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Angular Displacement Formula

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Angular Displacement is studied when an object performs circular motion or curvilinear motion. As we know when an object performs linear motion, the difference between the initial and final point is termed displacement. Similarly while performing the circular motion, the difference in the position of an object is measured as Angular Displacement. It is represented by the Greek letter θ. Angular displacement is measured using the unit degree or radian. In this article, we will learn in detail about angular displacement.

What is Angular Displacement?

The angle sketched out by the radius vector at the center of the circular route at a certain time is defined as angular displacement of the object moving around a circular path in that time.

Angular Displacement is a vector quantity as it has both magnitude and direction. It is depicted as a circular arrow pointing in either a clockwise or anti-clockwise direction from the starting point to the endpoint.

Angular Displacement Definition

Angular Displacement is defined with the help of the following examples. Let’s take a body performing a circular motion. Angle created by the body from its rest position to any position in rotational motion is known as Angular Displacement. 

Now, angular displacement can also be defined as the shortest angle between the initial and the final position for an object performing circular motion around a fixed point. It is a Vector quantity.

Angular Displacement Formula

Angular Displacement Formula helps us to calculate angular displacement it is given as:

Angular Displacement = θf – θi

where,

  • θf is final angular position
  • θi is initial angular position

and θ angular displacement is measured as,

θ = s / r

where, 

  • r is radius of curved path
  • s is distance travelled by object on the circular path

Unit of Angular Displacement

Angular Displacement is either measured in Radians or in Degrees. We know that 2Ï€ radians equal 360 degrees. SI unit for measuring Angular Displacement is Radians. Angular Displacement is a dimension less quantity.

Derivation of Angular Displacement Formula

Let’s take an object performing linear motion with initial velocity u and acceleration a. After time t the final velocity of the object is v and total displacement during this time is s then,

The image added below shows the angular displacement and how it is calculated,

Angular-Displacement

a = dv / dt

dv = a×dt

Integrating both sides, we get,

∫vudv = a×∫dt

v – u = at

Also,

a = dv / dt

a= (dv / dx) / (dx/ dt)

As we know v = dx/dt, 

a = v (dv / dx)

v dv = a dx

After integrating both sides of the equation,

∫vu vdv = a∫dx

v2 – u2 = 2as

Now, substituting the value of u from v = u + at

v2 − (v − at)2 = 2as

2vat – a2t2 = 2as

Now, by dividing both sides of the equation by 2a, we have,

s = vt – 1/2at2

Finally replacing  v with u we get,

s = ut + 1/2at2

Measurement of Angular Displacement

Angular Displacement of an object in rotational motion can easily be calculated using the formula,

image of Angular Displacement Measurement

θ = s / r

where,

  • θ is angular displacement
  • s is distance travelled by the body
  • r is radius of the circular path

Thus, we can say that the angular displacement of an object is the distance travelled by the object around the circumference of a circle divided by its radius.

Read More,

Solved Examples on Angular Displacement

Example 1: Minakshi travels around a 12 m diameter circular track. What is her angular displacement if she runs around the entire track for 70 m?

Solution:

Given,

  • s = 70 m
  • d = 12 m
  • r = 6 m

We have, θ = s / r

θ = 70/6

θ = 11.66 radians

Example 2: Dhanraj purchased a pizza with a radius of 0.3 meters. A fly lands on the pizza and wanders 60 cm around the edge. Calculate the fly’s angular displacement.

Solution:

Given,

  • r = 0.3 m
  • s = 60 cm = 0.06 m

We have, θ = s / r

θ = 0.06 / 0.3

θ = 0.2 rad

Example 3: In a certain case Angular displacement is 0.267 radians and the radius is 6 m. Find the distance traveled by the object on the circular path.

Solution:

Given,

  • θ = 0.267 radians
  • r = 6 m

We have, θ = s / r

 s = θ × r

 s = 0.267 × 6

 s = 1.602 m

Example 4: In a certain case Angular displacement is 34.2 radians and the distance travelled by the object on the circular path is 23 m. Find the radius of curvature of the specified path.

Solution:

Given,

  • θ = 34.2 radians
  • s = 23 m

We have, θ = s / r

 r = s / θ

 r = 23/34.2

 r = 0.67 m

FAQs on Angular Displacement

1. Define Angular Displacement.

Angular Displacement is the angle measured in radians of a point or line which has been rotated from a fixed axis.

2. What is Formula for Angular Displacement?

The formula of Angular Displacement is θf – θi where θ = s/r where S is the distance travelled on circular path and r is the radius of circular path.

3.What is the Unit and Dimension of Angular Displacement?

Angular Displacement is measured in Radians. It is a dimension less quantity.

4. What is Rotational Motion?

Rotation of bodies through a fixed point is called rotational motion. In rotational motion, the position of the body does not changes.

5. What is Rate of Change of Angular Displacement?

The time rate of change of angular displacement is called Angular Velocity.

6. What are Linear and Angular Displacements?

Linear displacement is the change in the position of an object whereas, angular displacement is the change in the angular position of an object.

7. Is Angular Displacement considered Scalar or Vector?

Angular displacement is considered a vector quantity as it has both magnitude and direction.

8. Is Angular Displacement a Vector Quantity?

Yes Angular Displacement is a Vector Quantity as it has both magnitude and direction.

9. What is Angular Displacement Meaning?

Angular Displacement simply means the displacement of an object in terms of angles. When a body moves on a circular track, the initial and final position on the track when joined with the centre of the circular path then an angle is formed. This is what we call Angular Displacement.



Last Updated : 14 Sep, 2023
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