Centripetal Acceleration Problems- Practice Guide

What Is Centripetal Acceleration?

Centripetal acceleration is the acceleration that keeps an object moving in a circular path. It's always directed toward the center of the circle. That's the whole deal.

If something moves in a circle at constant speed, it's still accelerating because velocity is a vector — direction changes, so velocity changes, and that means acceleration exists. 🔄

The Core Formula

Here's the equation you need to memorize:

a_c = v² / r

Where:

• a_c = centripetal acceleration (m/s²)
• v = linear velocity (m/s)
• r = radius of the circular path (m)

You can also write this using angular velocity:

a_c = ω²r

Where ω (omega) is angular velocity in rad/s.

Types of Centripetal Acceleration Problems

Finding Acceleration from Velocity and Radius

This is the simplest version. You get v and r, plug into a_c = v²/r.

Finding Acceleration from Period and Radius

Sometimes you get the period T instead of velocity. Use this relationship:

v = 2πr / T

Substitute that into the main formula:

a_c = (4π²r) / T²

Finding Acceleration from Angular Velocity

When you have angular velocity, just use a_c = ω²r. No conversion needed.

Finding Force from Centripetal Acceleration

Newton's second law still applies:

F_c = m × a_c

That's the centripetal force — the net force pointing toward the center.

Practice Problems

Problem 1: Basic Calculation

A car travels around a circular track with radius 50 m at a constant speed of 20 m/s. What is the centripetal acceleration?

Solution:

a_c = v² / r

a_c = (20)² / 50

a_c = 400 / 50

a_c = 8 m/s²

Problem 2: Using Period

A satellite orbits Earth in a circular path with radius 7,000 km. It completes one orbit every 90 minutes. Find the centripetal acceleration.

Solution:

First convert T to seconds: T = 90 × 60 = 5400 s

Convert r to meters: r = 7,000,000 m

a_c = 4π²r / T²

a_c = 4 × π² × 7,000,000 / (5400)²

a_c = 4 × 9.87 × 7,000,000 / 29,160,000

a_c = 276,360,000 / 29,160,000

a_c ≈ 9.48 m/s²

Problem 3: Finding the Force

A 2 kg ball swings in a horizontal circle with radius 1.5 m at 4 m/s. What centripetal force is required?

Solution:

First find acceleration: a_c = v²/r = 16/1.5 = 10.67 m/s²

Then find force: F = m × a_c = 2 × 10.67

F_c = 21.3 N

Quick Comparison: Linear vs. Angular Formulas

How to Solve Any Centripetal Acceleration Problem

Follow this step-by-step approach:

1. Identify what you know. Write down v, r, ω, or T. Convert units if needed.
2. Identify what you need. Acceleration? Force? Velocity? Keep the goal clear.
3. Pick the right formula. Use a_cc = ω²r if you have ω. Use a_c = 4π²r/T² if you have T.
4. Solve algebraically first. Don't plug numbers until you've rearranged the equation.
5. Check your units. Make sure m/s² is what you expect.

Common Mistakes to Avoid

• Using diameter instead of radius. The formula requires r, not d. If you only have diameter, divide by 2 first.
• Forgetting to square the velocity. a_c = v²/r, not v/r. This trips up a lot of people.
• Using frequency instead of period. Period is time for one revolution. Frequency is revolutions per second. If you have f, use T = 1/f.
• Mixing up centripetal and centrifugal. Centripetal force points toward the center. Centrifugal is a "fake" force that appears in a rotating frame. You won't need centrifugal force for these problems.
• Skipping unit conversion. Radians are dimensionless, but angular velocity must be in rad/s. Period must be in seconds.

When Centripetal Force Comes From Friction or Tension

In real problems, the centripetal force isn't some magic force. It's tension in a string, friction between tires and road, or normal force. You solve these the same way:

F_provided = m × v² / r

Example: A car rounding a flat curve has friction providing the centripetal force. If μ = 0.5 and the car mass is 1000 kg, the maximum friction force is μmg. Set that equal to mv²/r and solve for maximum safe speed.

Key Takeaways

• Centripetal acceleration always points toward the center of the circular path.
• The three main formulas are a_c = v²/r, a_c = ω²r, and a_c = 4π²r/T².
• Centripetal force equals mass times centripetal acceleration.
• Units matter. Always check that radius is in meters and velocity in m/s.
• Practice switching between velocity-based and angular-based equations.

That's it. Memorize the formulas, identify what you're given, plug and solve. The problems are straightforward once you stop overthinking them.