Circular Motion
Circular motion is the motion of an object along the circumference of a circle or a circular path at a constant or varying speed. In uniform circular motion, the speed is constant but the velocity vector continuously changes direction, requiring a centripetal acceleration and force directed toward the center. Circular motion is fundamental to understanding planetary orbits, rotating machinery, and many natural phenomena.
Key Formula
v = rω, T = 2πr / v = 2π / ω
LaTeX: v = r\omega, \quad T = \frac{2\pi r}{v} = \frac{2\pi}{\omega}
| Symbol | Meaning | Unit |
|---|---|---|
| v | Linear (tangential) speed | m/s |
| r | Radius of circular path | m |
| \omega | Angular velocity | rad/s |
| T | Period (time for one revolution) | s |
Worked Example
Problem
A wheel of radius 0.50 m rotates at 120 rpm (revolutions per minute). Find the angular velocity, linear speed of a point on the rim, and the period of revolution.
Solution
Step 1: Convert rpm to rad/s — ω = 120 × (2π/60) = 4π rad/s ≈ 12.57 rad/s. Step 2: Linear speed v = rω = 0.50 × 4π = 2π ≈ 6.28 m/s. Step 3: Period T = 2π/ω = 2π/(4π) = 0.5 s.
Answer
ω ≈ 12.57 rad/s, v ≈ 6.28 m/s, T = 0.5 s.
Key quantities in uniform circular motion and their relationships
| Quantity | Symbol | Unit | Formula | Description |
|---|---|---|---|---|
| Angular displacement | θ | rad | θ = s/r | Angle swept by radius |
| Angular velocity | ω | rad/s | ω = dθ/dt | Rate of angle change |
| Linear speed | v | m/s | v = rω | Speed along arc |
| Period | T | s | T = 2π/ω | Time for one revolution |
| Frequency | f | Hz | f = 1/T | Revolutions per second |
| Centripetal acc. | ac | m/s² | ac = v²/r | Inward acceleration |
Interactive Tools
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Related Terms
Centripetal Force
Centripetal force is the net inward force that acts on an object moving in a circular path, always directed toward the center of the circle. It is not a new type of force but rather the resultant of existing forces (tension, gravity, friction, normal force) that provides the necessary centripetal acceleration. Without this inward force, an object would continue in a straight line by Newton's first law.
Centripetal Acceleration
Centripetal acceleration is the acceleration experienced by an object moving in a circular path, always directed toward the center of the circle. Although the object's speed may remain constant, its direction changes continuously, producing a non-zero acceleration perpendicular to the velocity. This acceleration is responsible for the continuous change in the direction of motion.
Angular Velocity
Angular velocity is the rate of change of angular displacement of a rotating object with respect to time. It is a vector quantity whose direction is given by the right-hand rule along the axis of rotation. Angular velocity is the rotational analogue of linear velocity and is central to the analysis of rotating machinery, celestial bodies, and rigid body dynamics.
From Latin "circularis" (circular) and "motio" (movement, from "movere" to move). The systematic study of circular motion was established by Galileo Galilei and later formalized by Isaac Newton in "Principia Mathematica" (1687).