# Circular Motion

## Circular Motion Revision

**Circular Motion**

Many objects around us move in **circular motion**. Satellites orbiting the Earth, roundabouts and tools all move in **circular motion**. This unit looks at some of the calculations needed when objects are moving in **circular motion**.

**The Mechanics of 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.

**Using Radians**

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]**

To convert between degrees and radians, the following conversions can be used:

\text{angle in} \degree \times \dfrac{\pi}{180}=\text{angle in rad}

Or:

\text{angle in rad} \times \dfrac{180}{\pi} = \text{angle in} \degree

**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}

**Angular Velocity**

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}

**Centripetal Acceleration**

As previously seen, the **centripetal acceleration** (a) is the acceleration towards the centre of the circle when an object is spinning in uniform **circular motion**. **Centripetal acceleration** can be calculated using:

a=\dfrac{v^2}{r}

- a=
**centripetal acceleration**in radians per second squared \text{(rads}^{-2}\text{)} - v=
**linear speed**in metres per second (\text{ms}^{-1}) - r= the
**radius**in metres \text{(m)}

And as v=r\omega, substituting into the above equation gives:

a=r\omega ^2

- a=
**centripetal acceleration**in radians per second squared \text{(rads}^{-2}\text{)} - r= the
**radius**in metres \text{(m)} - \omega= the
**angular velocity**in radians per second \text{(rads}^{-1}\text{)}

**Example: **A fairground ride spins children on a swing in circular motion at a radius of 4 \text{ m}. If the children complete one full circle in 3 \text{ s}, what is their angular velocity?

**[3 marks]**

\begin{aligned} \omega &= \dfrac{2\pi}{T} \\ &= \dfrac{2\pi}{\textcolor{f21cc2}{3}} \\ &= \bold{2.095} \textbf{ rads} \bold{^{-1}} \\ \bold{a} &= \bold{r \omega^2} \\ &= \textcolor{f95d27}{4} \times 2.095^2 \\ &= \bold{18} \textbf{ ms} \bold{^{-2}} \end{aligned}

**Centripetal Force**

The **centripetal force** is the resultant force in Newtons \text{(N)} acting towards the centre of a circle, keeping the object moving in **circular motion**.

As F=ma and a=\dfrac{v^2}{r}:

F=\dfrac{mv^2}{r}= m\omega^2r

As objects stay in circular motion for different reasons, the **c****entripetal force** may exist as **tension** in a string, friction of car tyres going around a corner, or gravity acting on an object in orbit.

**Example: **A car of mass 1200 \text{ kg} is travelling around a circular racing track of radius 200 \text{ m}. The car is moving at a constant speed of 40 \text{ ms}^{-1}. What is the centripetal force needed to keep the car in circular motion?

**[2 marks]**

## Circular Motion Example Questions

**Question 1:** Why is an object in uniform circular motion considered to be accelerating, even if its speed does not change?

**[2 marks]**

**Velocity is a vector quantity meaning it has a magnitude and direction**. As the **direction is constantly changing, the velocity is constantly changing**. Therefore the object is accelerating as acceleration is change in velocity per unit of time.

**Question 2: **A child’s roundabout moves in circular motion. If the children complete one full circle in 2 \text{ s}, what is their angular velocity?

**[2 marks]**

**Question 3: **A swingball game in circular motion at a radius of 0.8 \text{ m}. If the ball completes one full circle in 1.2 \text{ s}, what is the ball’s angular acceleration?

**[3 marks]**

**Question 4:** A go cart of mass 300 \text{ kg} is travelling around a circular racing track of radius 150 \text{ m}. The go cart is moving at a constant speed of 30 \text{ ms}^{-1}. What is the centripetal force needed to keep the car in circular motion?

**[2 marks]**

## Circular Motion Worksheet and Example Questions

### Circular Motion Questions

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