Centripetal Force

The video lesson for this topic is coming soon.

Essential Question

What force keeps an object moving in a circle?

Anything moving in a circle is accelerating toward the center, and Newton’s second law says that takes a net force. By the end of this lesson, you should be able to identify the real force that keeps an object turning, draw free-body diagrams for circular motion, and use F_c = mv²/r.

Force Toward the Center

Circular motion: a points to the center
Newton’s 2nd law: needs F_net
F_net toward center = centripetal force

A top view of a puck on a string moving in a dashed circle around a center pin. A red tension arrow labeled F T points from the puck along the string toward the center, and a blue velocity arrow labeled v points tangent to the circle. Caption: the net force points to the center.

In the last lesson we saw that an object in uniform circular motion accelerates toward the center. Newton’s second law says every acceleration is caused by a net force in the same direction. So an object moving in a circle must have a net force pointing toward the center. That inward net force is called the centripetal force. For a puck swung on a string, the string’s tension pulls the puck inward and keeps it on the circle.

The Centripetal Force Equation

Start from Newton’s second law and use a_c = v²/r:

F_net = ma_c
F_c = mv² / r
F_c in newtons (N), m in kg, v in m/s, r in m

The equations F net = m a c and F c = m v squared over r in a yellow box. Below are four cards: F c is centripetal force in newtons, m is mass in kilograms, v is speed in meters per second, and r is radius in meters.

The centripetal force grows with mass and with the square of the speed, and it gets bigger for tighter circles. Doubling your speed on a curve takes four times as much inward force. Check the units: kg × (m/s)² ÷ m = kg·m/s², which is a newton.

Sources of Centripetal Force

Not a new kind of force
A real force pulls or pushes inward
Always name the real force

Four cards. Tension: a ball swung on a string. Friction: a car turning on a flat road. Gravity: the Moon orbiting Earth. Normal force: a rider pressed against the wall of a spinning rotor ride.

Centripetal force is not a new type of force like gravity or friction. It is a job that some real force has to do: pull or push the object toward the center. Tension does the job for a ball on a string. Static friction from the road does it for a car turning on a flat curve. Gravity does it for the Moon orbiting Earth. The normal force from the wall does it for a rider in a spinning rotor ride. When you solve a problem, always identify which real force is providing the centripetal force.

Free-Body Diagram: Car on a Curve

Draw only real forces
Friction points to the center
No extra “F_c” arrow

A free-body diagram of a car turning left on a flat road, seen from behind. A small box represents the car, with every arrow starting at its center. The normal force F N points up and gravity F g points down with equal lengths. Friction F f points left, toward the center of the curve. Caption: friction is the centripetal force.

Here is a free-body diagram for a car turning left on a flat road, seen from behind. Vertically, the normal force and gravity are balanced, because the car is not moving up or down. Horizontally, the only force is static friction from the road on the tires, pointing toward the center of the curve. That friction is the centripetal force, so F_f = mv²/r. Do not add a separate arrow labeled F_c; the centripetal force is the friction arrow you already drew. If the road is icy, or the car goes too fast, friction cannot supply enough force and the car slides off along the tangent.

Gravity in Orbit

Gravity pulls the Moon toward Earth
Gravity is the centripetal force
The Moon keeps falling around Earth

Earth at the center of a dashed circular orbit. The Moon is on the orbit with a red gravity arrow labeled F g pointing toward Earth and a blue velocity arrow labeled v pointing along the orbit. Caption: gravity bends the Moon's path into a circle.

The Moon moves in a nearly circular orbit around Earth. Nothing touches it, so the only force on it is Earth’s gravity, which always points toward Earth’s center. Gravity is the centripetal force for the Moon: F_g = mv²/r. You can think of the Moon as constantly falling toward Earth while also moving sideways so fast that it keeps missing. Satellites and the International Space Station orbit the same way.

Free-Body Diagram: Rotor Ride

Wall pushes rider toward the center
Normal force is the centripetal force
Friction holds the rider up

A side view of a spinning cylinder ride with a dashed vertical axis in the middle. A rider, shown as a box, is pressed against the right wall. The normal force F N points left, toward the axis. Friction F f points up and gravity F g points down with equal lengths.

In a rotor ride, riders stand against the wall of a spinning cylinder, and then the floor drops away. The free-body diagram shows why they do not fall. The wall pushes each rider toward the center (the axis), so the normal force is the centripetal force: F_N = mv²/r. That strong normal force also makes a large friction force possible, and friction pointing up balances gravity. Riders feel pressed into the wall, but the real force on them points inward.

Worked Examples

Try each problem on your own first, then check your work against the solution.

Example 1: Puck on a String

A 0.50 kg puck on an air hockey table is tied to a string and slides in a circle of radius 1.2 m at 6.0 m/s. What is the tension in the string?

A top view of a 0.50 kg puck on a string moving in a dashed circle of radius 1.2 m. A blue arrow labeled 6.0 m/s points along the circle, and a red arrow along the string toward the center is labeled F T = ?.

Source of F_c: The string’s tension pulls the puck toward the center, so F_T = mv²/r.
Knowns: m = 0.50 kg, v = 6.0 m/s, r = 1.2 m
Substitute: F_T = (0.50 kg)(6.0 m/s)² / 1.2 m = (0.50 kg)(36 m²/s²) / 1.2 m
Answer: F_T = 15 N toward the center

Example 2: Friction on a Curve

A 1,350 kg car rounds a flat curve of radius 45.0 m at 15.0 m/s. How much friction force must the road provide?

A top view of a car on a flat curved road. A blue arrow labeled 15.0 m/s points along the road, and a red arrow labeled F f = ? points toward the center of the curve. A dashed radius line is labeled r = 45.0 m.

Source of F_c: On a flat road, static friction points toward the center, so F_f = mv²/r.
Knowns: m = 1,350 kg, v = 15.0 m/s, r = 45.0 m
Substitute: F_f = (1,350 kg)(15.0 m/s)² / 45.0 m = (1,350 kg)(225 m²/s²) / 45.0 m
Answer: F_f = 6,750 N toward the center of the curve

Example 3: Rotor Ride Speed

A 60.0 kg rider stands against the wall of a rotor ride with a radius of 2.50 m. The wall pushes on her with a normal force of 1,350 N. How fast is she moving?

A side view of the rotor ride cylinder with a dashed axis. A 60.0 kg rider against the right wall has a red normal force arrow of 1,350 N pointing toward the axis. A double arrow from the axis to the wall is labeled r = 2.50 m, and a label asks v = ?.

Source of F_c: The wall’s normal force points toward the center, so F_N = mv²/r.
Rearrange: v² = F_Nr / m, so v = √(F_Nr / m)
Substitute: v = √((1,350 N)(2.50 m) / 60.0 kg) = √(56.25 m²/s²)
Answer: v = 7.50 m/s

Example 4: Toy Car Track

A 0.200 kg toy car moves at a constant 3.00 m/s around a flat circular track. Friction provides a 1.20 N force toward the center. What is the radius of the track?

A top view of a toy car on a circular track. A blue arrow labeled 3.00 m/s points along the track, and a red arrow labeled 1.20 N points toward the center. A dashed radius line is labeled r = ?.

Source of F_c: Friction points toward the center, so F_f = mv²/r.
Rearrange: Multiply both sides by r and divide by F_f: r = mv² / F_f
Substitute: r = (0.200 kg)(3.00 m/s)² / 1.20 N = (0.200 kg)(9.00 m²/s²) / 1.20 N
Answer: r = 1.50 m

Big Ideas

Circular motion needs F_net toward the center
F_c = mv²/r
Tension, friction, gravity, or normal force does the job
FBDs: real forces only, no extra F_c

Back to the essential question: a net force pointing toward the center, called the centripetal force, F_c = mv²/r. It is not a new kind of force. It is the job done by a real force such as tension, friction, gravity, or a normal force. This wraps up Unit 3 on 2D dynamics.