Here we will consider the motion of a body along a circular path. This is the rotational motion. Such motions have displacement which is different from the displacement as on linear motion. Displacement in such motion is in the form of angle and hence known as angular displacement. In this topic, we will discuss the angular displacement formula with examples. Let us begin it!

**Angular Displacement Formula**

**Concept of Angular Motion and Displacement:**

Angular displacement is defined as the shortest angle between the initial and the final positions for a given object having a circular motion about a fixed point. Here angular displacement is a vector quantity.

Thus it will have the magnitude as well as the direction. The direction is represented by a circular arrow pointing from the initial position to the final position. It may be either clockwise or anti-clockwise in direction

### The Formula for Angular Displacement

The angular displacement of a point is as follows:

**Angular displacement = \(\theta _{f}- \theta _{i}\)**

And,

**\(\theta = \frac {s}{r}\)**

When the acceleration of the object, the initial velocity and the time are given then angular displacement may also be calculated by using the formula:

**\(\theta = wt + 1/2 \alpha t^{2}\)**

\(\theta\) | the angular displacement of the object |

s | distance covered by the object on the circular path |

r | the radius of curvature of the given path |

\(\omega\) | initial angular velocity |

t | time |

\(\alpha\) | angular acceleration |

**Derivation**

Let us consider an object having a linear motion with initial velocity u and acceleration a. If after time t the final velocity of the object is v with total displacement s then,

**\(a = \frac{\mathrm{d} v}{\mathrm{d} t}\)**

This is the rate of change in the velocity.

We can also write it as,

**dv = a dt**

Integrating both the sides, we get,

\(\int_{u}^{v} dv = a \int_{0}^{t} dt\)

v – u = at

Also,

\(a = \frac{\mathrm{d} v}{\mathrm{d} t}\)

\(a = \frac{\mathrm{d} v}{\mathrm{d} x} / \frac{\mathrm{d} x}{\mathrm{d} t}\)

\(As we know v=\frac{dx}{dt}, we can write,\)

\(a = v \frac{\mathrm{d} v}{\mathrm{d} x}\)

v dv=a dx

After integrating both the sides of the equation, we get,

\(\int_{u}^{v} v dv = a \int_{0}^{s} dx\)

\(v^{2} – u^{2} =2as\)

Now, substituting the value of u from the equation-1 into the equation-2, we get,

\(v^{2} -(v- at)^{2} = 2as\)

\(2vat – a^{2}t^{2}= 2as\)

Now, by dividing both the sides of the equation by 2a, we have,

\(s = vt – \frac{1}{2} at^{2}\)

Finally substituting the value of v instead of u we will get,

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

**Solved Examples**

Q.1: Ram goes around a circular track that has a diameter of 8.5 m. If he runs around the complete track for a distance of 60 m, then determine his angular displacement?

Solution:

According to the question, Ram’s linear displacement, s = 60 m.

Also, the diameter of the circular, d = 8.5

So, radius of the path will be,

r = \(\frac {d}{2}\)

r = \(\frac {8.5}{2}\)

r = 4.25 mAs we know that, d = 2r, so r = 4.25 m.

Therefore, we now use the formula for angular displacement as follows:

\(\theta = \frac{ s } { r }\)

\(\theta = \frac {60 } { 4.25 }\)

\(\theta = 14.12 radians.\)

Therefore his angular displacement will be angle 14.12 radians.

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