Uniform Circular Motion

The motion of an object in a circle at constant speed

Uniform → acceleration is constant → speed is constant

Direction is constantly changing for there to be an acceleration

The direction of the resultant force must be towards the centre of the circle

Acceleration is therefore towards the centre of the circle

$F_{net}=ma$

Force is a vector so acceleration is a vector in the same direction

Velocity is always tangent to the circle - perpendicular to the force at any moment

The magnitude of the work done by the force on the object is zero because the object never moves any closer to the centre of the circle

$F\cos\left(90\right)\cdot S$ and $\cos\left(90\right)=0$ so the work done is zero

The energy only goes to changing the direction of the object

The Force towards the centre is called the Centripetal force — Centripetal force is a force that keeps a body moving with a constant speed in a circular path

Angular Velocity ($\omega$) is the rate of change of angle for an object moving in a circular path

$\omega=\frac{\Delta\theta}{\Delta t}$ where $\theta$ is the angular displacement and its unit is radians

if the length of the arc and the radius are the same, then $\theta$ is 1 radian

a radian is the angle subtended by an arc with a length equal to the radius of the circle

$S\left(arclength\right)\operatorname{}=r\theta$

the circumference of the circle is $2\pi r$ giving $\theta=2\pi$

Period T of circular Motion

When an object travels with constant speed in a circular orbit, the time it takes to complete on complete revolution is called the period, $T$, of rotation.

The frequency, $f$, is the number of rotations per unit time (per second)

$f=\frac{1}{T}$

speed is distance over time, distance is $2\pi r$ and time taken to complete one orbit is $T$ (in periods)

so speed $speed=\frac{2\pi r}{T}$

$V=r\left(\frac{2\pi}{T}\right),V=kr$ , so if $r$ is larger then $V$ is larger

If two moons have the same radius of orbit, but A has a period double that of B, state and explain which would have grater orbital speed.

Moon B would have a greater orbital speed as its velocity would be given by $\frac{2\pi r}{T}$ which would give a greater value than Moon A’s velocity which is given by $\frac{2\pi r}{2T}$ which is the same as $\frac12\cdot\frac{2\pi r}{T}$ .

Angular velocity ($\omega$) is the rate of change of angle

$\omega=\frac{2\pi}{T}=2\pi f$ since $f=\frac{1}{T}$

$V=r\omega$

As $r$ increases, $V$ increases but $\omega$ is always constant

• $180^{o}$ in radians $=\pi$

• $45^{o}$ in radians $=\frac{\pi}{4}$

• a roundabout which completes 1 revolution in 30s has an angular velocity of $\frac{\pi}{15}rads^{-1}$

• a spinning top with a time period of 0.10s has an angular velocity of $20\pi$ $rads^{-1}$