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# Centripetal force

A centripetal force (from Latin centrum, "center" and petere, "to seek" Extract of page 291) is a force that makes a body follow a curved path. Its direction 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 responsible for 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

The magnitude of the centripetal force on an object of mass m moving at tangential speed v along a path with radius of curvature r is:{{cite book | title = Facts and Practice for A-Level: Physics | author = Chris Carter | year = 2001 | isbn = 978-0-19-914768-7 | page = 30 | publisher = Oxford University Press | location = S.l. }} F = ma_c = \frac{m v^2}{r} where a_c is the centripetal acceleration. 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. The inverse relationship with the radius of curvature shows that half the radial distance requires twice the force. 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 = m r \omega^2 \,. Expressed using the orbital period T for one revolution of the circle, \omega = \frac{2\pi}{T} \, the equation becomes F = 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 = \frac{\gamma m v^2}{r} where \gamma = \frac{1}{\sqrt{1-v^2/c^2}} is called the Lorentz factor. More intuitively: F = \gamma m v \omega which is the rate of change of relativistic momentum (\gamma m v)

## Sources of centripetal force

requires a centripetal force, towards the axis as shown, to maintain its circular path.]] In the case of an object that is swinging around on the end of a rope in a horizontal plane, the centripetal force on the object is supplied by the tension of the rope. The rope example is an example involving a 'pull' force. The centripetal force can also be supplied as a 'push' force, such as in the case where the normal reaction of a wall supplies the centripetal force for a wall of death rider. Newton's idea of a centripetal force corresponds to what is nowadays referred to as a central force. When a satellite is in orbit around a planet, gravity is considered to be a centripetal force even though in the case of eccentric orbits, the gravitational force is directed towards the focus, and not towards the instantaneous center of curvature. Another example of centripetal force arises in the helix that is traced out when a charged particle moves in a uniform magnetic field in the absence of other external forces. In this case, the magnetic force is the centripetal force that acts towards the helix axis.

## 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 \hat{x} and \hat{y}: : \textbf{r} = r \cos(\theta) \hat{x} + r \sin(\theta) \hat{y}. Assume uniform circular motion, which 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.
Now find the velocity \textbf{v} and acceleration \textbf{a} of the motion by taking derivatives of position with respect to time. ::: \textbf{r} = r \cos(\omega t) \hat{x} + r \sin(\omega t) \hat{y} : \dot{\textbf{r}} = \textbf{v} = - r \omega \sin(\omega t) \hat{x} + r \omega \cos(\omega t) \hat{y} : \ddot{\textbf{r}} = \textbf{a} = - r \omega^2 \cos(\omega t) \hat{x} - r \omega^2 \sin(\omega t) \hat{y} ::: \textbf{a} = - \omega^2 (r \cos(\omega t) \hat{x} + r \sin(\omega t) \hat{y}) Notice that the term in parenthesis 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

and magnitude /dt.]] 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_{\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.

### 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·ur, 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 i and j respectively:Note: unlike the Cartesian unit vectors i and j, which are constant, in polar coordinates the direction of the unit vectors u'r and u depend on θ, and so in general have non-zero time derivatives. ur = cosθ i + sinθ j and uθ = -sinθ i + cosθ j. One can differentiate to find velocity: \mathbf{v} = r \frac {\mathrm{d} \mathbf{u}_\mathrm{r}}{\mathrm{d}t} = r \frac {\mathrm{d}}{\mathrm{d}t} \left( \mathrm{cos}\ \theta \ \mathbf{i} + \mathrm{sin}\ \theta \ \mathbf{j}\right) : = r \frac {d \theta} {dt} \left( -\mathrm{sin}\ \theta \ \mathbf{i} + \mathrm{cos}\ \theta \ \mathbf{j}\right)\, : = r \frac{\mathrm{d}\theta}{\mathrm{d}t} \mathbf{u}_\mathrm{\theta} \, : = \omega r \mathbf{u}_\mathrm{\theta} \, 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 . Differentiating again, and noting that {\frac {\mathrm{d}\mathbf{u}_\mathrm{\theta}}{\mathrm{d}t} = -\frac{\mathrm{d}\theta}{\mathrm{d}t} \mathbf{u}_\mathrm{r} = - \omega \mathbf{u}_\mathrm{r}} \ , we find that the acceleration, a is: \mathbf{a} = r \left( \frac {\mathrm{d}\omega}{\mathrm{d}t} \mathbf{u}_\mathrm{\theta} - \omega^2 \mathbf{u}_\mathrm{r} \right) \ . Thus, the radial and tangential components of the acceleration are: \mathbf{a}_{\mathrm{r}} = - \omega^{2} r \ \mathbf{u}_\mathrm{r} = - \frac{|\mathbf{v}|^{2}}{r} \ \mathbf{u}_\mathrm{r} \    and   \ \mathbf{a}_{\mathrm{\theta}} = r \ \frac {\mathrm{d}\omega}{\mathrm{d}t} \ \mathbf{u}_\mathrm{\theta} = \frac {\mathrm{d} | \mathbf{v} | }{\mathrm{d}t} \ \mathbf{u}_\mathrm{\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.

### General planar motion

• {{cite book
| author = Tipler, Paul | title = Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics | edition = 5th | publisher = W. H. Freeman | year = 2004 | isbn = 0-7167-0809-4 }}