Linear speed and angular speed are both used to measure the motion of objects, but they refer to different types of motion.
Linear speed refers to the speed that an object moves in a straight path along a trajectory. For example, linear speed could refer to a car driving along a straight road.
Angular speed refers to the speed at which an object rotates around a certain point. An example of angular speed could be the spinning of a wheel or a gear in a machine.
One of the primary differences between linear speed and angular speed besides the type of motion they describe is the units they are measured in.
Linear speed is measured in linear units such as miles or kilometers, while angular speed is measured in radians and degrees.
Another differences is that the linear speed is concerned with the distance traveled by an object, while the angular speed is concerned with the angle through which an object rotates.
Formulas
Linear Speed:
In the equations below, v represents the linear speed, d represents the distance, t represents the time, r represents the radius of the circular path, and omega (ω) represents the angular speed.
\text{v} = \frac{\text{d}}{\text{t}} \text{ or } \text{v} = \text{r}\omegaAngular Speed:
In the equation below, omega (ω) represents the angular speed, theta (θ) represents the angle covered (in degrees or radians), and t represents the time taken.
\omega = \frac{\theta}{\text{t}}, \ \theta \text{ in radians} Examples
1. Suppose that a point P is on a circle with radius r, and ray OP is rotating with angular speed. r = 4in, ω= pi/3 rad per min, t = 6 min.
a) What is the angle generated by P in time t?
First, because we can see the given constants, we know which equation we should be using. We have an r, a ω, and a t, and we know that we are looking for an angle θ.
The r is irrelevant to our equation, and we can simply use our values for ω and t to fill in the equation for angular speed.
We can change to the equation to isolate θ on one side by multiplying both sides by the time.
\theta = \text{w}\cdot\text{t} \\ = \frac{\pi rad}{3 min}\cdot\frac{6 min}{1} \\ =\frac{\pi \cdot 6}{3} \\ = 2\pi \text{rad}b) What is the distance traveled by P along the circle in time t?
Use the formula for arc length!
\text{s} = \text{r}\theta \\ = 4 \ \cdot \ 2\pi \\
= 8\pi\text{inches}2. Find the angular speed of a wind turbine with blades turning at a rate of 13 revolutions per minute.
\text{1 revolution} = 2\pi\text{radians} \\ \omega = \frac{\text{13revolutions}}{1\text{min}}\cdot\frac{2\pi\text{radians}}{1\text{revolution}} \\ = 26\pi \ \text{radians per minute or 81.64 radians per minute}