Angular displacement is the change in the angle made as something is revolving about a constant point or axis. It quantifies how far and in which direction an object has rotated from an initial position to a final position. Imagine you are turning a steering wheel or a door handle; the angle through which you turn it is called angular displacement.

Formula of Angular Displacement

The angular displacement of a point can be given by using the following formula,

Where,

θ = s/r

Here, θ is the angular displacement of the object through which the movement has occurred, s is the distance covered by the object on the circular path and r is the radius of curvature of the given path.

When α, ω, and t are known, where the acceleration of the object α, the initial angular velocity ω, and the time t is at which to calculate the displacement.

=t2-1/2t2

Derivation of Angular Displacement Formula

Take some object 'A' moving linearly; then it must have some initial velocity 'u' and acceleration 'a'. Again, we assume that after the time t, the final velocity of the object is 'v', and the final displacement of the object is 's'.

We know that acceleration is defined as the rate of change of velocity. Therefore,

a = dv/dt

dv = adt

Integrating on both sides,

Also,

Since v=dx/dt, we can 

a=vdvdx

v dv=a dx

Taking integrations of both the sides, we get

Substituting the value of u from the equation 1 into the second equation, we get,

Dividing both the sides of the equation by 2a, we get

On substituting the value of v instead of u we get,

Solved Problems

Problem 1: Neena travels along a circular path of radius 7 m. She covers the entire length of the path, which equals 50 m. Determine her angular displacement.

Solution:

As per the question, Neena's linear displacement, s = 50 m.

Also, the diameter of the curved path, d = 7 m

As we know that, d = 2r, so r =7/2= 3.5 m

And making use of the formula of angular displacement,

θ = s/r

θ =50m/3.5m

θ = 14.28 radians

Hence, angular displacement is 14.28 radians

Problem 2: Rohit has bought a pizza whose radius is 0.5 m. A fly has landed on the pizza and walked around the edge for a distance of 80 cm. Calculate the angular displacement of the fly?

Solution:

From the problem, we know that s = 80 cm = 0.08 m is the length of the fly on the pizza.

The radius of the pizza is given as r = 0.5 m.

Now using the angular displacement in the formula,

θ = s/r,

θ = 0.08m/0.5 m

θ = 0.16 radians

 

Other Related Sections

NCERT Solutions | Sample Papers | CBSE SYLLABUS| Calculators | Converters | Stories For Kids | Poems for kids| Learning Concepts I Practice Worksheets I Formulas | Blogs | Parent Resource