Angular Displacement - Definition, Explanation, Examples, and FAQs

Introduction

Angular displacement and its measurement are key concepts of Dynamics. The particle is considered to be in motion when it changes its position with respect to time. That means the particle is displaced from one position to another. The particle makes linear displacement if it is moving in linear motion. Similarly, the particle makes angular displacement if it is moving in a circular motion. That means circularly moving particles make angular displacement.

Laws of Motion

There are laws describing the motion of particles. According to the first law, the particle does not change its state of motion or rest without any external force.

The second law indicates that the force acting on the particle depends upon the mass of the particle and acceleration of the particle.

$$\mathrm{F=ma}$$

Then the third law shows that if a force is applied to the particle, then there should be an equal force produced in the opposite direction. If the particles are moving in a certain path, then they should be displaced from one position to another. If the particle moves in a straight path, it makes a linear displacement. If a particle moves in a circular path, then it makes an angular displacement.

What is Angular Displacement?

Definition

Angular displacement is the angle through which the radius vector of the circular path changes with respect to time. That means radius vector shows the angular displacement with time. Angular displacement is the angle between the position vector of A and B.

Angular displacement

$$\mathrm{s = rӨ}$$

Here

s - angular displacement

r - radius vector of the circular path

Ө - the angle of displacement

Unit Of Angular Displacement

As angular displacement occurs only for the particle moving in a rotational motion, the properties of the circle are taken into consideration. The complete cycle of circular motion is 360 degrees. Half rotation denotes 180 degrees. Angular displacement is denoted in degrees or radians.

The formula for angular displacement: It is the ratio between the distance travelled in one second and the radius of the circular path.

$$\mathrm{Ө =\frac{s}{r}}$$

Here

s - the distance travelled in one-second

r - radius of the circular path

For one rotation Ө =2Π rad

Derivation for Angular Displacement

We know that Angular displacement is the angle between the initial and final position of the particle which undergoes circular motion with a fixed axis. As it is a vector quantity it has both magnitude and direction. Clockwise rotational motion is denoted as positive and anticlockwise rotation is denoted as negative.

Angular displacement

$$\mathrm{Ө =\frac{s}{r}}$$

It can also be written as

$$\mathrm{Ө =Ө_2-Ө_1}$$

Ө₂ = final angle

Ө₁ = initial angle

Angular displacement is also calculated using the below formula

$$\mathrm{θ=\omega t+1/2\alpha t^2}$$

As the particle in motion, it processes velocity and acceleration. Acceleration is defined as the ratio between the change in velocity and time.

Acceleration

$$\mathrm{a=\frac{dv}{dt}}$$

$$\mathrm{dv = a\:dt}$$

By integrating this equation, we get,

$$\mathrm{\int_{u}^{v}\:dv\:=a\int\:s\:dt}$$

$$\mathrm{a=\frac{dv}{dt}}$$

The acceleration can also be written as

$$\mathrm{a =\frac{dv}{dx} \frac{dx}{dt}}$$

And also, we know that velocity is denoted as the rate of change of distance.

So,

$$\mathrm{v =\frac{dx}{dt}}$$

$$\mathrm{\int_{u}^v dv=\int dx}$$

$$\mathrm{v_2–u_2=2as}$$

Substituting the value for u as v-at

$$\mathrm{v^2-(v-at)^2 = 2as}$$

$$\mathrm{v^2-v^2-a^2 t^2+2vat = 2as}$$

$$\mathrm{2vat-a^2t^2 = 2as}$$

Divide this equation by 2a on both sides we get,

$$\mathrm{s= vt-1/2 at^2}$$

$$\mathrm{s = \omega t + 1/2 at^2}$$

This equation gives the relation between angular velocity, angular acceleration, and displacement of the particle undergoing circular motion.

Solved problem

1. Rani travelled in a circular path of radius 5m and completed a distance of 65m, then find her angular displacement.

Ans: Given

the Radius of the path $\mathrm{r=\frac{d}{2}=\frac{5}{2}=2.5m}$

Linear displacement s=65m

Angular displacement $\mathrm{Ө = \frac{s}{r}}$

$\mathrm{Ө =\frac{65}{2.5}= 26\:rad.}$

Conclusion

This article explains linear motion and rotational motion. The particle which moves in a straight line is said to be in linear motion and which moves in a circular path is said to be in a circular motion. The particle which undergoes moves along with circular path processes angular displacement. Angular displacement is measured in rad or degrees. Angular displacement has both magnitude and direction as it is a vector quantity. Angular displacement is the angle between the position vector of the particle undergoing circular motion.

FAQs

Q1. Is angular displacement a vector quantity?

Ans. Yes. Angular displacement is a vector quantity. It has both direction and magnitude. For complete rotation, it takes 360 degrees. It may be clockwise or anticlockwise.

Q2. Give an example of angular displacement.

Ans. A Pole dancer is a good example of angular displacement. If the pole dancer makes full rotation, they cover 360 degrees. If they rotate half a circle, they cover 180 degrees. Here is also direction is taken into consideration. Also, the wall clock is an example of angular displacement in which the hands of minute, hour, and second make rotating about the center point.

Q3. Is the angular displacement the same for all paths?

Ans. Angular displacement may not be the same for all. Angular displacement is the ratio of distance travelled in a particular time. If the angular displacement is the same for all particles, then it is said that the particle undergoes simple periodic motion. Simple harmonic or periodic motion means that the displacement is the same for the same period of time. It is not necessary for a body to make an equal displacement for an equal period of time. It may also make changes in the angular displacement in accordance with time.

Q4. An object moving in a circular path of radius 12m makes an angular displacement of about 60 degrees. Then find the displacement of the object.

Ans. Given

the Radius of the circular path r=12m

Angle through which the object is displaced is 60degrees.

To find the displacement of the object

Angular displacement $\mathrm{Ө =\frac{s}{r}}$

$$\mathrm{s=Ө r}$$

$$\mathrm{s =\frac{\Pi}{3}\times 12}$$

$$\mathrm{s = 4\pi = 4 × 3.14 = 12.56m}$$

Q5. Differentiate linear displacement and angular displacement.

Ans.

Linear Displacement	Angular Displacement
The particle is moving in a straight path	The particle is moving in a circular path
It is the linear distance travelled in one second	It is the angular displacement of the particle
Its unit is metre (m).	Its unit is degree or rad.

Table-1: Difference between linear displacement and Angular displacement