Worked Examples

Example 1: Satellite Orbiting Earth

A satellite completes one full orbit of Earth at a radius of 7,000 km (7,000,000 m) in 5,820 seconds. Find its circular velocity and centripetal acceleration.

Given: r = 7,000,000 m, T = 5,820 s

Step 1 – Circular Velocity: v = (2 × π × r) / T = (2 × 3.1416 × 7,000,000) / 5,820 = 43,982,297 / 5,820 ≈ 7,557 m/s

Step 2 – Centripetal Acceleration: a = v² / r = (7,557)² / 7,000,000 = 57,107,449 / 7,000,000 ≈ 8.16 m/s²

Answer: v ≈ 7,557 m/s; a ≈ 8.16 m/s²

Example 2: Car on a Circular Track

A car travels around a circular track of radius 50 m, completing one lap in 20 seconds. Find its circular velocity and centripetal acceleration.

Given: r = 50 m, T = 20 s

Step 1 – Circular Velocity: v = (2 × π × r) / T = (2 × 3.1416 × 50) / 20 = 314.16 / 20 = 15.71 m/s

Step 2 – Centripetal Acceleration: a = v² / r = (15.71)² / 50 = 246.80 / 50 = 4.94 m/s²

Answer: v ≈ 15.71 m/s; a ≈ 4.94 m/s²

Example 3: Ball on a String

A ball is swung in a horizontal circle on a string of length 0.8 m, completing 5 full revolutions per second. Find its circular velocity and centripetal acceleration.

Given: r = 0.8 m, f = 5 rev/s, so T = 1/f = 1/5 = 0.2 s

Step 1 – Circular Velocity: v = (2 × π × r) / T = (2 × 3.1416 × 0.8) / 0.2 = 5.0265 / 0.2 = 25.13 m/s

Step 2 – Centripetal Acceleration: a = v² / r = (25.13)² / 0.8 = 631.52 / 0.8 = 789.4 m/s²

Answer: v ≈ 25.13 m/s; a ≈ 789.4 m/s²

Common Mistakes to Avoid

• Using diameter instead of radius. Many students plug the full diameter of the circle into the formula. The formula v = (2 × π × r) / T requires the radius — half the diameter. Always halve the diameter before substituting.
• Forgetting to convert units before calculating. Mixing kilometers with seconds, or minutes with meters, produces wrong answers. Convert all distances to meters and all times to seconds before using any formula.
• Confusing period (T) and frequency (f). T is the time for one full revolution; f is the number of revolutions per second. They are reciprocals: f = 1/T. Substituting frequency directly where period is needed will give an incorrect velocity.
• Thinking centripetal acceleration points outward. Centripetal acceleration always points toward the center of the circle, not away from it. The outward feeling you sense in a turning car is a reaction effect, not the direction of acceleration.
• Assuming faster speed means less centripetal acceleration. Because a = v² / r, centripetal acceleration increases with the square of speed. Doubling the speed quadruples the centripetal acceleration, not halves it.

Real-World Applications

• Banked highway curves: Engineers design curved road sections with a tilted surface so that vehicles traveling at the posted speed experience centripetal acceleration directed toward the center of the curve, reducing reliance on friction and improving safety at high speeds.
• Washing machine spin cycle: The drum spins clothes in a circular path at high frequency. The very large centripetal acceleration forces water outward through the drum’s holes, effectively wringing clothes dry without heat.
• Artificial gravity in space stations: A rotating space station uses circular motion so that astronauts on the inner wall experience a centripetal acceleration directed inward, simulating the effect of gravity and preventing the bone and muscle loss caused by prolonged weightlessness.
• Particle accelerators: In a cyclotron or synchrotron, charged particles are steered in circular paths by magnetic fields. Scientists calculate the required centripetal acceleration precisely to keep particles on track at speeds approaching the speed of light.

Frequently Asked Questions

What is the difference between circular velocity and centripetal acceleration?

Circular velocity is the speed at which an object travels along the circumference of a circular path, measured in m/s. Centripetal acceleration is the rate of change of direction of that velocity, always pointing toward the center, measured in m/s². An object can have constant speed yet still accelerate because its direction continuously changes.

Does centripetal acceleration change if the radius is doubled?

Yes. Since a = v² / r, doubling the radius while keeping speed constant will halve the centripetal acceleration. A larger circular path requires less inward acceleration to maintain the same speed.

Why is centripetal acceleration directed toward the center?

For an object to follow a curved path, a net force — and therefore an acceleration — must act perpendicular to its velocity, pulling it inward. Without this inward acceleration, the object would travel in a straight line, as described by Newton’s first law.

Can circular velocity be zero?

If the object is stationary on a circular path, its speed is zero and there is no centripetal acceleration. Circular velocity is only meaningful when the object is actually moving along the circular path.

How does period relate to circular velocity?

Period T is the time to complete one full revolution. Since the distance traveled in one revolution equals the circumference (2 × π × r), dividing that distance by T gives the circular velocity: v = (2 × π × r) / T. A shorter period means higher velocity for the same radius.

Is centripetal acceleration the same as gravitational acceleration?

Not always, but gravity can provide centripetal acceleration. For a satellite in orbit, gravity is the force that causes the centripetal acceleration keeping it on a circular path. In other situations, tension, friction, or a normal force plays that role instead.

Sources & References

• Wikipedia — Circular motion

Written and fact-checked by Dr. Irfan Mansuri. External links open in a new tab and are provided for further reading and verification.