Circular Motion Equations- Essential Physics Formulas

What Is Circular Motion?

Circular motion happens when an object moves along a circular path. That's it. No fancy definitions needed. The object can move at constant speed or with changing speed—both count as circular motion.

There are two types you need to know:

• Uniform circular motion — speed stays constant, direction keeps changing
• Non-uniform circular motion — speed changes as the object moves around the circle

Most physics problems you'll encounter deal with uniform circular motion first. Get that down, then add complications later.

The Core Circular Motion Equations

These are the formulas you'll use constantly. Memorize them or keep them handy until they stick.

Angular Velocity (ω)

Angular velocity measures how fast something rotates. It tells you the rate of change of angle.

Formula:

ω = Δθ / Δt

Where:

• ω = angular velocity (radians per second, rad/s)
• Δθ = change in angle (radians)
• Δt = change in time (seconds)

For one complete revolution: ω = 2π / T

Period (T) and Frequency (f)

The period is the time for one complete revolution. The frequency is how many revolutions happen per second.

Formulas:

• T = 2πr / v
• f = 1 / T
• ω = 2πf

Where:

• T = period (seconds)
• f = frequency (hertz, Hz)
• r = radius of the circular path
• v = linear/tangential velocity

Centripetal Acceleration (a_c)

Acceleration always points toward the center of the circle. This is what keeps an object moving in a curved path instead of flying off tangentially.

Formula:

a_c = v² / r = ω²r

Where:

• a_c = centripetal acceleration (m/s²)
• v = linear velocity (m/s)
• r = radius (m)
• ω = angular velocity (rad/s)

Both versions work. Pick whichever gives you the least grief with the information you have.

Centripetal Force (F_c)

Net force pointing toward the center. Without it, objects fly off in straight lines—this is Newton's first law in action.

Formula:

F_c = m × v² / r = m × ω²r

Where:

• F_c = centripetal force (newtons, N)
• m = mass (kg)
• v = linear velocity (m/s)
• r = radius (m)

Combine with Newton's second law (F = ma) and you can solve almost any circular motion problem.

Tangential Velocity (v)

Velocity along the tangent to the circle. Perpendicular to the radius at any given point.

Formula:

v = 2πr / T = ωr

Angular Acceleration (α)

Used when rotational speed changes—non-uniform circular motion territory.

Formula:

α = Δω / Δt

Where α = angular acceleration (rad/s²)

Linear acceleration from rotation: a = αr

Quick Reference Table: Main Circular Motion Formulas

Centrifugal Force: The Fake Force

Skip this section if your physics class ignores it. If you need it, here it is.

Centrifugal force doesn't exist. It's a fictitious force—an illusion you feel in a rotating reference frame. When you're in a car taking a sharp turn, you feel pushed outward, but no such force acts on you. You're just following a straight line while the car turns beneath you.

The "force" you feel is your body's inertia resisting the change in direction.

In an inertial frame (what physicists normally use), only centripetal force exists. It pulls you inward. Your inertia makes you think something pushes you outward.

Mathematically, some textbooks assign it a value: F_cf = mv²/r, but it's just the negative of the real centripetal force from a rotating observer's perspective.

How To Solve Circular Motion Problems

Here's the process. Follow it every time.

Step 1: Identify the Circle

Find the center of rotation and the radius of the path. Draw a diagram if needed—your future self will thank you.

Step 2: List What You Know

Write down all given quantities. Mass, radius, velocity, period, frequency—whatever the problem gives you.

Step 3: Identify What's Being Asked

Force, acceleration, velocity, time? Know your target before hunting for formulas.

Step 4: Choose the Right Formula

Match your knowns to the formula that solves for your target. If you have v and r but need a_c, use a_c = v²/r. Simple.

Step 5: Plug In and Solve

Units matter. Convert everything to SI units (kg, m, s, rad/s) before calculating. One missed conversion ruins everything.

Step 6: Check Your Work

Does the answer make physical sense? A car taking a turn at 20 m/s with a 10m radius needs massive force. If your answer is tiny, you messed up.

Example Problem

A 1000 kg car turns a corner with a radius of 20 m at 15 m/s. What centripetal force acts on it?

Given: m = 1000 kg, r = 20 m, v = 15 m/s

Formula: F_c = mv²/r

Calculation: F_c = (1000)(15)² / 20 = (1000)(225) / 20 = 225,000 / 20 = 11,250 N

The car experiences 11,250 newtons of force pulling it toward the inside of the turn.

Common Mistakes to Avoid

• Using the wrong radius—measure from the center, not the edge of the path
• Confusing period and frequency—T is seconds per cycle, f is cycles per second
• Forgetting that centripetal force is not a new type of force—it's tension, gravity, or friction providing the inward pull
• Mixing up angular velocity units—radians aren't degrees (2π rad = 360°)
• Ignoring the direction of forces—centripetal always points to the center

Real-World Applications

These formulas show up everywhere:

• Car turns — friction provides the centripetal force
• Satellites orbiting Earth — gravity provides the centripetal force
• Roller coasters — track forces keep cars on the loop
• CD/DVD spinning — angular velocity and period calculations
• Washing machine spin cycle — centripetal acceleration pushes water through holes

Any time something moves in a curve, these equations apply.

Connecting to Other Physics Concepts

Circular motion doesn't exist in isolation. It ties to:

• Newton's laws — F = ma applies to the centripetal force equation
• Gravitation — orbital motion combines circular motion with gravitational force
• Energy — kinetic energy calculations use the same velocity terms
• Momentum — angular momentum L = mvr for circular motion

Master circular motion and you build a foundation for rotation, orbits, and advanced mechanics.