Angular Motion

In Physics, just as there are formulas to calculate linear velocity and displacement, there are also equivalent formulas to calculate angular movement. This is great because you have an angular counterpart for many of the linear motion equations. In this article, we will be discussing the angular motion and its relationship with linear motion.

How is Angular Motion Defined?

Angular motion is defined as,

The motion of a body about a fixed point or fixed axis. It is equal to the angle passed over at the point or axis by a line drawn to the body.

Relationship Between Linear And Angular Motion

Here are a few variable substitutions you can make to get the angular motion formulas:

• Displacement – In linear motion, we use ‘s to quantify the linear distance travelled. In angular motion, we use ‘θfor the same to quantify the angular distance, and it is measured in radians.
• Velocity – In linear motion, we use v to denote velocity while in angular motion, we use ‘ω to indicate angular velocity. Angular velocity is the number of radians covered per second.
• Acceleration – We use ‘a to denote linear acceleration, while we use ‘α to mean the angular acceleration. The unit for angular acceleration is radians per second2.

Now that we have discussed the variations in linear and angular motions let us look at the formulas for both linear and angular motion.

 Formula Linear Angular Velocity $$\begin{array}{l}v=\frac{\Delta s}{\Delta t}\end{array}$$ $$\begin{array}{l}\omega =\frac{\Delta \Theta}{\Delta t}\end{array}$$ Acceleration $$\begin{array}{l}a=\frac{\Delta v}{\Delta t}\end{array}$$ $$\begin{array}{l}a=\frac{\Delta \omega}{\Delta t}\end{array}$$ Displacement $$\begin{array}{l}s=v_{i}t+\frac{1}{2}at^{2}\end{array}$$ $$\begin{array}{l}\Theta =v_{i}t+\frac{1}{2}\alpha t^{2}\end{array}$$ Motion with time cancelled out $$\begin{array}{l}v^2_{f}-v^2_i=2as\end{array}$$ $$\begin{array}{l}\omega ^2_{f}-\omega ^2_i=2\alpha \Theta\end{array}$$

Let us consider an example to understand the relationship between linear and angular velocity better. Say, for example, you have a ball tied to a string. It makes a complete circle, 2π radians in 0.5 seconds. The angular velocity of the ball can easily be found out using the formula

$$\begin{array}{l}\omega= \frac{\Delta \Theta}{\Delta t}=\frac{2\pi\,radian}{0.5\,s}=4\pi\,rad/s\end{array}$$

Another demonstration of the usefulness of radians in measuring angles is that the linear speed can easily be related to the angular speed.

The linear speed can be easily related to the angular speed as follows:

Consider the equation,

$$\begin{array}{l}\omega= \frac{\Delta \Theta}{\Delta t}\end{array}$$

Multiply both the sides by the radius r,

$$\begin{array}{l}r\omega = \frac {r\Delta \Theta}{\Delta t}\end{array}$$

The term

$$\begin{array}{l}r\Delta \Theta\end{array}$$
denotes the distance travelled by an object moving in a circle of radius r; hence, this equation becomes

$$\begin{array}{l}r\omega =\frac{\Delta s}{\Delta t}\end{array}$$

You may recognize the right side of this equation as the equation for speed. The relationship between linear and angular speed can be written as follows:

$$\begin{array}{l}r\omega = v\end{array}$$