Uniform Circular Motion

The video lesson for this topic is coming soon.

Essential Question

How can something move at a constant speed and still accelerate?

A car on a curve or a rider on a Ferris wheel can keep a steady speed and still be accelerating. By the end of this lesson, you should be able to explain centripetal acceleration and use v = 2πr/T and a_c = v²/r.

Moving in a Circle

Path: a circle of radius r
Speed stays constant
Ferris wheels, merry-go-rounds, curves

An object moving counterclockwise around a circle. A radius line from the center to the object is labeled r, and a curved arrow outside the circle shows the direction of motion. Caption: constant speed around a circle.

Uniform circular motion means moving in a circle at a constant speed. The circle has a radius r, the distance from the center to the moving object. Riders on a Ferris wheel, a kid on a merry-go-round, a car going around a curve at a steady speed, and the Moon orbiting Earth are all close to uniform circular motion.

Velocity Keeps Changing

Velocity is tangent to the circle
Same speed, new direction
New velocity means acceleration

A circle with an object shown at the top, left, bottom, and right. At each spot a blue velocity arrow of the same length points along the circle, tangent to it: left at the top, down at the left, right at the bottom, and up at the right. Caption: same length, different directions.

At every point, the velocity of an object moving in a circle points along the circle, tangent to it. In the picture, every arrow has the same length, because the speed is constant. But the arrows point in different directions. Velocity is a vector, so a change in direction is a change in velocity, and any change in velocity is an acceleration. That is how an object can have a constant speed and still accelerate.

Centripetal Acceleration

Points toward the center
“Centripetal” = center-seeking
Always ⊥ to the velocity

A circle with an object at three positions. At each position a blue velocity arrow points tangent to the circle and a green acceleration arrow points straight toward the center, at a right angle to the velocity. Caption: a c always points to the center.

To keep turning onto a circle, the object’s velocity must keep bending toward the center. So its acceleration points toward the center of the circle. This is called centripetal acceleration, a_c, from Latin words meaning “center-seeking.” It is always perpendicular to the velocity. Because it is sideways to the motion, it changes only the direction of the velocity, not the speed.

Period and Speed

v = 2πr / T

v = speed, in meters per second (m/s)
r = radius of the circle, in meters (m)
T = period, the time for one full lap, in seconds (s)

A circle with a radius line labeled r and a curved arrow around the outside labeled 2 pi r, one full lap. Below it, the equation v = 2 pi r / T in a yellow box.

The period T is the time it takes to go around the circle once. In one lap, the object travels the circumference of the circle, 2πr. Speed is distance divided by time, so the speed is 2πr divided by T.

The Centripetal Acceleration Equation

a_c = v^2 / r

a_c = centripetal acceleration, in m/s², pointing toward the center
v = speed, in m/s
r = radius, in m

The equation a c = v squared over r in a yellow box. Below it, three cards: a c is centripetal acceleration in meters per second squared, v is speed in meters per second, and r is radius in meters.

The faster you go around a curve, the more sharply your velocity has to turn, and the acceleration grows with the square of the speed. Double the speed and the acceleration becomes four times as large. A tighter circle (smaller r) also means a larger acceleration. The units work out: (m/s)² ÷ m = m/s².

When the Pull Stops

It keeps going straight
It leaves along the tangent
It does NOT fly straight outward

A ball swung counterclockwise on a string around a dashed circle. At the right side of the circle the string breaks. A green arrow shows the ball moving straight up along the tangent, labeled real path. A red dashed arrow pointing straight outward from the center is crossed out and labeled not this way.

If the force pulling an object toward the center suddenly disappears, nothing bends its path anymore. By Newton’s first law, it keeps moving in a straight line in whatever direction it was going at that instant, which is along the tangent. It does not fly straight outward away from the center. The feeling of being “thrown outward” on a turn is really your body’s inertia trying to keep going straight.

Worked Examples

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

Example 1: Merry-Go-Round

A child sits 2.0 m from the center of a merry-go-round. It makes one full turn every 4.0 s. What is the child’s speed?

A top view of a merry-go-round: a circle with a child at the edge, a radius line labeled r = 2.0 m, and a label T = 4.0 s.

Knowns: r = 2.0 m, T = 4.0 s, v = ?
Equation: v = 2πr / T
Substitute: v = 2π(2.0 m) / 4.0 s = 12.6 m / 4.0 s
Answer: v = 3.1 m/s

Example 2: Car on a Curve

A car drives around a curve of radius 50.0 m at a constant 15.0 m/s. What is the car’s centripetal acceleration?

A top view of a curved road. A car on the curve has a blue velocity arrow labeled v = 15.0 m/s pointing along the road, tangent to the curve, and a green arrow labeled a c = ? pointing toward the center of the curve. A dashed radius line is labeled r = 50.0 m.

Knowns: v = 15.0 m/s, r = 50.0 m, a_c = ?
Equation: a_c = v² / r
Substitute: a_c = (15.0 m/s)² / 50.0 m = (225 m²/s²) / 50.0 m
Answer: a_c = 4.50 m/s² toward the center of the curve

Example 3: Ferris Wheel

A Ferris wheel has a radius of 20.0 m and turns once every 40.0 s. Find a rider’s speed and centripetal acceleration.

A Ferris wheel drawn as a circle with spokes and a support stand. A rider at the edge has a radius line labeled r = 20.0 m, and a label says T = 40.0 s.

Knowns: r = 20.0 m, T = 40.0 s
Speed: v = 2πr / T = 2π(20.0 m) / 40.0 s = 125.7 m / 40.0 s = 3.14 m/s
Acceleration: a_c = v² / r = (3.14 m/s)² / 20.0 m = (9.87 m²/s²) / 20.0 m
Answer: v = 3.14 m/s and a_c = 0.493 m/s² toward the center

Example 4: Ball on a String

A ball on a string moves in a horizontal circle of radius 0.50 m. Its centripetal acceleration is 32 m/s². How fast is the ball moving?

A top view of a ball on a string moving in a dashed circle. The string from the center to the ball is labeled 0.50 m. A green arrow from the ball toward the center is labeled 32 m/s squared, and a blue tangent arrow is labeled v = ?.

Knowns: r = 0.50 m, a_c = 32 m/s², v = ?
Equation: a_c = v² / r, rearranged to v = √(a_c r)
Substitute: v = √((32 m/s²)(0.50 m)) = √(16 m²/s²)
Answer: v = 4.0 m/s

Big Ideas

Constant speed, but changing velocity
a_c points toward the center
v = 2πr/T and a_c = v²/r
No pull: object moves off along the tangent

Back to the essential question: velocity is a vector, so changing direction changes the velocity even when the speed stays the same. An object in uniform circular motion is always turning toward the center, so it has a centripetal acceleration a_c = v²/r that points to the center. Next up, we’ll find the force that causes that acceleration: centripetal force.