Linear and Angular Motion Formulas Explained

Physics Formula, Angular Velocity and Linear Velocity - Formula Quest Mania

Physics of Angular and Linear Speed: A Complete Guide

In circular motion, two critical measurements describe how objects move: angular velocity and linear velocity. Angular velocity measures how fast something spins, while linear velocity measures how fast it moves along a path. Though different, the two are closely linked — understanding their relationship provides powerful insight into rotating systems, from spinning wheels to planetary orbits.

This article dives deep into the formulas, definitions, derivations, and real-world examples of angular and linear velocity. Using MathJax to display LaTeX equations, we aim to provide clear mathematical foundations, step-by-step problem-solving, and practical applications.

What is Angular Velocity?

Angular velocity ($ \omega $) is defined as the rate of change of angular displacement ($ \theta $) over time. It describes how quickly an object rotates around a central point or axis. Angular velocity is a vector quantity, with its direction determined by the right-hand rule.

Formula for Angular Velocity

$$ \omega = \frac{\theta}{t} $$

Where:

$ \omega $: Angular velocity (rad/s)
$ \theta $: Angular displacement (radians)
$ t $: Time (seconds)

Conversion of Units

Angular velocity is often given in revolutions per minute (RPM). To convert to radians per second:

$$ \omega = \text{RPM} \times \frac{2\pi}{60} $$

Example: Convert 120 RPM to rad/s

$$ \omega = 120 \times \frac{2\pi}{60} = 4\pi \ \text{rad/s} \approx 12.57 \ \text{rad/s} $$

What is Linear Velocity?

Linear velocity ($ v $) is the rate at which an object moves along a path. In circular motion, the object may rotate around a center, but each point on its edge is moving in a straight line tangent to the circle at any moment.

Formula for Linear Velocity

For general motion: $$ v = \frac{d}{t} $$

For circular motion, the linear velocity is also expressed as:

$$ v = r \cdot \omega $$

Where:

$ r $: Radius of the circular path
$ \omega $: Angular velocity in rad/s

Tangential vs Angular Quantities

While angular velocity refers to rotational motion, tangential velocity is the linear speed of a point on the edge of the rotating body. The relationship:

$$ v_{\text{tangential}} = r \cdot \omega $$

Thus, even though two points on a rotating disc have the same angular velocity, the one farther from the center travels faster in linear terms.

What About Centripetal Acceleration?

An object moving in a circle is constantly accelerating towards the center — this is centripetal acceleration, defined as:

$$ a_c = \frac{v^2}{r} = r \omega^2 $$

This acceleration causes centripetal force: $$ F_c = m a_c = m r \omega^2 $$

Where:

$ m $: Mass of the object
$ r $: Radius
$ \omega $: Angular velocity

Applications of Angular and Linear Velocity

1. Compact Discs and Rotating Storage

In CDs or DVDs, the inner tracks spin faster (higher angular velocity) while the outer tracks cover more distance (higher linear velocity). Devices must balance both to read data accurately.

2. Wind Turbines

Blades of a turbine have the same angular velocity, but the tips have greater linear velocity. Understanding this helps engineers optimize power output and blade design.

3. Vehicle Tires

When a car moves, the rotation of its wheels (angular velocity) determines how fast it travels (linear velocity). Tire circumference and RPM affect the speedometer reading.

4. Orbital Mechanics

Planets orbiting the sun follow circular or elliptical paths. Angular velocity governs their period, while linear velocity determines orbital speed.

5. Industrial Machinery

In conveyor belts, gears, and pulleys, angular and linear velocities determine speed, torque, and mechanical advantage.

More Practice Examples

Example 1: CD Rotation

A CD spins at 600 RPM. Its radius is 0.06 m. What is the linear velocity at the edge?

Convert to rad/s: $$ \omega = 600 \times \frac{2\pi}{60} = 62.83 \ \text{rad/s} $$

Then, $$ v = r \cdot \omega = 0.06 \times 62.83 = 3.77 \ \text{m/s} $$

Example 2: Bike Wheel

A bicycle wheel has a radius of 0.35 m and spins at 4 revolutions per second. Find:

Angular velocity: $ \omega = 4 \times 2\pi = 8\pi \approx 25.13 \ \text{rad/s} $
Linear velocity: $ v = r \cdot \omega = 0.35 \times 25.13 \approx 8.8 \ \text{m/s} $

Example 3: Planetary Orbit

Earth orbits the Sun at an approximate angular velocity of $ \omega = \frac{2\pi}{365.25 \times 24 \times 3600} \approx 1.99 \times 10^{-7} \ \text{rad/s} $, and radius of orbit is ~$ 1.5 \times 10^{11} $ m.

Then linear velocity: $$ v = r \cdot \omega = (1.5 \times 10^{11}) \cdot (1.99 \times 10^{-7}) \approx 29,850 \ \text{m/s} $$

Key Differences Between Angular and Linear Velocity

Aspect	Angular Velocity	Linear Velocity
Definition	Rate of change of angle	Rate of change of displacement
Unit	rad/s	m/s
Direction	Along axis of rotation	Tangent to path
Formula	$ \omega = \frac{\theta}{t} $	$ v = \frac{d}{t} $
Connection	$ v = r \cdot \omega $

Tips for Solving Problems

Always convert RPM to rad/s before using in formulas.
Ensure radius is in meters for SI consistency.
Use the right-hand rule to find vector direction of $ \omega $.
Watch units: seconds, meters, and radians.
For complex systems, analyze both angular and linear quantities separately, then connect using $ v = r \cdot \omega $.

Conclusion

Angular and linear velocity are foundational to the understanding of motion in both natural and mechanical systems. Angular velocity explains how fast something spins, while linear velocity tells us how fast it moves across a distance. The key linking equation $ v = r \cdot \omega $ lets us transition between these views seamlessly.

From spinning CDs to orbiting planets, the concepts of angular and linear velocity reveal the hidden mechanics behind everyday motion. Mastering these relationships provides a powerful tool for students, engineers, and scientists alike.