Circular motion and centripetal force

Key ideas

An object moving in a circle at constant speed is still accelerating — because its velocity's direction keeps changing. That acceleration points toward the centre of the circle:

$a_c = \frac{v^2}{r}$

By Newton's second law, something must supply the force toward the centre — the centripetal force:

$F_c = \frac{mv^2}{r}$

A ball on a circular path with its velocity arrow along the tangent and the centripetal force arrow pointing from the ball toward the centre of the circle.

Centripetal force is not a new kind of force. It's the job description filled by a real force: friction (a car cornering), tension (a ball on a string), or gravity (the Moon orbiting Earth). At Level 2, one single force provides the centripetal force — that's stated in the standard.

Worked ExampleCentripetal force on a cornering car

A $900$ kg car takes a corner of radius $50$ m at $20$ m s⁻¹. Find the centripetal acceleration and the friction force needed.

Step 1 — Centripetal acceleration

$a_c = \frac{v^2}{r} = \frac{20^2}{50} = 8.0 \text{ m s}^{-2}$

Step 2 — The force that produces it

$F_c = ma_c = (900)(8.0) = 7200 \text{ N}$

Friction between the tyres and road supplies this $7200$ N.

Practice question

A $0.20$ kg ball is whirled on a string in a horizontal circle of radius $0.80$ m at $4.0$ m s⁻¹. Find the tension in the string.

Worked solution: $F_c = \dfrac{mv^2}{r} = \dfrac{(0.20)(4.0)^2}{0.80} = 4.0$ N. The tension supplies the centripetal force.

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

A $0.30$ kg ball on a string is swung in a horizontal circle of radius $1.2$ m at a speed of $4.0$ m s⁻¹.

Calculate the centripetal force acting on the ball.

Merit

A car of mass $1200$ kg rounds a bend of radius $40$ m at a constant speed of $15$ m s⁻¹.

Calculate the centripetal force required, and name the force that provides it.

Excellence

A car rounds a corner at a constant speed.

Explain fully, using physics principles, why the car is accelerating even though its speed is constant, and describe what would happen if the road suddenly became icy.

Circular motion and centripetal force

Covers AS91171

Key ideas

An object moving in a circle at constant speed is still accelerating — because its velocity's direction keeps changing. That acceleration points toward the centre of the circle:

$a_c = \frac{v^2}{r}$

By Newton's second law, something must supply the force toward the centre — the centripetal force:

$F_c = \frac{mv^2}{r}$

A ball on a circular path with its velocity arrow along the tangent and the centripetal force arrow pointing from the ball toward the centre of the circle.

Worked ExampleCentripetal force on a cornering car

A $900$ kg car takes a corner of radius $50$ m at $20$ m s⁻¹. Find the centripetal acceleration and the friction force needed.

Step 1 — Centripetal acceleration

$a_c = \frac{v^2}{r} = \frac{20^2}{50} = 8.0 \text{ m s}^{-2}$

Step 2 — The force that produces it

$F_c = ma_c = (900)(8.0) = 7200 \text{ N}$

Friction between the tyres and road supplies this $7200$ N.

Practice question

A $0.20$ kg ball is whirled on a string in a horizontal circle of radius $0.80$ m at $4.0$ m s⁻¹. Find the tension in the string.

Worked solution: $F_c = \dfrac{mv^2}{r} = \dfrac{(0.20)(4.0)^2}{0.80} = 4.0$ N. The tension supplies the centripetal force.

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

A $0.30$ kg ball on a string is swung in a horizontal circle of radius $1.2$ m at a speed of $4.0$ m s⁻¹.

Calculate the centripetal force acting on the ball.

Merit

A car of mass $1200$ kg rounds a bend of radius $40$ m at a constant speed of $15$ m s⁻¹.

Calculate the centripetal force required, and name the force that provides it.

Excellence

A car rounds a corner at a constant speed.

Explain fully, using physics principles, why the car is accelerating even though its speed is constant, and describe what would happen if the road suddenly became icy.