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# Centripetal Acceleration

## Objects moving in a circle must experience acceleration and force perpendicular to their direction of travel.

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Centripetal Acceleration

If a mass m\begin{align*} m \end{align*} is traveling with velocity v\begin{align*} \vec{v} \end{align*} and experiences a centripetal --- always perpendicular --- force Fc\begin{align*} \vec{F_c} \end{align*}, it will travel in a circle of radius

r=mv2|F| [1]\begin{align*}r = \frac{m v^2}{|\vec{F}|}\text{ [1]}\end{align*}Alternatively, to keep this mass moving at this velocity in a circle of this radius, one needs to apply a centripetal force of

Fc=mv2r [2]\begin{align*}\vec{F_c} = \frac{mv^2}{r} \text{ [2]}\end{align*}By Newton's Second Law, this is equivalent to a centripetal acceleration of

Fc=mac=mv2r [3]\begin{align*}\vec{F_c} =\cancel{m}\vec{a_c} = \cancel{m}\frac{v^2}{r} \text{ [3]}\end{align*}

Key Equations

Centripetal ForceFC=mv2rmvrmass (in kilograms, kg)speed (in meters per second, m/s)radius of circle\begin{align*} F_C = \frac{mv^2}{r} \begin{cases} m & \text{mass (in kilograms, kg)}\\ v & \text{speed (in meters per second, m/s}\text{)}\\ r & \text{radius of circle} \end{cases}\end{align*}

Centripetal Acceleration

aC=v2r{vrspeed (in meters per second, m/s)radius of circle\begin{align*} a_C = \frac{v^2}{r} \begin{cases} v & \text{speed (in meters per second, m/s}\text{)}\\ r & \text{radius of circle} \end{cases}\end{align*}

#### Example

If you are 4m from the center of a Merry-Go-Round that is rotating at 1 revolution every 2 seconds, what is your centripetal acceleration?

First, we need to find your tangential velocity. We can do this using the given angular velocity.

ωωωvvv=2π rad2 s=π rad/s=vr=ωr=πrad/s4m=4πm/s\begin{align*} \omega&=\frac{2\pi\text{ rad}}{2\text{ s}}\\ \omega&=\pi\text{ rad/s}\\ \omega&=\frac{v}{r}\\ v&=\omega r\\ v&=\pi\;\text{rad/s}*4\;\text{m}\\ v&=4\pi\;\text{m/s} \end{align*}

acacac=v2r=(4πm/s)24m=4π2m/s2\begin{align*} a_c&=\frac{v^2}{r}\\ a_c&=\frac{(4\pi\;\text{m/s})^2}{4\;\text{m}}\\ a_c&=4\pi^2\;\text{m/s}^2\\ \end{align*}

### Review

1. A 6000 kg roller coaster goes around a loop of radius 30m at 6 m/s. What is the centripetal acceleration?

1. For the Gravitron ride above, assume it has a radius of 18 m and a centripetal acceleration of 32 m/s2. Assume a person is in the graviton with 180 cm height and 80 kg of mass. What is the speed it is spinning at? Note you may not need all the information here to solve the problem.

1. 1.2 m/s2

2. 24 m/s

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