Circular Motion
Radians, angular speed, centripetal acceleration, and centripetal force.
Spec Points Covered
- Convert between degrees and radians and use the equation for angular displacementThe distance moved in a particular direction from a starting point. A vector quantity. Measured in metres (m)..
- Define angular speed and relate it to linear speed, periodThe time taken for one complete oscillation or wave cycle. Measured in seconds (s)., and frequencyThe number of complete oscillations passing a point per unit time. Measured in hertz (Hz)..
- Derive and apply the three forms of centripetal accelerationThe rate of change of velocityThe rate of change of displacement. A vector quantity. Measured in m s⁻¹.. A vector quantity. Measured in m s⁻².The accelerationThe rate of change of velocityThe rate of change of displacement. A vector quantity. Measured in m s⁻¹.. A vector quantity. Measured in m s⁻². directed towards the centre of the circular path, required to maintain circular motion..
- Apply the centripetal forceThe resultant force directed towards the centre of a circular path that causes an object to move in a circle. It is not a separate force but the net force providing circular motion. equations and identify the real force providing it.
- Explain why no work is done during uniform circular motion.
Notes
01
An object in circular motion is always accelerating
3.6.1.1
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02
Angular displacement is measured in radians
Radian
3.6.1.1
→
03
Angular speed is the rate of change of angular displacement
Angular speed
3.6.1.1
→
04
Centripetal acceleration has three equivalent forms
Centripetal acceleration
3.6.1.1
→
05
Centripetal force is the resultant force towards the centre
Centripetal force
3.6.1.1
→
06
No work is done during uniform circular motion
3.6.1.1
→
07
Solving circular motion problems: equate centripetal force to the real force
3.6.1.1
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On Data Sheet
Not on Data Sheet
Angular speed
$$\begin{aligned}
\omega &= \frac{\Delta \theta}{\Delta t} \\
&= \frac{v}{r} \\
&= 2\pi f \\
&= \frac{2\pi}{T}
\end{aligned}$$
- Where:
- $\omega$ = angular speed (rad \(s^{-1}\))
- $v$ = linear speed (m \(s^{-1}\))
- $r$ = radius (m)
- $f$ = frequency (Hz)
- $T$ = time period (s)
All four forms are equivalent. Choose based on given data.
Angular displacement
$$\Delta \theta = \frac{\Delta s}{r}$$
- Where:
- $\Delta \theta$ = angular displacement (rad)
- $\Delta s$ = arc length (m)
- $r$ = radius (m)
Both arc length and radius must be in the same units.
Centripetal acceleration
$$\begin{aligned}
a &= \frac{\(v^{2}\)}{r} \\
&= r\omega^2 \\
&= v\omega
\end{aligned}$$
- Where:
- $a$ = centripetal acceleration (m \(s^{-2}\))
- $v$ = linear speed (m \(s^{-1}\))
- $r$ = radius (m)
- $\omega$ = angular speed (rad \(s^{-1}\))
All three forms on the data sheet. a = v omega is often overlooked but useful.
Centripetal force
$$\begin{aligned}
F &= \frac{mv^2}{r} \\
&= mr\omega^2 \\
&= mv\omega
\end{aligned}$$
- Where:
- $F$ = centripetal force (N)
- $m$ = mass (kg)
- $v$ = linear speed (m \(s^{-1}\))
- $r$ = radius (m)
- $\omega$ = angular speed (rad \(s^{-1}\))
From Newton's second law: F = ma applied to each acceleration form.
Q1. Define a radianThe angle subtended at the centre of a circle by an arc equal in length to the radius. One complete revolution = 2π radians..
The angle subtended at the centre of a circle by an arc equal in length to the radius of the circle.
Q2. How many radians are in a full circle?
2 pi radians (approximately 6.28 rad).
Q3. Define angular speed and state its unit.
- The rate of change of angular displacementThe distance moved in a particular direction from a starting point. A vector quantity. Measured in metres (m). with respect to time.
- Measured in rad \(s^{-1}\).
Q4. Write three equivalent expressions for angular speed.
omega = delta theta / delta t = v/r = 2 pi f = 2 pi / T.
Q5. State the relationship between linear speed and angular speed.
- v = r omega.
- Linear speed equals radius times angular speed.