Radians and angular speed Cambridge International AS & A Level Physics revision
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In plain words
For things going round in circles, it is easier to measure how fast the angle changes than how fast the object moves. Every point on a spinning wheel turns through the same angle each second, even though the rim moves much faster than the hub. The angles are measured in radians, which make the formulas come out simple.
Three things to know
- One radian is the angle at the centre of a circle made by an arc as long as the radius. Angle in radians = arc length ÷ radius. A full circle is 2π radians, which is 360°.
- Angular speed ω = angle turned ÷ time taken = 2π ÷ T = 2πf, in rad/s.
- Linear speed v = rω: points further from the centre move faster.
Worked example
A wheel of radius 0.30 m turns 5.0 times a second. Find its angular speed and the speed of a point on its rim.
- ω = 2πf = 2π × 5.0 = 31.4 rad/s.
- v = rω = 0.30 × 31.4.
- = 9.4 m/s.
Tips and tricks
- Degrees to radians: multiply by π ÷ 180. Put your calculator into radian mode whenever the angle is in radians.
- Revolutions per minute to rad/s: multiply by 2π ÷ 60.
It lands in your notebook with its questions as flashcards.
Radians and angular speed: 5 questions and answers
These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.
How many radians are there in a full circle?
360° is 2π radians.
A point moves in a circle of radius 2.0 m with an angular speed of 3.0 rad/s. What is its speed?
v = rω.
An object goes round a circle once every 0.50 s. What is its angular speed?
ω = 2π ÷ T.
Two points on a spinning disc are at different distances from the centre. What is the same for both?
Every point turns through the same angle in the same time.
What is 180° in radians?
Half a full circle.
Quiz
5 questions
Tap an answer and you’ll see straight away whether it’s right, and why.
Worksheet
3 questions, 6 marks. Write your answers on paper, then check them.
Radians and angular speed
Cambridge International AS & A Level Physics 9702 · 6 marks · papermunch.org
Name ______________________________ Date ______________
Convert 90° to radians, and 1.0 radian to degrees.[2]
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π/2 radians (1.57 rad), and 57.3°.
Find the angular speed of the Earth's spin. Its period is 24 hours.[2]
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7.3 × 10⁻⁵ rad/s. 2π ÷ 86 400.
A disc turns at 300 revolutions per minute. Find its angular speed.[2]
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31 rad/s. 300 × 2π ÷ 60.
Answers: Radians and angular speed
- 1. π/2 radians (1.57 rad), and 57.3°.
- 2. 7.3 × 10⁻⁵ rad/s. 2π ÷ 86 400.
- 3. 31 rad/s. 300 × 2π ÷ 60.