Angular Displacement Formula
Last Updated : 14 Sep, 2023
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 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 2π radians equal 360 degrees. SI unit for measuring Angular Displacement is Radians. Angular Displacement is a dimension less quantity.
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,
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.
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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,
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,
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,
We have, θ = s / r
r = s / θ
r = 23/34.2
r = 0.67 m
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