Uniform Circular Motion- Example Problems and Solutions
What Is Uniform Circular Motion?
Uniform circular motion happens when an object moves in a circle at a constant speed. That's it. No acceleration in speed, just constant velocity magnitude while the direction keeps changing.
This confuses people. They hear "constant speed" and think no forces are involved. Wrong. The velocity vector is always changing because direction changes. That's why centripetal acceleration exists.
If you're struggling with physics homework or exams, here's everything you need to solve uniform circular motion problems correctly.
Key Variables You Must Know
Before touching any problem, memorize these terms. They're not optional.
- v = linear (tangential) speed
- ω = angular velocity (radians per second)
- r = radius of the circular path
- T = period (time for one complete revolution)
- f = frequency (revolutions per second)
- ac = centripetal acceleration
- Fc = centripetal force
The Core Formulas
These equations are your toolkit. Know when to use each one.
| Formula | Use When |
|---|---|
| v = 2πr / T | Finding speed from period |
| ω = 2π / T = 2πf | Finding angular velocity |
| ac = v² / r | Finding centripetal acceleration |
| Fc = m × v² / r | Finding required centripetal force |
| v = ωr | Relating linear and angular speed |
Common Mistakes That Kill Your Grade
- Forgetting that centripetal force is not a new type of force. It's the net force pointing toward the center—gravity, tension, friction, whatever.
- Using the wrong radius. If a car goes around a banked curve, the radius matters.
- Mixing up period and frequency. T = 1/f, not the other way around without inverting.
- Calculating centripetal acceleration with speed in km/h instead of m/s. Always convert first.
Example Problems and Solutions
Problem 1: Satellite Orbiting Earth
A satellite orbits Earth at a constant speed of 7,800 m/s. The radius of its orbit is 7,000 km. Find the centripetal acceleration.
Step 1: Convert radius to meters.
7,000 km × 1,000 = 7,000,000 m
Step 2: Apply the formula.
ac = v² / r
ac = (7,800)² / 7,000,000
ac = 60,840,000 / 7,000,000
Answer: 8.69 m/s²
Problem 2: Car Taking a Turn
A 1,200 kg car rounds a curve with radius 50 m at 20 m/s. What centripetal force is required?
Step 1: Plug into the force formula.
Fc = m × v² / r
Fc = 1,200 × (20)² / 50
Fc = 1,200 × 400 / 50
Fc = 480,000 / 50
Answer: 9,600 N
This force comes from friction between tires and road. If friction can't provide this force, the car skids.
Problem 3: Finding Period from Angular Velocity
A wheel rotates at 120 rpm. What is the period?
Step 1: Convert rpm to rps (revolutions per second).
120 rpm ÷ 60 = 2 revolutions per second
This is the frequency: f = 2 Hz
Step 2: Find the period.
T = 1 / f = 1 / 2 = 0.5 seconds
Problem 4: Tension in a Rope (Conical Pendulum)
A 2 kg mass swings on a 1.5 m rope, making a 30° angle with the vertical. Find the tension in the rope.
Step 1: Draw the forces. Tension has two components: one canceling weight, one providing centripetal force.
T × cos(30°) = mg
T × sin(30°) = mv² / r
Step 2: Solve for T from the vertical equation.
T = mg / cos(30°)
T = (2 × 9.8) / 0.866
T = 19.6 / 0.866
Answer: 22.6 N
Step 3 (optional): Find the speed.
From the horizontal equation: v = √[(T × sin(30°) × r) / m]
Where r = L × sin(30°) = 1.5 × 0.5 = 0.75 m
v = √[(22.6 × 0.5 × 0.75) / 2] = √(8.48 / 2) = √4.24 = 2.06 m/s
How to Approach Any Circular Motion Problem
- Identify what's moving and the radius of its path.
- List known quantities—mass, speed, radius, period, whatever you're given.
- Choose the right formula. If you have speed and radius, you can find acceleration. If you have mass, you can find force.
- Check your units. Convert everything to meters, seconds, kilograms before calculating.
- Identify the source of centripetal force. Is it tension? Gravity? Friction? Normal force? This is where most students lose points.
Quick Reference: What to Remember
The centripetal force equation is just Newton's second law applied to circular motion.
Fnet = mac
Fnet = m × v² / r
Whatever force points toward the center—gravity, tension, friction, normal force—it's providing the centripetal force. If there's no such force, there's no circular motion.
Speed is constant but velocity isn't because direction changes. That's why there's still acceleration.
Centripetal acceleration points toward the center, not outward. The "centrifugal force" people mention? It's not real in an inertial reference frame. It's just the sensation of being pushed outward when your inertia wants to go straight.