# Angular Displacement Formula

Angular displacement is the angle measured in radians and is defined as the shortest angle between the initial and the final points for a given object undergoing circular motion about a fixed point. Angular Displacement is a vector quantity. It has both direction and magnitude. It is represented by a circular arrow pointing from the initial point to the final point, which is either clockwise or anti-clockwise in direction.

## Formula of Angular Displacement

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

$$\begin{array}{l}AngularÂ displacement = \theta _{f}- \theta _{i}\end{array}$$

Where,

$$\begin{array}{l}\theta = s/r\end{array}$$

Here, Î¸ 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. Â

When the acceleration of the object (Î±), the initial angular velocity (Ï‰) and the time (t) at which the displacement is to be calculated is known, we can use the following formula.

$$\begin{array}{l}\theta = wt + 1/2 \alpha t^{2}\end{array}$$

### Derivation of Angular Displacement Formula

Let us consider an object â€˜Aâ€™ undergoing linear motion with initial velocity â€˜uâ€™ and acceleration â€˜aâ€™. Let us say, after time t, the final velocity of the object is â€˜vâ€™ and the total displacement of the object is â€˜sâ€™.

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

$$\begin{array}{l}a = \frac{\mathrm{d} v}{\mathrm{d} t}\end{array}$$

$$\begin{array}{l}dv = a dt\end{array}$$

Integrating on both sides,

$$\begin{array}{l}\int_{u}^{v} dv = a \int_{0}^{t} dt\end{array}$$

$$\begin{array}{l}v – u = at\end{array}$$

Also,

$$\begin{array}{l}a = \frac{\mathrm{d} v}{\mathrm{d} t}\end{array}$$

$$\begin{array}{l}a = \frac{\mathrm{d} v}{\mathrm{d} x} *\frac{\mathrm{d} x}{\mathrm{d} t}\end{array}$$

sinceÂ Â v=dx/dt, we can write,

$$\begin{array}{l}a = v \frac{\mathrm{d} v}{\mathrm{d} x}\end{array}$$

v dv=a dx

Upon integrating both the sides,we get

$$\begin{array}{l}\int_{u}^{v} v dv = a \int_{0}^{s} dx\end{array}$$

$$\begin{array}{l}v^{2} – u^{2} =2as\end{array}$$

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

$$\begin{array}{l}v^{2} -(v- at)^{2} = 2as\end{array}$$

$$\begin{array}{l}2vat – a^{2}t^{2}= 2as\end{array}$$

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

$$\begin{array}{l}s = vt – \frac{1}{2} at^{2}\end{array}$$

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

$$\begin{array}{l}s = ut + \frac{1}{2}at^{2}\end{array}$$

## Numericals

Problem 1:

1) Neena goes around a circular track that has a diameter of 7 m. If she runs around the entire track for a distance of 50 m, what is her angular displacement?

According to 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 according to the formula for angular displacement,

$$\begin{array}{l}\theta = \frac{s}{r}\end{array}$$

Î¸ = 50m /3.5 m

Problem 2

2) Rohit bought a pizza of a radius of 0.5 m. A fly lands on the pizza and walks around the edge for a distance of 80 cm. Calculate the angular displacement of the fly?

$$\begin{array}{l}\theta = \frac{s}{r}\end{array}$$