Circular Motion

An object in uniform circular motion has a constant linear velocity. However, as velocity is a vector quantity, an object in uniform circular motion is considered to be accelerating, as direction is constantly changing. 

The object in uniform circular motion is kept in circular motion by a centripetal force (F) which always acts towards the centre of the circle. The centripetal force causes a centripetal acceleration (a) which also acts towards the centre of the circle. 

Often calculations for this unit can be made using radians, instead of degrees. It is therefore important to understand what radians are and how to convert between degrees and radians. 

The change in angle of an object as it rotates is known as angular displacement (\theta) and is usually stated as an angle in radians. This can be calculated using the equation:

\theta=\dfrac{l}{r}

• \theta = angular displacement in radians \text{(rad)}
• l= arc length (distance travelled around the circle) in metres \text{(m)}
• r= radius of the circle in metres \text{(m)}

Example: An object is in uniform circular motion with a radius of 2 \text{ m}. Calculate the angular displacement if the object moves 3 \text{ m} around its circular path.

[2 marks]

\theta = \dfrac{\textcolor{f95d27}{3}}{\textcolor{10a6f3}{2}} = 1.5 \text{ rad}

Example: An object spins in uniform circular motion with a radius of 0.5 \text{ m}. Calculate the angular displacement in degrees if the object moves 0.25 \text{ m} around its circular path.

[3 marks]

\theta = \dfrac{\textcolor{ffad05}{0.25}}{\textcolor{00d865}{0.5}} = \bold{0.5} \textbf{ rad} \\ \begin{aligned} \text{angle in radians} \times \dfrac{180}{\pi} &= \text{angle in }\degree \\ &= 0.5 \times \dfrac{180}{\pi} \\ &= \bold{28.6 \degree}\end{aligned}

The rate of change of angular displacement is known as the angular velocity (\omega). The units for angular velocity are radians per second \text{(rads}^{-1}\text{)} making it important to remember the equations above for converting to radians. 

To calculate angular velocity, the following equation is used:

\omega = \dfrac{\Delta \theta}{\Delta t}

• \omega= the angular velocity in radians per second \text{(rads}^{-1}\text{)}
• \Delta \theta= the angular displacement in radians \text{(rad)}
• \Delta t= time in seconds \text{(s)}

For an object completing a full circle of circular motion in time period, the angular velocity can be calculated using the equation:

\omega = \dfrac{2\pi}{T}= 2\pi f

• \omega= the angular velocity in radians per second \text{(rads}^{-1}\text{)}
• T= the time period in seconds \text{(s)}
• f= the frequency in hertz \text{(Hz)} 

Example: A fairground ride spins children on a swing in circular motion. If the children complete one full circle in 5 \text{ s}, what is their angular velocity?

[2 marks]

\begin{aligned} \bold{\omega} &= \bold{\dfrac{2\pi}{T}} \\ &= \dfrac{2\pi}{\textcolor{aa57ff}{5}} \\ &= \bold{1.3} \textbf{ rads} \bold{^{-1}} \end{aligned}