Circular Motion Calculator – v = ωr & a = v²/r | OmniCalculator.Space

  • Adam
  • November 27, 2025

Calculate circular motion parameters instantly with OmniCalculator.Space’s free calculator. Find velocity, centripetal acceleration, centripetal force, period & frequency using physics formulas. Step-by-step solutions for uniform circular motion problems.

Circular Motion Calculator - v = ωr & a = v²/r | OmniCalculator.Space

The circular motion calculator from OmniCalculator.Space helps you analyze objects moving in circular paths by calculating velocity, acceleration, period, frequency, and centripetal force. Whether studying uniform circular motion or solving physics problems, this calculator provides accurate results using fundamental formulas like v = ωr and ac = v²/r.

Calculate Circular Motion Parameters

Results:

What is Circular Motion?

Circular motion occurs when an object moves along a circular path at a constant or varying speed. In uniform circular motion, the object maintains constant speed while its velocity direction continuously changes due to centripetal acceleration directed toward the center. This fundamental concept in physics applies to everything from electrons orbiting nuclei to planets orbiting stars, making the OmniCalculator.Space circular motion calculator an essential tool for students and professionals.

Key Characteristics of Circular Motion:

  • Centripetal acceleration: Always directed toward the center of the circular path
  • Constant speed (uniform): Magnitude of velocity remains constant, but direction changes continuously
  • Tangential velocity: Velocity vector is always tangent to the circular path
  • Angular velocity: Measures how fast the object rotates around the center
  • Centripetal force: Net inward force required to maintain circular motion
  • Period and frequency: Time for one complete revolution and number of revolutions per unit time

Circular Motion Formulas

Understanding circular motion requires mastering several interconnected formulas that relate linear and angular quantities. The OmniCalculator.Space calculator uses these fundamental equations:

Linear Velocity Formula

v = ωr

Where:

  • v = linear (tangential) velocity (m/s)
  • ω = angular velocity (rad/s)
  • r = radius of circular path (m)

Centripetal Acceleration Formula

ac = v² / r

or equivalently

ac = ω²r

Where:

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

Centripetal Force Formula

Fc = mv² / r

or equivalently

Fc = mω²r

Where:

  • Fc = centripetal force (Newtons, N)
  • m = mass (kg)
  • v = linear velocity (m/s)
  • r = radius (m)
  • ω = angular velocity (rad/s)

Period and Frequency Formulas

Period (time for one revolution):

T = 2πr / v = 2π / ω

Frequency (revolutions per second):

f = 1/T = v / (2πr) = ω / (2π)

Relationship:

ω = 2πf = 2π/T

How to Calculate Circular Motion

Follow these systematic steps using the OmniCalculator.Space circular motion calculator to solve any circular motion problem:

  1. Identify known quantities: Determine which variables you have (radius, velocity, angular velocity, mass, etc.).
  2. Choose the appropriate formula: Select based on what you're calculating (velocity, acceleration, force, period).
  3. Convert units if necessary: Ensure all values are in SI units (m, kg, s, rad/s).
  4. Substitute values: Plug known values into the selected formula.
  5. Calculate the result: Perform the mathematical operations.
  6. Verify reasonableness: Check if the answer makes physical sense.
  7. Find related quantities: Use relationships to calculate other parameters if needed.

Calculation Example 1: Linear Velocity

Problem: A point on a rotating disk has angular velocity 5 rad/s at a distance of 0.4 m from the center. Find the linear velocity.

Solution:

Given: ω = 5 rad/s, r = 0.4 m

Using formula: v = ωr

v = 5 × 0.4

v = 2.0 m/s

Additional calculations:

• Period: T = 2π/ω = 2π/5 = 1.26 seconds

• Frequency: f = 1/T = 0.796 Hz

Calculation Example 2: Centripetal Acceleration

Problem: A car travels at 20 m/s around a curve with radius 50 m. Calculate the centripetal acceleration.

Solution:

Given: v = 20 m/s, r = 50 m

Using formula: ac = v²/r

ac = (20)² / 50

ac = 400 / 50

ac = 8.0 m/s²

Interpretation: The car experiences acceleration of 8 m/s² toward the center, or about 0.82 g.

Calculation Example 3: Centripetal Force

Problem: A 1500 kg car navigates a circular track of radius 100 m at 25 m/s. Find the required centripetal force.

Solution:

Step 1: Calculate centripetal acceleration

ac = v²/r = (25)²/100 = 625/100 = 6.25 m/s²

Step 2: Calculate centripetal force

Fc = mac = 1500 × 6.25

Fc = 9,375 N = 9.375 kN

Alternative direct calculation:

Fc = mv²/r = 1500 × 625/100 = 9,375 N ✓

Types of Circular Motion

Circular motion can be classified into two main categories, each with distinct characteristics:

Uniform Circular Motion

In uniform circular motion, the object moves at constant speed along a circular path. Key features include:

  • Constant speed: Magnitude of velocity remains unchanged
  • Variable velocity: Direction changes continuously, so velocity is not constant
  • Constant acceleration magnitude: Centripetal acceleration has constant magnitude ac = v²/r
  • Perpendicular acceleration: Acceleration is always perpendicular to velocity
  • No tangential acceleration: All acceleration is directed radially inward

Non-Uniform Circular Motion

When speed changes along the circular path, the motion is non-uniform:

  • Variable speed: Both magnitude and direction of velocity change
  • Tangential acceleration: Component along the velocity direction causes speed changes
  • Radial acceleration: Centripetal component directed toward center
  • Total acceleration: Vector sum of tangential and radial components

Circular Motion Parameters Comparison

Parameter Formula Units Description
Linear Velocity v = ωr m/s Tangential speed along path
Angular Velocity ω = v/r rad/s Rate of angular displacement
Centripetal Acceleration ac = v²/r m/s² Inward acceleration
Centripetal Force Fc = mv²/r N (Newton) Inward force requirement
Period T = 2πr/v seconds Time for one revolution
Frequency f = 1/T Hz Revolutions per second

Applications of Circular Motion

The OmniCalculator.Space circular motion calculator is invaluable for analyzing numerous real-world applications:

  • Astronomy: Planetary orbits, satellite motion, and galactic rotation calculations
  • Automotive Engineering: Designing safe curves, banking angles, and analyzing vehicle dynamics
  • Amusement Parks: Calculating forces on roller coasters, Ferris wheels, and spinning rides
  • Sports: Analyzing hammer throw, discus, shot put, and ball trajectories in curved paths
  • Machinery: Centrifuges, washing machines, turbines, and rotating equipment design
  • Electronics: Cyclotrons, particle accelerators, and electron beam deflection
  • Navigation: Aircraft turning radius calculations and maritime course plotting
  • Physics Education: Demonstrating fundamental concepts of motion and forces

Frequently Asked Questions

What is the formula for circular motion velocity?
The linear velocity formula for circular motion is v = ωr, where v is the tangential velocity in m/s, ω is the angular velocity in rad/s, and r is the radius in meters. This formula shows that velocity increases linearly with both angular velocity and radius. For example, if ω = 10 rad/s and r = 0.5 m, then v = 5 m/s. The OmniCalculator.Space circular motion calculator automatically computes this and related quantities.
How do you calculate centripetal acceleration?
Centripetal acceleration is calculated using ac = v²/r where v is linear velocity and r is radius. Alternatively, use ac = ω²r with angular velocity. For example, an object moving at 10 m/s in a 5 m radius circle has ac = 100/5 = 20 m/s². This acceleration is always directed toward the center of the circular path and represents the rate of change of velocity direction, not speed magnitude in uniform circular motion.
What is centripetal force and how is it calculated?
Centripetal force is the net inward force required to maintain circular motion, calculated as Fc = mv²/r or Fc = mω²r. It's not a separate type of force but rather the net result of real forces like tension, gravity, or friction. For a 2 kg object moving at 8 m/s in a 4 m radius circle: Fc = 2 × 64/4 = 32 N. Use OmniCalculator.Space to quickly calculate centripetal force for any circular motion scenario.
How do you find the period of circular motion?
The period (time for one complete revolution) is calculated using T = 2πr/v or T = 2π/ω. For example, if an object travels at 5 m/s in a circle of radius 10 m, then T = 2π(10)/5 = 4π = 12.57 seconds. The period is inversely related to frequency: f = 1/T. Higher speeds result in shorter periods. The OmniCalculator.Space calculator provides instant period calculations from velocity and radius.
What is the difference between angular velocity and linear velocity?
Angular velocity (ω) measures how fast an object rotates in rad/s, while linear velocity (v) measures tangential speed in m/s. They're related by v = ωr. All points on a rigid rotating object share the same angular velocity, but points farther from the center have higher linear velocities. For example, at ω = 2 rad/s, a point at r = 3 m has v = 6 m/s, while a point at r = 1 m has v = 2 m/s.
Why is acceleration directed toward the center in circular motion?
In uniform circular motion, acceleration is centripetal (center-seeking) because the velocity direction continuously changes even though speed is constant. Acceleration measures the rate of change of velocity (a vector), and changing direction requires acceleration. The centripetal acceleration formula ac = v²/r quantifies this inward acceleration. Without this center-directed acceleration, the object would travel in a straight line due to inertia (Newton's first law).
How do you convert RPM to angular velocity?
To convert revolutions per minute (RPM) to angular velocity in rad/s, use ω = 2πn/60 where n is RPM. Each revolution is 2π radians, and we divide by 60 to convert minutes to seconds. For example, 300 RPM converts to ω = 2π(300)/60 = 10π = 31.42 rad/s. Once you have ω, use v = ωr to find linear velocity. The OmniCalculator.Space calculator handles these conversions automatically.
What provides the centripetal force in different situations?
The centripetal force comes from various sources depending on the situation: tension in a string for swinging objects, gravity for orbital motion, friction between tires and road for turning vehicles, normal force on banked curves, and magnetic force for charged particles in cyclotrons. The key is that the net inward force equals mv²/r. If insufficient centripetal force is available, the object cannot maintain circular motion and moves outward.
Can you have circular motion without centripetal force?
No, circular motion requires centripetal force. Newton's first law states objects move in straight lines unless acted upon by a force. To curve an object's path into a circle requires continuous inward force (centripetal force). The magnitude must be exactly Fc = mv²/r. Too little force and the object spirals outward; too much and it spirals inward. Without any centripetal force, the object immediately departs on a straight tangent line at its current velocity.
How does radius affect circular motion parameters?
Radius significantly affects all circular motion parameters. For constant angular velocity: linear velocity increases with radius (v = ωr), centripetal acceleration increases (ac = ω²r), and centripetal force increases (Fc = mω²r). For constant linear velocity: centripetal acceleration decreases inversely (ac = v²/r), meaning larger circles require less inward acceleration. Period increases with radius (T = 2πr/v). Use OmniCalculator.Space to explore these relationships interactively.

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