# Angular Displacement Formula

We’ll look at how a body moves in a circular direction in this section. The rotational motion is shown in this diagram. These motions have a displacement that is not the same as a linear motion’s displacement. Because the displacement in such motion takes the form of an angle, it’s called angular displacement. The angular displacement formula will be discussed in this topic with examples. Now let’s get started!

### 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 the angular displacement of the object moving around a circular path in that time.

A vector quantity is an Angular Displacement. 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 ending point.

**Unit of Angular Displacement: **Radian or Degrees is the unit of Angular Displacement. 2π radians equal 360 degrees. The SI unit for displacement is the meter. Because Angular Displacement is concerned with curvilinear motion, the SI unit for it is Degrees or Radian.

### Angular Displacement Formula

The following formula can be used to calculate a point’s angular displacement:

Angular Displacement = θ_{f}– θ_{i}where,

θ = s/rHere, r is the radius of curvature of the specified path, s is the distance travelled by the object on the circular path, and is the angular displacement of the object through which the movement happened.

We can use the following formula if we know the object’s acceleration (α), starting angular velocity (ω), and the time (t) at which the displacement is to be determined.

θ = ωt + 1/2(αt^{2})

### Derivation of Angular Displacement

Consider an item ‘A’ moving in a straight line with an initial velocity of ‘u’ and an acceleration of ‘a’. Let us state that the object’s ultimate velocity is ‘v’ and its total displacement is ‘s’ after time t.

We all know that acceleration is the rate at which velocity changes. Therefore,

a = dv/dt

∴dv = a × dt

On both sides, integrating

∫

_{u}^{v}dv = a ∫_{0}^{t}dt∴ v – u = at

Also,

a = dv/dt

∴ a = (dv/dx) × (dx/dt)

we have, v = dx/dt

∴ a = (dv/dx)v

∴ v dv = a dx

On both sides, integrating

∴ ∫

_{u}^{v}v dv = ∫_{0}^{s}a dx∴ v

^{2}– u^{2}= 2as ⇢(Equation 1)we have, v – u =at

∴ u = v – at

put value of u in equation 1,

v

^{2}– (v – at)^{2}= 2as∴ 2vat – a

^{2}t^{2}= 2asDivide both sides by 2a,

∴ s = vt – 1/2(at

^{2})we use the value of v instead of u, we get

∴

s = ut + 1/2(at^{2})

### Sample Questions

**Question 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?**

**Answer:**

Given : s = 70 m, d = 12 m = r = 6 m

Find : θ

Solution :

We have,

θ = s/r

∴ θ = 70/6

∴

θ = 11.66 radians

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

**Answer:**

Given : r = 0.3 m, s = 60 cm = 0.06 m

Find : θ

Solution :

We have,

θ = s/r

∴ θ = 0.06/0.3

∴

θ = 0.2 radians

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

**Answer:**

Given : θ = 0.267 radians, r = 6 m

Find : s

Solution :

We have,

θ = s/r

∴ s = θ × r

∴ s = 0.267 × 6

∴

s = 1.602 m

**Question 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.**

**Answer:**

Given : θ = 34.2 radians, s = 23 m

Find : r

Solution :

We have,

θ = s/r

∴ r = s/θ

∴ r = 23/34.2

∴

r = 0.67 m

**Question 5: What is the definition of Angular Displacement as a Vector?**

**Answer:**

There is no such thing as an angular displacement vector.

A vector is a quantity that has both direction and magnitude and satisfies the rules of vector algebra. Although angular displacement appears to be a quantity that can only be expressed in one direction, you can specify directions to establish conventions like the right-hand rule of thumb. The term “magnitude” refers to the quantity of spin. However, it does not follow all of the principles of vector algebra, particularly the commutative law: u + v = v + u for the vector and u and v. Choose a 3D object, such as a cell phone, with the screen facing you upright. Rotate it clockwise so that the screen is still horizontal but faces you (landscape adjustment). Rotate the screen once more so that it faces the ceiling. Due to the sum of two angular displacements, this occurs. The ultimate orientation will be different if the rotation order is changed. Angular displacement in a different order will provide various outcomes that violate commutability.

**Question 6: What Are Some Angular Displacement Examples?**

**Answer:**

Example of angular displacement: If a dancer dances around a pole in a full rotation, their angular rotation will be 360 degree. The displacement will be 1800 if the rotation is half. This will be a vector quantity, which implies that it will have both a magnitude and a direction. A 360-degree displacement done clockwise versus anticlockwise, for example, is extremely different.