Calculate centripetal acceleration
Use a = rω² or a = v²/r to calculate inward acceleration in uniform circular motion.
A Level; Cambridge International Physics 9702, 2025–2027 syllabus, Topic 12. These original calculations are study aids; full timed assessment and human subject-teacher review remain outstanding.
Before you begin
- Use radians, angular speed, velocity, acceleration, mass and resultant force with consistent SI units.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which expression gives centripetal acceleration from tangential speed?
What you will learn
- Recall and use a = rω²
- Recall and use a = v²/r
- Choose a formula that matches the supplied speed or angular speed
Angular and tangential forms describe the same acceleration
For uniform circular motion, centripetal acceleration has magnitude a = rω². Since v = rω, substituting ω = v/r gives a = v²/r.
Both expressions give inward acceleration in m s⁻². Use r in metres, v in m s⁻¹, and ω in rad s⁻¹. The direction remains toward the centre even when only the magnitude is calculated.
Predict how radius and speed change acceleration
At fixed angular speed, a is proportional to radius. At fixed tangential speed, a is inversely proportional to radius.
At fixed radius, doubling v makes a four times larger because speed is squared. State which quantities are held constant before making a proportional comparison.
Pause and explain
What is a for v = 6.0 m s⁻¹ and r = 3.0 m?
Worked example
A point moves at 4.0 m s⁻¹ in a circle of radius 2.0 m. Find centripetal acceleration.
Show the worked solution
- Use a = v²/r.
- a = 4.0²/2.0.
- Give m s⁻² and direction.
Answer a = 8.0 m s⁻² toward the centre.
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Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A stone moves at 5.0 m s⁻¹ in a circle of radius 2.5 m. Find centripetal acceleration.
Give me a hint
Square the speed before dividing by radius.
Compare my reasoning
- a = v²/r.
- a = 25/2.5.
- State the inward direction and unit.
10 m s⁻² toward the centre.
Look for these in your work
- Correct equation
- Speed squared
- Direction and unit stated
An object moves with angular speed 8.0 rad s⁻¹ on radius 0.50 m. Find centripetal acceleration.
Give me a hint
Use the radius and square the angular speed in a = rω².
Compare my reasoning
- a = 0.50 × 8.0².
- a = 0.50 × 64.
- Give m s⁻².
32 m s⁻² inward.
Look for these in your work
- Angular relation used
- Angular speed squared
- Acceleration unit stated
A vehicle's speed doubles on the same circular path. Compare its centripetal acceleration with the original value.
Give me a hint
Hold radius fixed in a = v²/r.
Compare my reasoning
- New speed = 2v.
- New a = (2v)²/r = 4v²/r.
- The acceleration magnitude is four times the original.
Centripetal acceleration quadruples when speed doubles at fixed radius.
Look for these in your work
- Fixed radius stated
- Speed square applied
- Factor of four derived
Common mistakes
- Using a = v/r without squaring speed
- Forgetting acceleration is directed inward
- Changing radius while assuming both ω and v stay fixed
What can you explain now?
Which expression gives centripetal acceleration from tangential speed?
Compare with the explanation
a = v²/r
The speed is squared and divided by radius.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Why does doubling speed at fixed radius quadruple centripetal acceleration?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.