Centripetal Force Examples: 47 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Centripetal Force.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Physics.

Concept Recap

The net inward force required to keep an object moving along a circular path, directed toward the centre of the circle, equal to $mv^2/r$ where $m$ is the object's mass, $v$ its speed, and $r$ the radius of the circle.

The force that pulls you toward the center when you go around a curve.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Centripetal Force asks students to choose the object, list external interactions, and reason from the resulting force or torque pattern.

Common stuck point: Students often know a formula related to centripetal force but skip the recognition step: Have I isolated one system and listed the external forces or torques acting on it before applying a law? That leads to a correct-looking substitution attached to the wrong physical model.

Sense of Study hint: Ask: Have I isolated one system and listed the external forces or torques acting on it before applying a law?

Worked Examples

Example 1

medium

0.5 \text{ kg}

ball on a

1.2 \text{ m}

string is swung in a horizontal circle at

4 \text{ m/s}

. What is the centripetal force?

F_c ⊥ v. F_c = mv²/r = 0.5 × 16 / 1.2 ≈ 6.67 N toward the center.

Answer

F_{c} \approx 6.67 N

First step

Recall the centripetal force formula:

F_{c} = \frac{m v^{2}}{r}

, where

m

is mass,

v

is speed, and

r

is radius.

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Example 2

hard

A car of mass

1000 \text{ kg}

rounds a curve of radius

50 \text{ m}

. If the maximum static friction force is

8000 \text{ N}

, what is the maximum safe speed?

f_s = mv²/r → v_max = √(f_s r/m) = √(8000 × 50/1000) = 20 m/s.

Example 3

medium

1200\text{ kg}

car rounds a flat curve of radius

80\text{ m}

20\text{ m/s}

. What friction force keeps it on the curve?

f = F_c = mv²/r = 1200 × 400 / 80 = 6000 N directed toward the center.

Example 4

medium

A roller coaster car of mass

400\text{ kg}

passes through the bottom of a vertical loop of radius

10\text{ m}

15\text{ m/s}

. Find the normal force from the track on the car. Use

g = 10\text{ m/s}^2

N − mg = mv²/r = 9000 N. Track pushes up with 13 000 N.

Example 5

hard

0.5\text{ kg}

ball is attached to a

1\text{ m}

string and swung in a vertical circle. Find the minimum speed at the top so the string stays taut. Use

g = 10\text{ m/s}^2

T → 0 at minimum speed: mg = mv²/r, so v_min = √(gr) ≈ 3.16 m/s.

Example 6

hard

A car drives over a hilltop (curve radius

20\text{ m}

). Find the maximum speed at which the car maintains contact with the road. Use

g = 10\text{ m/s}^2

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium

A satellite orbits Earth at radius

r = 7 \times 10^6 \text{ m}

with speed

v = 7500 \text{ m/s}

. What is the centripetal acceleration?

Example 2

medium

0.25 \text{ kg}

ball moves in a circle at

6 \text{ m/s}

and experiences a centripetal force of

9 \text{ N}

. What is the radius of the circle?

Example 3

easy

2

kg object moves in a circle of radius

4

m at

4

m/s. Find the centripetal force.

Example 4

easy

An object needs

20

N of centripetal force in a

5

m circle at

10

m/s. Find its mass.

Example 5

easy

1

kg ball moves in a circle of radius

2

m at

6

m/s. Find the centripetal force.

Example 6

easy

In which direction does the centripetal force on an object in circular motion point?

Example 7

easy

If the speed of an object in a fixed circle doubles, how does the centripetal force change?

Example 8

easy

0.5

kg ball on a string moves at

4

m/s in a circle of radius

1

m. Find the string tension (horizontal circle).

T = F_c = mv²/r

Example 9

easy

A car of mass

1000

kg rounds a curve of radius

50

m at

10

m/s. Find the centripetal force.

Example 10

easy

Is centripetal force a new kind of force, or the net of existing forces?

Example 11

medium

0.2

kg ball on a string moves in a horizontal circle of radius

0.5

m, completing

2

revolutions per second. Find the centripetal force.

Example 12

medium

A car rounds a flat curve of radius

40

m. The max friction force is

4000

N and the car is

1000

kg. Find the maximum safe speed.

f_max = mv²/r → v_max = √(f_max r/m)

Example 13

medium

0.5

kg ball swings in a vertical circle of radius

1

m. At the top, what minimum speed keeps the string taut (

g=10

m/s

^2

At minimum speed: T = 0, so mg = mv²/r. This gives v_min

Example 14

medium

1000

kg car rounds a curve of radius

50

m at

15

m/s. What friction force keeps it on the road?

Example 15

medium

2

kg ball moves in a circle with centripetal acceleration

8

m/s

^2

. Find the centripetal force.

Example 16

medium

A conical pendulum: a

1

kg ball swings so the string makes

30^\circ

with vertical (

g=10

m/s

^2

). Find the string tension.

Tension T decomposes: T cos 30° balances weight; T sin 30° is the centripetal force.

Example 17

medium

A satellite needs centripetal force from gravity. A

500

kg satellite orbits at

7000

m/s with radius... given gravity provides

2500

N, find the orbital radius.

Example 18

medium

0.3

kg mass on a

0.4

m string is whirled horizontally; the string breaks at

48

N tension. Find the maximum speed.

T_max = mv²/r = 48 N → v = √(T_max r/m)

Example 19

medium

2

kg object moves in a circle of radius

0.5

m at

3

m/s. Find its centripetal acceleration.

Example 20

challenge

A car rounds a banked curve (radius

50

m) designed for

20

m/s with no friction (

g=10

m/s

^2

). Find the banking angle.

N sin θ = F_c (centripetal); N cos θ

Example 21

challenge

2

kg ball swings in a vertical circle (radius

1

m) at

5

m/s at the bottom (

g=10

m/s

^2

). Find the tension at the bottom.

T − mg = mv²/r

Example 22

challenge

A coin sits

0.1

m from the center of a turntable. The max static friction gives

a = 4

m/s

^2

. Find the maximum angular speed before it slips.

Example 23

easy

3\text{ kg}

object moves in a circle of radius

2\text{ m}

4\text{ m/s}

. Find the centripetal force.

Example 24

easy

If the radius of an object's circular path is doubled while its speed stays the same, by what factor does the centripetal force change?

Example 25

easy

0.4\text{ kg}

stone is whirled in a horizontal circle of radius

0.8\text{ m}

5\text{ m/s}

. Find the tension in the string.

T = mv²/r

Example 26

easy

1.5\text{ kg}

ball moves in a circle of radius

0.5\text{ m}

with centripetal acceleration

20\text{ m/s}^2

. Find the centripetal force.

Example 27

easy

0.1\text{ kg}

puck moves in a circle of radius

0.25\text{ m}

2\text{ m/s}

. Find the centripetal force.

Example 28

medium

0.6\text{ kg}

ball on a

0.9\text{ m}

string is swung horizontally and completes one revolution in

1.2\text{ s}

. Find the tension. Use

g = 10\text{ m/s}^2

Example 29

medium

70\text{ kg}

cyclist leans into a turn of radius

25\text{ m}

10\text{ m/s}

. Find the required centripetal force.

Example 30

medium

0.25\text{ kg}

ball on a string moves in a horizontal circle of radius

0.5\text{ m}

. The string can stand a maximum tension of

50\text{ N}

. Find the maximum speed.

T_max = mv²/r → v_max = √(T_max r/m)

Example 31

medium

On a flat curve of radius

30\text{ m}

, the coefficient of static friction is

\mu_s = 0.5

. Find the maximum safe speed. Use

g = 10\text{ m/s}^2

Example 32

medium

0.5\text{ kg}

ball swings in a vertical circle of radius

0.8\text{ m}

5\text{ m/s}

at the top of the loop. Find the tension in the string at the top. Use

g = 10\text{ m/s}^2

T + mg = mv²/r

Example 33

medium

0.3\text{ kg}

ball at the end of a

1\text{ m}

string moves in a horizontal circle making

1.5\text{ rev/s}

. Find the centripetal force.

Example 34

medium

A car on a banked curve relies only on the horizontal component of the normal force. The banking angle is

20^\circ

and the radius is

40\text{ m}

. Find the design speed. Use

g = 10\text{ m/s}^2

N sin 20° = F_c; N cos 20°

Example 35

medium

An astronaut in a centrifuge sits

5\text{ m}

from the rotation axis. To feel

3g

of centripetal acceleration, what angular speed is needed? Use

g = 10\text{ m/s}^2

Example 36

hard

1.2\text{ kg}

ball swings in a vertical circle on a

0.6\text{ m}

string. At the bottom, the tension is

30\text{ N}

. Find the speed at the bottom. Use

g = 10\text{ m/s}^2

T − mg

Example 37

hard

A conical pendulum: a

0.5\text{ kg}

ball swings in a horizontal circle on a

1\text{ m}

string making

30^\circ

with vertical. Find the period. Use

g = 10\text{ m/s}^2

Example 38

hard

A coin sits on a turntable

0.15\text{ m}

from the axis. The coefficient of static friction is

0.4

. Find the maximum rotation rate in rev/s before the coin slips. Use

g = 10\text{ m/s}^2

Example 39

hard

0.2\text{ kg}

ball swings on a

0.5\text{ m}

string in a vertical circle at constant speed

4\text{ m/s}

. Find the tension when the string is horizontal. Use

g = 10\text{ m/s}^2

At the horizontal position: tension alone equals F_c = mv²/r

Example 40

challenge

A satellite of mass

m

orbits Earth at radius

r

where gravity gives acceleration

g(r)

. Show that orbital speed is

v = \sqrt{g(r)\,r}

and compute it for

r = 6.6\times 10^6\text{ m}

with

g(r) = 9.2\text{ m/s}^2

Example 41

challenge

A car rounds a banked curve (radius

50\text{ m}

, bank angle

20^\circ

) at

20\text{ m/s}

. The required friction coefficient (along the slope, preventing slipping up) satisfies

v^2 > gr\tan\theta

. Find the minimum

\mu_s

. Use

g = 10\text{ m/s}^2

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Related Concepts

Circular Motion Force Orbital Motion Angular Momentum

Background Knowledge

These ideas may be useful before you work through the harder examples.

circular motionforce