(s)^2 + y'(s)^2}} = y'(s) \ ;$$ and $$\cos \theta = \frac{x'(s)}{\sqrt{x'(s)^2 + y'(s)^2}} = x'(s) \ .$$

The radius of curvature is introduced completely formally (without need for geometric interpretation) as: $$\frac{1}{\rho} = \frac{\mathrm{d}\theta}{\mathrm{d}s}\. $$

The derivative of θ can be found from that for sinθ: $$\frac{\mathrm{d} \sin\theta}{\mathrm{d}s} = \cos \theta \frac {\mathrm{d}\theta}{\mathrm{d}s} = \frac{1}{\rho} \cos \theta \ = \frac{1}{\rho} x'(s)\. $$

Now: $$\frac{\mathrm{d} \sin \theta }{\mathrm{d}s} = \frac{\mathrm{d}}{\mathrm{d}s} \frac{y'(s)}{\sqrt{x'(s)^2 + y'(s)^2}} = \frac{y(s)x'(s)^2-y'(s)x'(s)x(s)} {\left(x'(s)^2 + y'(s)^2\right)^{3/2}}\, $$ in which the denominator is unity. With this formula for the derivative of the sine, the radius of curvature becomes: $$\frac {\mathrm{d}\theta}{\mathrm{d}s} = \frac{1}{\rho} = y(s)x'(s) - y'(s)x(s) = \frac{y(s)}{x'(s)} = -\frac{x(s)}{y'(s)} \ ,$$ where the equivalence of the forms stems from differentiation of Eq. 1: $$x'(s)x(s) + y'(s)y(s) = 0 \. $$ With these results, the acceleration can be found: $$\begin{align} \mathbf{a}(s) &= \frac{\mathrm{d}}{\mathrm{d}t}\mathbf{v}(s) = \frac{\mathrm{d}}{\mathrm{d}t}\left[\frac{\mathrm{d}s}{\mathrm{d}t} \left( x'(s), \ y'(s) \right) \right] \\ & = \left(\frac{\mathrm{d}^2s}{\mathrm{d}t^2}\right)\mathbf{u}_\mathrm{t}(s) + \left(\frac{\mathrm{d}s}{\mathrm{d}t}\right) ^2 \left(x(s),\ y(s) \right) \\ & = \left(\frac{\mathrm{d}^2s}{\mathrm{d}t^2}\right)\mathbf{u}_\mathrm{t}(s) - \left(\frac{\mathrm{d}s}{\mathrm{d}t}\right) ^2 \frac{1}{\rho} \mathbf{u}_\mathrm{n}(s) \end{align}$$ as can be verified by taking the dot product with the unit vectors ut(s) and un(s). This result for acceleration is the same as that for circular motion based on the radius ρ. Using this coordinate system in the inertial frame, it is easy to identify the force normal to the trajectory as the centripetal force and that parallel to the trajectory as the tangential force. From a qualitative standpoint, the path can be approximated by an arc of a circle for a limited time, and for the limited time a particular radius of curvature applies, the centrifugal and Euler forces can be analyzed on the basis of circular motion with that radius.

This result for acceleration agrees with that found earlier. However, in this approach, the question of the change in radius of curvature with s is handled completely formally, consistent with a geometric interpretation, but not relying upon it, thereby avoiding any questions the image above might suggest about neglecting the variation in ρ.

Example: circular motion
To illustrate the above formulas, let x, y be given as:
 * $$x = \alpha \cos \frac{s}{\alpha} \ ; \ y = \alpha \sin\frac{s}{\alpha} \ .$$

Then:
 * $$x^2 + y^2 = \alpha^2 \, $$

which can be recognized as a circular path around the origin with radius α. The position s = 0 corresponds to [α, 0], or 3 o'clock. To use the above formalism, the derivatives are needed:

 * $$y^{\prime}(s) = \cos \frac{s}{\alpha} \ ; \ x^{\prime}(s) = -\sin \frac{s}{\alpha} \, $$
 * $$y^{\prime\prime}(s) = -\frac{1}{\alpha}\sin\frac{s}{\alpha} \ ; \ x^{\prime\prime}(s) = -\frac{1}{\alpha}\cos \frac{s}{\alpha} \ . $$

With these results, one can verify that:
 * $$ x^{\prime}(s)^2 + y^{\prime}(s)^2 = 1 \ ; \ \frac{1}{\rho} = y^{\prime\prime}(s)x^{\prime}(s)-y^{\prime}(s)x^{\prime\prime}(s) = \frac{1}{\alpha} \ . $$

The unit vectors can also be found:
 * $$\mathbf{u}_\mathrm{t}(s) = \left[-\sin\frac{s}{\alpha} \, \ \cos\frac{s}{\alpha} \right] \ ; \ \mathbf{u}_\mathrm{n}(s) = \left[\cos\frac{s}{\alpha} \ , \ \sin\frac{s}{\alpha} \right] \ , $$

which serve to show that s = 0 is located at position [ρ, 0] and s = ρπ/2 at [0, ρ], which agrees with the original expressions for x and y. In other words, s is measured counterclockwise around the circle from 3 o'clock. Also, the derivatives of these vectors can be found:
 * $$\frac{\mathrm{d}}{\mathrm{d}s}\mathbf{u}_\mathrm{t}(s) = -\frac{1}{\alpha} \left[\cos\frac{s}{\alpha} \, \ \sin\frac{s}{\alpha} \right] = -\frac{1}{\alpha}\mathbf{u}_\mathrm{n}(s) \ ; $$
 * $$ \ \frac{\mathrm{d}}{\mathrm{d}s}\mathbf{u}_\mathrm{n}(s) = \frac{1}{\alpha} \left[-\sin\frac{s}{\alpha} \, \ \cos\frac{s}{\alpha} \right] = \frac{1}{\alpha}\mathbf{u}_\mathrm{t}(s) \ . $$

To obtain velocity and acceleration, a time-dependence for s is necessary. For counterclockwise motion at variable speed v(t):
 * $$s(t) = \int_0^t \ dt^{\prime} \ v(t^{\prime}) \, $$

where v(t) is the speed and t is time, and s(t = 0) = 0. Then:
 * $$\mathbf{v} = v(t)\mathbf{u}_\mathrm{t}(s) \ ,$$
 * $$\mathbf{a} = \frac{\mathrm{d}v}{\mathrm{d}t}\mathbf{u}_\mathrm{t}(s) + v\frac{\mathrm{d}}{\mathrm{d}t}\mathbf{u}_\mathrm{t}(s) = \frac{\mathrm{d}v}{\mathrm{d}t}\mathbf{u}_\mathrm{t}(s)-v\frac{1}{\alpha}\mathbf{u}_\mathrm{n}(s)\frac{\mathrm{d}s}{\mathrm{d}t} $$
 * $$\mathbf{a} = \frac{\mathrm{d}v}{\mathrm{d}t}\mathbf{u}_\mathrm{t}(s)-\frac{v^2}{\alpha}\mathbf{u}_\mathrm{n}(s) \, $$

where it already is established that α = ρ. This acceleration is the standard result for non-uniform circular motion.

title: Centripetal force


text:
Commodore

Commodore may refer to:

Ranks

 * Commodore (rank), a naval rank
 * Commodore (Royal Navy), in the United Kingdom
 * Commodore (India), in India
 * Commodore (United States)
 * Commodore (Canada)
 * Commodore (Finland)
 * Commodore (Germany) or Kommodore
 * Air commodore, a rank in the Royal Air Force and other Commonwealth air forces
 * Commodore (yacht club), an officer of a yacht club
 * Commodore (Sea Scouts), a position in the Boy Scouts of America's Sea Scout program
 * Convoy commodore, a civilian in charge of a shipping convoy during the Second World War

Fiction

 * The Commodore, a Horatio Hornblower novel by C. S. Forester
 * The Commodore (book), a novel in the Aubrey–Maturin series by Patrick O'Brian

Music and music venues

 * Commodore Ballroom, a nightclub and music venue in Vancouver, British Columbia
 * Commodore Records, a jazz and swing music record label
 * Commodores, an American soul/funk band

People

 * "The Commodore", the nickname of American entrepreneur Cornelius Vanderbilt (1794–1877)
 * "Commodore Kuehnle", the nickname of American entrepreneur and politician Louis Kuehnle (1857–1934)
 * Aaron Commodore (1819/20 – 1892), African American politician
 * Mike Commodore (born 1979), a Canadian ice hockey player
 * Commodore P. Vedder (1838–1910), New York politician

Places

 * Commodore, Pennsylvania, United States
 * Commodore Island, a Canadian island in Hudson Bay
 * Phoenix Park Hotel, hotel built in Washington, D.C., in 1922, known originally as "The Commodore"

Automobiles

 * Hudson Commodore, an automobile produced from 1941 to 1952 in the US by Hudson Motor Car Company
 * Opel Commodore, an automobile produced from 1967 to 1982 in Germany by Adam Opel AG
 * Holden Commodore, an automobile produced from 1978 to 2017 in Australia by the Holden division of General Motors

Ships and boats

 * Commodore (shipwreck), an American steamboat shipwrecked off the coast of Florida
 * Commodore 17, an American sailboat design
 * USS Commodore, the name of various United States Navy ships

Aircraft

 * Consolidated Commodore, a flying boat used for passenger travel in the 1930s

Computers

 * Commodore International (also named Commodore Business Machines), a computer company that operated from 1954 to 1994
 * Commodore 64, an 8-bit home computer introduced in January 1982 by Commodore International
 * Commodore USA, a computer company that operated from 2010 to 2013 after purchasing the classic brand name

Other uses

 * Several genera or species of brush-footed butterflies, in particular:
 * Precis
 * Auzakia danava
 * Junonia artaxia
 * Vanderbilt Commodores, intercollegiate athletics teams for Vanderbilt University

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Commodore


text:
Conditioning

Conditioning may refer to:

Science, computing, and technology

 * Air conditioning, the removal of heat from indoor air for thermal comfort
 * Automobile air conditioning, air conditioning in a vehicle
 * Ice storage air conditioning, air conditioning using ice storage
 * Solar air conditioning, air conditioning using solar power
 * Beer conditioning, maturation, clarification, and stabilisation of beer
 * Bottle conditioning, above conditioning after bottling
 * Chemical conditioning, improvement and stabilization chemical components
 * Data conditioning, the use of data management techniques in a computer system
 * Flow conditioning, the study of the movement of fluids in pipes
 * Signal conditioning, manipulating an analog signal in such a way that it meets the requirements of the next stage

Biology and physical fitness

 * Aerobic conditioning, exercise which trains the heart and lungs to pump blood more efficiently
 * Body conditioning via physical exercise

Learning

 * Classical conditioning or Pavlovian conditioning, a behavioral mechanism in which one stimulus comes to signal the occurrence of a second stimulus
 * Eyeblink conditioning, classical conditioning involving pairing of a stimulus with an eyeblink-eliciting stimulus
 * Fear conditioning, classical conditioning involving aversive stimuli
 * Second-order conditioning, a two-step process in classical conditioning
 * Covert conditioning, classical and operant conditioning in mental health treatment
 * Operant conditioning or instrumental conditioning, a form of learning in which behavior is modified by its consequences
 * Social conditioning, operant conditioning training individuals to act in a society
 * Evaluative conditioning, a form of learning in which attitude towards one stimulus is learnt by its pairing with a second stimulus

Mathematics

 * Condition number also known as Conditioning (numerical analysis), a quantity describing whether or not a numerical problem is well-behaved
 * Conditioning (probability), a concept in probability theory

title: Conditioning


text:
Checksum

A checksum is a small-sized block of data derived from another block of digital data for the purpose of detecting errors that may have been introduced during its transmission or storage. By themselves, checksums are often used to verify data integrity but are not relied upon to verify data authenticity.

The procedure which generates this checksum is called a checksum function or checksum algorithm. Depending on its design goals, a good checksum algorithm usually outputs a significantly different value, even for small changes made to the input. This is especially true of cryptographic hash functions, which may be used to detect many data corruption errors and verify overall data integrity; if the computed checksum for the current data input matches the stored value of a previously computed checksum, there is a very high probability the data has not been accidentally altered or corrupted.

Checksum functions are related to hash functions, fingerprints, randomization functions, and cryptographic hash functions. However, each of those concepts has different applications and therefore different design goals. For instance, a function returning the start of a string can provide a hash appropriate for some applications but will never be a suitable checksum. Checksums are used as cryptographic primitives in larger authentication algorithms. For cryptographic systems with these two specific design goals, see HMAC.

Check digits and parity bits are special cases of checksums, appropriate for small blocks of data (such as Social Security numbers, bank account numbers, computer words, single bytes, etc.). Some error-correcting codes are based on special checksums which not only detect common errors but also allow the original data to be recovered in certain cases.

Parity byte or parity word
The simplest checksum algorithm is the so-called longitudinal parity check, which breaks the data into "words" with a fixed number $n$ of bits, and then computes the bitwise exclusive or (XOR) of all those words. The result is appended to the message as an extra word. In simpler terms, for $n$=1 this means adding a bit to the end of the data bits to guarantee that there is an even number of '1's. To check the integrity of a message, the receiver computes the bitwise exclusive or of all its words, including the checksum; if the result is not a word consisting of $n$ zeros, the receiver knows a transmission error occurred.

With this checksum, any transmission error which flips a single bit of the message, or an odd number of bits, will be detected as an incorrect checksum. However, an error that affects two bits will not be detected if those bits lie at the same position in two distinct words. Also swapping of two or more words will not be detected. If the affected bits are independently chosen at random, the probability of a two-bit error being undetected is $1/n$.

Sum complement
A variant of the previous algorithm is to add all the "words" as unsigned binary numbers, discarding any overflow bits, and append the two's complement of the total as the checksum. To validate a message, the receiver adds all the words in the same manner, including the checksum; if the result is not a word full of zeros, an error must have occurred. This variant, too, detects any single-bit error, but the pro modular sum is used in SAE J1708.

Position-dependent
The simple checksums described above fail to detect some common errors which affect many bits at once, such as changing the order of data words, or inserting or deleting words with all bits set to zero. The checksum algorithms most used in practice, such as Fletcher's checksum, Adler-32, and cyclic redundancy checks (CRCs), address these weaknesses by considering not only the value of each word but also its position in the sequence. This feature generally increases the cost of computing the checksum.

Fuzzy checksum
The idea of fuzzy checksum was developed for detection of email spam by building up cooperative databases from multiple ISPs of email suspected to be spam. The content of such spam may often vary in its details, which would render normal checksumming ineffective. By contrast, a "fuzzy checksum" reduces the body text to its characteristic minimum, then generates a checksum in the usual manner. This greatly increases the chances of slightly different spam emails producing the same checksum. The ISP spam detection software, such as SpamAssassin, of co-operating ISPs, submits checksums of all emails to the centralised service such as DCC. If the count of a submitted fuzzy checksum exceeds a certain threshold, the database notes that this probably indicates spam. ISP service users similarly generate a fuzzy checksum on each of their emails and request the service for a spam likelihood.

General considerations
A message that is $m$ bits long can be viewed as a corner of the $m$-dimensional hypercube. The effect of a checksum algorithm that yields an $n$-bit checksum is to map each $m$-bit message to a corner of a larger hypercube, with dimension $m + n$. The $2^{m + n}$ corners of this hypercube represent all possible received messages. The valid received messages (those that have the correct checksum) comprise a smaller set, with only $2^{m}$ corners.

A single-bit transmission error then corresponds to a displacement from a valid corner (the correct message and checksum) to one of the $m$ adjacent corners. An error which affects $k$ bits moves the message to a corner which is $k$ steps removed from its correct corner. The goal of a good checksum algorithm is to spread the valid corners as far from each other as possible, to increase the likelihood "typical" transmission errors will end up in an invalid corner.

title: Checksum


text:
Cultural evolution (disambiguation)

Cultural evolution is cultural change viewed from an evolutionary perspective.

It may also refer to:
 * Behavioral ecology, the study of the evolutionary basis for animal behavior due to ecological pressures
 * Cultural selection theory, studies of cultural change modelled on evolutionary biology
 * Dual-inheritance theory, a specific framework for studying cultural evolution
 * Memetics, neo-Darwinist view of the transmission of cultural traits
 * Sociocultural evolution, the change of cultures and societies over time as studied in anthropology

title: Cultural evolution (disambiguation)


text:
City University of New York

The City University of New York (CUNY, spoken, ) is the public university system of New York City. It is the largest urban university system in the United States, comprising 25 campuses: eleven senior colleges, seven community colleges, and seven professional institutions. In 1960, John R. Everett became the first chancellor of the Municipal College System of New York City, later known as the City University of New York (CUNY). CUNY, established by New York State legislation in 1961 and signed into law by Governor Nelson Rockefeller, was an amalgamation of existing institutions and a new graduate school.

The system was governed by the Board of Higher Education of the City of New York, created in 1926, and later renamed the Board of Trustees of the CUNY in 1979. The institutions merged into CUNY included the Free Academy (later City College of New York), the Female Normal and High School (later Hunter College), Brooklyn College, and Queens College. CUNY has historically provided accessible education, especially to those excluded or unable to afford private universities. The first community college in New York City was established in 1955 with shared funding between the state and the city, but unlike the senior colleges, community college students had to pay tuition.

The integration of CUNY's colleges into a single university system took place in 1961, under a chancellor and with state funding. The Graduate Center, serving as the principal doctorate-granting institution, was also established that year. In 1964, Mayor Robert F. Wagner Jr. extended the senior colleges' free tuition policy to community colleges.The 1960s saw student protests demanding more racial diversity and academic representation in CUNY, leading to the establishment of Medgar Evers College and the implementation of the Open Admissions policy in 1970. This policy dramatically increased student diversity but also introduced challenges like low retention rates. The 1976 fiscal crisis ended the free tuition policy, leading to the introduction of tuition fees for all CUNY colleges. The university enrolls more than 275,000 students and counts thirteen Nobel Prize winners and twenty-four MacArthur Fellows among its alumni.

Founding
In 1960, John R. Everett became the first chancellor of the Municipal College System of the City of New York, later renamed CUNY, for a salary of $25,000 ($0 in current dollar terms). CUNY was created in 1961, by New York State legislation, signed into law by Governor Nelson Rockefeller. The legislation integrated existing institutions and a new graduate school into a coordinated system of higher education for the city, under the control of the "Board of Higher Education of the City of New York", which had been created by New York State legislation in 1926. By 1979, the Board of Higher Education had become the "Board of Trustees of the CUNY".

The institutions that were merged to create CUNY were:

 * The Free Academy – Founded in 1847 by Townsend Harris, it was fashioned as "a Free Academy for the purpose of extending the benefits of education gratuitously to persons who have been pupils in the common schools of the city and county of New York." The Free Academy later became the City College of New York.
 * The Female Normal and High School – Founded in 1870, and later renamed the Normal College. It would be renamed again in 1914 to Hunter College. During the early 20th century, Hunter College expanded into the Bronx, with what became Herbert Lehman College.
 * Brooklyn College – Founded in 1930.
 * Queens College – Founded in 1937.

Accessible education
CUNY has served a diverse student body, especially those excluded from or unable to afford private universities. Its four-year colleges offered a high-quality, tuition-free education to the poor, the working class, and the immigrants of New York City who met the grade requirements for matriculated status. During the post-World War I era, when some Ivy League universities, such as Yale University, discriminated against Jews, many Jewish academics and intellectuals studied and taught at CUNY. The City College of New York developed a reputation of being "the Harvard of the proletariat."

As New York City's population and public college enrollment grew during the early 20th century and the city struggled for resources, the municipal colleges slowly began adopting selective tuition, also known as instructional fees, for a handful of courses and programs. During the Great Depression, with funding for public colleges severely constrained, limits were imposed on the size of the colleges' free Day Sessions, and tuition was imposed upon students deemed "competent" but not academically qualified for the day program. Most of these "limited matriculation" students enrolled in the Evening Sessions, and paid tuition. Additionally, as the population of New York grew, CUNY was not able to accommodate the demand for higher education. Higher and higher requirements for admission were imposed; in 1965, a student seeking admission to CUNY needed an average grade of 92 or A−. This helped to ensure that the student population of CUNY remained largely white and middle-class.

Demand in the United States for higher education rapidly grew after World War II, and during the mid-1940s a movement began to create community colleges to provide accessible education and training. In New York City, however, the community college movement was constrained by many factors including "financial problems, narrow perceptions of responsibility, organizational weaknesses, adverse political factors, and other competing priorities."

Community colleges would have drawn from the same city coffers that were funding the senior colleges, and city higher education officials were of the view that the state should finance them. It was not until 1955, under a shared-funding arrangement with New York State, that New York City established its first community college, on Staten Island. Unlike the day college students attending the city's public baccalaureate colleges for free, community college students had to pay tuition fees under the state-city funding formula. Community college students paid tuition fees for approximately 10 years.

Over time, tuition fees for limited-matriculated students became an important source of system revenues. In fall 1957, for example, nearly 36,000 attended Hunter, Brooklyn, Queens and City Colleges for free, but another 24,000 paid tuition fees of up to $300 a year ($0 in current dollar terms). Undergraduate tuition and other student fees in 1957 comprised 17 percent of the colleges' $46.8 million in revenues, about $7.74 million ($0 in current dollar terms).

Three community colleges had been established by early 1961 when New York City's public colleges were codified by the state as a single university with a chancellor at the helm and an infusion of state funds. But the city's slowness in creating the community colleges as demand for college seats was intensifying and had resulted in mounting frustration, particularly on the part of minorities, that college opportunities were not available to them.

In 1964, as New York City's Board of Higher Education moved to take full responsibility for the community colleges, city officials extended the senior colleges' free tuition policy to them, a change that was included by Mayor Robert F. Wagner Jr. in his budget plans and took effect with the 1964–65 academic year.

Calls for greater access to public higher education from the black and Puerto Rican communities in New York, especially in Brooklyn, led to the founding of "Community College Number 7," later Medgar Evers College, in 1966–1967. In 1969, a group of black and Puerto Rican students occupied City College and demanded the racial integration of CUNY, which at the time had an overwhelmingly white student body.

Student protests
Students at some campuses became increasingly frustrated with the university's and Board of Higher Education's handling of university administration. At Baruch College in 1967, over a thousand students protested the plan to make the college an upper-division school limited to junior, senior, and graduate students. At Brooklyn College in 1968, students attempted a sit-in to demand the admission of more black and Puerto Rican students and additional black studies curriculum. Students at Hunter College also demanded a Black studies program. Members of the SEEK program, which provided academic support for underprepared and underprivileged students, staged a building takeover at Queens College in 1969 to protest the decisions of the program's director, who would later be replaced by a black professor. Puerto Rican students at Bronx Community College filed a report with the New York State Division of Human Rights in 1970, contending that the intellectual level of the college was inferior and discriminatory. Hunter College was crippled for several days by a protest of 2,000 students who had a list of demands focusing on more student representation in college administration. Across CUNY, students boycotted their campuses in 1970 to protest a rise in student fees and other issues, including the proposed (and later implemented) open admissions plan.

Like many college campuses in 1970, CUNY faced a number of protests and demonstrations after the Kent State massacre and Cambodian Campaign. The Administrative Council of the City University of New York sent U.S. president Richard Nixon a telegram in 1970 stating, "No nation can long endure the alienation of the best of its young people." Some colleges, including John Jay College of Criminal Justice, historically the "college for cops," held teach-ins in addition to student and faculty protests.

In April of 2024, CUNY students joined other campuses across the United States in protests against the Israel–Hamas war. The student protestors demanded that CUNY divest from companies with ties to Israel and that CUNY officials cancel any upcoming trips to Israel and protect students involved in the demonstrations.

Open admissions
Under pressure from community activists and CUNY Chancellor Albert Bowker, the Board of Higher Education (BHE) approved an open admissions plan in 1966, but it was not scheduled to be fully implemented until 1975. However, in 1969, students and faculty across CUNY participated in rallies, student strikes, and class boycotts demanding an end to CUNY's restrictive admissions policies. CUNY administrators and Mayor John Lindsay expressed support for these demands, and the BHE voted to implement the plan immediately in the fall of 1970.

All high school graduates were guaranteed entrance to the university without having to fulfill traditional requirements such as exams or grades. The policy nearly doubled the number of students enrolled in the CUNY system to 35,000 (compared to 20,000 the year before). Black and Hispanic student enrollment increased threefold. Remedial education, to supplement the training of under-prepared students, became a significant part of CUNY's offerings. Additionally, ethnic and Black Studies programs and centers were instituted on many CUNY campuses, contributing to the growth of similar programs nationwide.

However, retention of students in CUNY during this period was low, with two-thirds of students enrolled in the early 1970s leaving within four years without graduating.

Financial crisis of 1976
In fall 1976, during New York City's fiscal crisis, the free tuition policy was discontinued under pressure from the federal government, the financial community that had a role in rescuing the city from bankruptcy, and New York State, which would take over the funding of CUNY's senior colleges. Tuition, which had been in place in the State University of New York system since 1963, was instituted at all CUNY colleges.

Meanwhile, CUNY students were added to the state's need-based Tuition Assistance Program (TAP), which had been created to help private colleges. Full-time students who met the income eligibility criteria were permitted to receive TAP, ensuring for the first time that financial hardship would deprive no CUNY student of a college education. Within a few years, the federal government would create its own need-based program, known as Pell Grants, providing the neediest students with a tuition-free college education. Joseph S. Murphy was Chancellor of the City University of New York from 1982 to 1990, when he resigned. CUNY at the time was the third-largest university in the United States, with over 180,000 students.

By 2011, nearly six of ten full-time undergraduates qualified for a tuition-free education at CUNY due in large measure to state, federal and CUNY financial aid programs. CUNY's enrollment dipped after tuition was re-established, and there were further enrollment declines through the 1980s and into the 1990s.

Financial crisis of 1995
In 1995, CUNY suffered another fiscal crisis when Governor George Pataki proposed a drastic cut in state financing. Faculty cancelled classes and students staged protests. By May, CUNY adopted deep cuts to college budgets and class offerings. By June, to save money spent on remedial programs, CUNY adopted a stricter admissions policy for its senior colleges: students deemed unprepared for college would not be admitted, this a departure from the 1970 Open Admissions program. That year's final state budget cut funding by $102 million, which CUNY absorbed by increasing tuition by $750 and offering a retirement incentive plan for faculty.

In 1999, a task force appointed by Mayor Rudolph Giuliani issued a report that described CUNY as "an institution adrift" and called for an improved, more cohesive university structure and management, as well as more consistent academic standards. Following the report, Matthew Goldstein, a mathematician and City College graduate who had led CUNY's Baruch College and briefly, Adelphi University, was appointed chancellor. CUNY ended its policy of open admissions to its four-year colleges, raised its admissions standards at its most selective four-year colleges (Baruch, Brooklyn, City, Hunter and Queens), and required new enrollees who needed remediation to begin their studies at a CUNY open-admissions community college.

2010 onward
CUNY's enrollment of degree-credit students reached 220,727 in 2005 and 262,321 in 2010 as the university broadened its academic offerings. The university added more than 2,000 full-time faculty positions, opened new schools and programs, and expanded the university's fundraising efforts to help pay for them. Fundraising increased from $35 million in 2000 to more than $200 million in 2012.

As of Autumn 2013, all CUNY undergraduates are required to take an administration-dictated common core of courses which have been claimed to meet specific "learning outcomes" or standards. Since the courses are accepted university-wide, the administration claims it will be easier for students to transfer course credits between CUNY colleges. It also reduced the number of core courses some CUNY colleges had required, to a level below national norms, particularly in the sciences. The program is the target of several lawsuits by students and faculty, and was the subject of a "no confidence" vote by the faculty, who rejected it by an overwhelming 92% margin.

Chancellor Goldstein retired on July 1, 2013, and was replaced on June 1, 2014, by James Milliken, president of the University of Nebraska, and a graduate of the University of Nebraska and New York University School of Law. Milliken retired at the end of the 2018 academic year and moved on to become the chancellor for the University of Texas system.

In 2018, CUNY opened its 25th campus, the CUNY School of Labor and Urban Studies, named after former president Joseph S. Murphy and combining some forms and functions of the Murphy Institute that were housed at the CUNY School of Professional Studies.

On February 13, 2019, the board of trustees voted to appoint Queens College president Felix V. Matos Rodriguez as the chancellor of the City University of New York. Matos became both the first Latino and minority educator to head the university. He assumed the post May 1.

Enrollment and demographics
CUNY is the fourth-largest university system in the United States by enrollment, behind the California State University, State University of New York (SUNY), and University of California systems. More than 271,000-degree-credit students, continuing, and professional education students are enrolled at campuses located in all five New York City boroughs.

The university has one of the most diverse student bodies in the United States, with students hailing from around the world, although most students live in New York City. The black, white and Hispanic undergraduate populations each comprise more than a quarter of the student body, and Asian undergraduates make up 18 percent. Fifty-eight percent are female, and 28 percent are 25 or older. In the 2017–2018 award year, 144,380 CUNY students received the Federal Pell Grant.

CUNY Citizenship Now!
Founded in 1997 by immigration lawyer Allan Wernick, CUNY Citizenship Now! is an immigration assistance organization that provides free and confidential immigration law services to help individuals and families on their path to U.S. citizenship. In 2021, CUNY launched a College Immigrant Ambassador Program in partnership with the New York City Department of Education.

Management structure
The forerunner of today's City University of New York was governed by the Board of Education of New York City. Members of the Board of Education, chaired by the president of the board, served as ex officio trustees. For the next four decades, the board members continued to serve as ex officio trustees of the College of the City of New York and the city's other municipal college, the Normal College of the City of New York.

In 1900, the New York State Legislature created separate boards of trustees for the College of the City of New York and the Normal College, which became Hunter College in 1914. In 1926, the legislature established the Board of Higher Education of the City of New York, which assumed supervision of both municipal colleges.

In 1961, the New York State Legislature established the City University of New York, uniting what had become seven municipal colleges at the time: the City College of New York, Hunter College, Brooklyn College, Queens College, Staten Island Community College, Bronx Community College and Queensborough Community College. In 1979, the CUNY Financing and Governance Act was adopted by the State and the Board of Higher Education became the City University of New York Board of Trustees.

Today, the City University is governed by the board of trustees composed of 17 members, ten of whom are appointed by the governor of New York "with the advice and consent of the senate," and five by the mayor of New York City "with the advice and consent of the senate." The final two trustees are ex officio members. One is the chair of the university's student senate, and the other is non-voting and is the chair of the university's faculty senate. Both the mayoral and gubernatorial appointments to the CUNY Board are required to include at least one resident of each of New York City's five boroughs. Trustees serve seven-year terms, which are renewable for another seven years. The chancellor is elected by the board of trustees, and is the "chief educational and administrative officer" of the City University.

The administrative offices are in Midtown Manhattan.

Faculty
CUNY employs 6,700 full-time faculty members and over 10,000 adjunct faculty members. Faculty and staff are represented by the Professional Staff Congress (PSC), a labor union and chapter of the American Federation of Teachers.

Notable faculty

 * André Aciman, writer, Graduate Center
 * Ali Jimale Ahmed, poet and professor of Comparative Literature, Queens College and Graduate Center
 * F. Murray Abraham, actor of stage and screen; professor of theater, winner of the Academy Award for Best Actor, Brooklyn College
 * Chantal Akerman, film director, City College of New York
 * Meena Alexander, poet and writer, Graduate Center and Hunter College
 * Hannah Arendt, philosopher and political theorist; author of The Origins of Totalitarianism (1951) and The Human Condition (1958), Brooklyn College
 * Talal Asad, anthropologist, Graduate Center
 * John Ashbery, poet, Pulitzer Prize for Poetry winner, Brooklyn College
 * William Bialek, biophysicist, Graduate Center
 * Edwin G. Burrows, historian and writer, Pulitzer Prize for History winner for co-writing Gotham: A History of New York City to 1898 with Mike Wallace, Brooklyn College
 * Ron Carter, jazz bassist, City College
 * Joe Chambers, jazz drummer, City College
 * Dee L. Clayman, classicist, Graduate Center
 * Margaret Clapp, scholar, winner of the Pulitzer Prize for Biography or Autobiography, president of Wellesley College, Brooklyn College
 * Ta-Nehisi Coates, writer, journalist, and activist, CUNY Graduate School of Journalism
 * Billy Collins, poet, U.S. Poet Laureate, Lehman College (retired)
 * Blanche Wiesen Cook, historian, John Jay College of Criminal Justice and Graduate Center
 * John Corigliano, composer, Graduate Center
 * Michael Cunningham, writer, winner of Pulitzer Prize for Fiction and PEN/Faulkner Award for The Hours, Brooklyn College
 * Roy DeCarava, artist and photographer, Hunter College
 * Carolyn Eisele, mathematician, Hunter College
 * Nancy Fraser, philosopher and political scientist, Graduate Center
 * Ruth Wilson Gilmore, geographer, Graduate Center
 * Allen Ginsberg, beat poet, Brooklyn College
 * Aaron Goodelman, sculptor
 * Joel Glucksman, Olympic saber fencer, Brooklyn College
 * Ralph Goldstein, Olympic épée fencer, Brooklyn College
 * Michael Grossman, economist, Graduate Center
 * Kimiko Hahn, poet, winner of PEN/Voelcker Award for Poetry, Queens College
 * David Harvey, geographer, Graduate Center
 * Jimmy Heath, jazz saxophonist, City College
 * Bell Hooks, educator, writer and critic, City College of New York
 * Karen Brooks Hopkins, president of the Brooklyn Academy of Music, Brooklyn College
 * John Hospers, first presidential candidate of the US Libertarian Party, Brooklyn College
 * Tyehimba Jess, poet, winner of Pulitzer Prize for Poetry, College of Staten Island
 * KC Johnson born (1967), Brooklyn College and Graduate Center
 * Sheila Jordan, jazz vocalist, City College
 * Michio Kaku, physicist, City College
 * Jane Katz, Olympian swimmer, John Jay College of Criminal Justice
 * Alfred Kazin, writer and critic, Hunter College and Graduate Center
 * Saul Kripke, philosopher, Graduate Center
 * Irving Kristol, journalist, City College
 * Paul Krugman, economist, Graduate Center
 * Peter Kwong, journalist, filmmaker, activist, Hunter College and Graduate Center
 * Nathan H. Lents, scientist, author, and science communicator, John Jay College of Criminal Justice
 * Ben Lerner, writer, MacArthur Fellow, Brooklyn College
 * Audre Lorde, poet and activist, City College, John Jay College of Criminal Justice
 * Cate Marvin, poet, Guggenheim Fellowship winner, College of Staten Island
 * Abraham Maslow, psychologist in the school of humanistic psychology, best known for his theory of human motivation which led to a therapeutic technique known as self-actualization, Brooklyn College
 * John Matteson, historian and writer, Pulitzer Prize winner, John Jay College of Criminal Justice
 * Maeve Kennedy McKean, attorney and public health official
 * Stanley Milgram, social psychologist, Graduate Center
 * Charles W. Mills, philosopher, Graduate Center
 * June Nash, anthropologist, Graduate Center
 * Ruth O'Brien, political scientist and disability studies writer, Graduate Center
 * Denise O'Connor, Olympic foil fencer, Brooklyn College
 * John Patitucci, jazz bassist, City College
 * Itzhak Perlman, violinist, Brooklyn College
 * Frances Fox Piven, political scientist, activist, and educator, Graduate Center
 * Roman Popadiuk, US Ambassador to Ukraine, Brooklyn College
 * Graham Priest, philosopher, Graduate Center
 * Inez Smith Reid, Senior Judge of the District of Columbia Court of Appeals, Brooklyn College
 * Adrienne Rich, poet and activist, City College of New York
 * David M. Rosenthal, philosopher, Graduate Center
 * Mark Rothko (born Markus Yakovlevich Rothkowitz), influential abstract expressionist painter, Brooklyn College
 * Arthur M. Schlesinger Jr., historian and social critic, Graduate Center
 * Flora Rheta Schreiber, journalist, John Jay College of Criminal Justice
 * Eve Kosofsky Sedgwick, literary critic, Graduate Center
 * Betty Shabazz, educator and activist, Medgar Evers College
 * Mark Strand, United States Poet Laureate, Pulitzer Prize for Poetry-winning poet, essayist, and translator, Brooklyn College
 * Dennis Sullivan, mathematician, Graduate Center
 * Harold Syrett (1913–1984), president of Brooklyn College
 * Katherine Verdery, anthropologist, Graduate Center
 * Michele Wallace, women's studies and film studies, City College and Graduate Center
 * Mike Wallace, historian and writer, John Jay College of Criminal Justice and Graduate Center
 * Ruth Westheimer (better known as Dr. Ruth; born Karola Ruth Siegel), sex therapist, media personality, author, radio, television talk show host, and Holocaust survivor, Brooklyn College
 * Elie Wiesel, novelist, political activist, winner of the Nobel Peace Prize, Presidential Medal of Freedom, and Congressional Gold Medal, City College
 * C. K. Williams, poet, won Pulitzer Prize for Poetry, Brooklyn College
 * Andrea Alu, engineer and physicist, Graduate Center
 * Robert Alfano, physicist, discovered the supercontinuum, City College
 * Branko Milanović, economist most known for his work on income distribution and inequality; a visiting presidential professor at the Graduate Center of the City University of New York, an affiliated senior scholar at the Luxembourg Income Study and former lead economist in the World Bank's research department.
 * Simi Linton, arts consultant, author, filmmaker, and activist. Focuses on disability in the arts, disability studies, and ways that disability rights and disability justice perspectives can be brought to bear on the arts.

Public Safety Department
CUNY has a unified public safety department, the City University of New York Public Safety Department, with branches at each of the 26 CUNY campuses. The New York City Police Department is the primary policing and investigation agency within the New York City as per the NYC Charter, which includes all CUNY campuses and facilities.

The Public Safety Department came under heavy criticism from student groups, after several students protesting tuition increases tried to occupy the lobby of the Baruch College. The occupiers were forcibly removed from the area and several were arrested on November 21, 2011.

Antisemitism at CUNY
In recent years, there have been a number of antisemitic incidents on CUNY campuses, including:

 * In March 2014, Brooklyn College settled the Title VI complaint that the Zionist Organization of America ("ZOA") had filed against its antisemitic discrimination.
 * In 2017, a CUNY admin was recorded saying that there were too many Jews on campus.
 * In 2020, a CUNY student was arrested for spray-painting antisemitic graffiti on a campus building.
 * In 2021, a survey found that nearly one in four CUNY students had experienced antisemitism on campus. The survey also found that Jewish students were more likely to report feeling unsafe on campus than students of other faiths.
 * In 2021, students and faculty at Hunter College disrupted two sessions of a required course by taking over the class to advocate for "the decolonization of Palestine."
 * In May 2021, a student at John Jay posted a picture of Adolf Hitler on Instagram with a message saying "We need another Hitler today." A group of Jewish students met with Karol Mason, the President of the college, who refused to condemn the action publicly.
 * The Equal Employment Opportunity Commission cited CUNY in 2021 for failing to protect a Jewish professor after the PSC discriminated against him and subjected him to a hostile work environment on the basis of his Jewish faith.

CUNY has taken steps to address antisemitism on its campuses. In 2020, the university created a task force to combat antisemitism. The task force has developed a number of initiatives, including training for faculty and staff on how to identify and address antisemitism.

In June 2024, the United States Department of Education concluded that CUNY has failed to protect Jewish students from discrimination following the October 7 attacks. CUNY's Hunter College also faced scrutiny for incidents dating back to 2021. In response, Chancellor Félix V. Matos Rodríguez stated that CUNY is dedicated to maintaining a discrimination-free and hate-free environment, and that new measures will ensure consistent and transparent investigation and resolution of complaints.

City University Television (CUNY TV)
CUNY also has a broadcast TV service, CUNY TV (channel 75 on Spectrum, digital HD broadcast channel 25.3), which airs telecourses, classic and foreign films, magazine shows, and panel discussions in foreign languages.

City University Film Festival (CUNYFF)
The City University Film Festival is CUNY's official film festival. The festival was founded in 2009.

Notable alumni
CUNY graduates include 13 Nobel laureates, 2 Fields Medalists, 2 U.S. Secretaries of State, a Supreme Court Justice, several New York City mayors, members of Congress, state legislators, scientists, artists, and Olympians.

title: City University of New York