Circular Motion Calculator

General Physics

Calculate variables of circular motion like centripetal force, acceleration, and velocity.

Practical Examples

Explore real-world scenarios of circular motion calculations.

Ferris Wheel Centripetal Force

Centripetal Force

Calculate the centripetal force on a 60 kg person riding a Ferris wheel with a radius of 15 meters at a velocity of 5 m/s.

Mass: 60

Radius: 15

Velocity: 5

Mass Unit: kg

Radius Unit: m

Velocity Unit: m/s

Satellite's Orbital Velocity

Velocity

A 1000 kg satellite experiences a centripetal force of 3400 N orbiting the Earth at a radius of 6600 km. Find its velocity.

Mass: 1000

Radius: 6600

Centripetal Force: 3400

Mass Unit: kg

Radius Unit: km

Force Unit: N

Car Turning a Corner

Acceleration

A car is turning a corner with a radius of 50 meters at a speed of 40 km/h. What is its centripetal acceleration?

Radius: 50

Velocity: 40

Radius Unit: m

Velocity Unit: km/h

Hammer Throw Radius

Radius

An athlete spins a 7.26 kg hammer. If the centripetal force is 800 N and the hammer's velocity is 10 m/s, what is the radius of the circle?

Mass: 7.26

Velocity: 10

Centripetal Force: 800

Mass Unit: kg

Velocity Unit: m/s

Force Unit: N

Other Titles
Understanding the Circular Motion Calculator: A Comprehensive Guide
An in-depth look at the principles of circular motion and how to apply them.

1. Introduction to Circular Motion

  • What is Circular Motion?
  • Uniform vs. Non-Uniform Circular Motion
  • Key Terminology
Circular motion is the movement of an object along the circumference of a circle or rotation along a circular path. It can be uniform, with a constant angular rate of rotation and constant speed, or non-uniform, with a changing rate of rotation. This concept is fundamental in physics and engineering, describing everything from planetary orbits to the operation of a centrifuge.
Key Terminology
Radius (r): The distance from the center of the circle to the object.
Velocity (v): The speed of the object in a given direction (tangential to the circle).
Angular Velocity (ω): The rate at which the object rotates, measured in radians per second.
Period (T): The time it takes for one full revolution.
Frequency (f): The number of revolutions per unit of time (f = 1/T).
Centripetal Acceleration (ac): The acceleration directed towards the center of the circle that keeps the object on its path.
Centripetal Force (Fc): The force responsible for the centripetal acceleration, directed towards the center.

2. Key Concepts and Formulas

  • Centripetal Acceleration Formula
  • Centripetal Force Formula
  • Relationships Between Linear and Angular Variables
The behavior of an object in circular motion is governed by a set of core mathematical formulas that link its mass, speed, radius, and the forces acting upon it.
Core Formulas
Centripetal Acceleration: ac = v² / r = ω² * r
Centripetal Force: Fc = m ac = m v² / r = m ω² r
Velocity: v = ω r = 2 π * r / T
Angular Velocity: ω = v / r = 2 π f

3. Step-by-Step Guide to Using the Calculator

  • Selecting the Calculation Type
  • Inputting Your Data
  • Interpreting the Results
Our calculator simplifies complex circular motion problems into a few easy steps.
1. Choose what to calculate: Use the dropdown menu to select the variable you want to find (e.g., Centripetal Force).
2. Enter the known values: Fill in the required input fields that appear. For example, to find force, you'll need mass, velocity, and radius.
3. Select units: Choose the appropriate unit for each input value from the adjacent dropdowns.
4. Click 'Calculate': The calculator will instantly provide the result, along with the formula used for the calculation.
5. Use an example: If you're unsure, load an example to see how the calculator works with pre-filled data.

4. Real-World Applications of Circular Motion

  • Astronomy and Space Exploration
  • Engineering and Technology
  • Amusement Parks and Recreation
The principles of circular motion are visible all around us in both natural phenomena and human-made technology.
Application Examples
Satellites and Planets: Gravity provides the centripetal force that keeps satellites in orbit around Earth and planets around the Sun.
Vehicles Turning: The friction between the tires and the road provides the centripetal force needed for a car to turn a corner.
Centrifuges: Used in labs to separate substances of different densities by spinning them at high speeds.
Roller Coasters: The loops and turns on a roller coaster are designed using the principles of circular motion to keep riders safely in their seats.

5. Mathematical Derivations and Advanced Topics

  • Derivation of Centripetal Acceleration
  • Work Done in Uniform Circular Motion
  • Non-Uniform Circular Motion
For those interested in the underlying mathematics, understanding the derivation of these formulas provides deeper insight.
Derivation of ac = v²/r
The derivation of centripetal acceleration involves vector calculus. By considering the change in the velocity vector over an infinitesimally small time interval dt, we can show through geometric analysis of similar triangles (formed by the position and velocity vectors) that the magnitude of the acceleration is v²/r and its direction is towards the center of the circle.
Work Done
In uniform circular motion, the centripetal force is always perpendicular to the direction of motion (the velocity vector). Since work done is the dot product of force and displacement (W = F · d), and the angle between them is 90 degrees, the work done by the centripetal force is zero. This means the kinetic energy of the object remains constant.