Angular Displacement Formula

Angular Displacement Formula

Angular displacement refers to the change in angular position of a rotating body or object over a specific period. It is a measure of the amount of rotation an object undergoes around a fixed point or axis and is typically represented in radians or degrees. The angular displacement formula calculates this change, allowing us to quantify rotational motion. Learn more about angular displacement, its definition, formula and more in this article by Extramarks

What is Angular Displacement?

Angular displacement refers to the change in angular position of a rotating body or object over a specific period. It is a measure of the amount of rotation an object undergoes around a fixed point or axis and is typically represented in radians or degrees. The angular displacement formula calculates this change, allowing us to quantify rotational motion.

Angular Displacement Definition

To define Angular Displacement, assume a body is moving in a circular motion. The angle created by a body from its point of rest at any point in rotational motion is referred to as angular displacement.The shortest angle between an object’s beginning and final positions in a circular motion around a fixed point is known as Angular Displacement, and it is a vector variable.

Angular Displacement Formula

For a body rotating through an angle θ the angular displacement Δθ is given by

Δθ = θ_final​−θ_initial

where, θ_final is the final angular position and θ_initial is the initial angular position. The angular displacement can be positive or negative, depending on the direction of rotation: counterclockwise (positive) or clockwise (negative).​​

Unit of Angular Displacement

Angular displacement is measured in angular units, typically in radians (rad) or degrees (°). These units describe the amount of rotation or angular change experienced by a rotating object relative to a fixed point or axis.

Measurement of Angular Displacement

It can be measured by using a simple formula. The formula is:

where,

• θ is the angular displacement,
• s is the distance travelled by the body, and
• r is the radius of the circle along which it is moving.

In simpler words, the displacement of an object is the distance travelled by it around the circumference of a circle divided by its radius.

Angular Displacement Formula Derivation

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,

a = dv / dt

dv = a×dt

Integrating both sides, we get,

∫^v_udv = 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,

∫^v_u vdv = a∫dx

v² – u² = 2as

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

v² − (v − at)² = 2as

2vat – a²t² = 2as

s = vt – 1/2at²

Finally replacing  v with u we get,

s = ut + 1/2at²

Solved Examples on Angular Displacement Formula

Example 1: Risabh travels around a 50 m diameter circular track. What is his angular displacement if she runs around the entire track for 150 m?

Solution:

Given,

• s = 150 m
• d = 50 m
• r = 25 m

We have, θ = s / r

θ = 150/25

θ = 6 radians

Example 2: Raj purchased a pizza with a radius of 0.4 meters. A fly lands on the pizza and wanders 120 cm around the edge. Calculate the fly’s angular displacement.

Solution:

Given,

• r = 0.4 m
• s = 120 cm = 0.12 m

We have, θ = s / r

θ = 0.3 rad

Example 3: A wheel rotates from an initial angle of \( 30^\circ \) to a final angle of \( 150^\circ \). Calculate the angular displacement of the wheel.\

Solution:

Given:
Initial angular position (\( \theta_{\text{initial}} \)) = \( 30^\circ \)
Final angular position (\( \theta_{\text{final}} \)) = \( 150^\circ \)

\[ \Delta \theta = \theta_{\text{final}} – \theta_{\text{initial}} \]

\[ \Delta \theta = 150^\circ – 30^\circ \]
\[ \Delta \theta = 120^\circ \]

The angular displacement of the wheel is \( 120^\circ \).