 # Angular Displacement Formula

Definition: Angular displacement 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 that means, it has a direction as well as magnitude and is represented by a circular arrow pointing from the initial point to the final point, that is either clockwise or anti clockwise in direction.

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

$Angular displacement = \theta _{f}- \theta _{i}$

Where,

$\theta = s/r$

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 the acceleration of the object, the initial velocity and the time at which the displacement is to be calculated is known, we can use the following formula.

$\theta = wt + 1/2 \alpha t^{2}$

Here, ω is the initial angular velocity, t is the time at which the angular displacement is to be calculated and α is the angular acceleration of the object.

Derivation:

Let us consider an object ‘A’ undergoing linear motion with initial velocity ‘u’ and acceleration ‘a’. Let us say, after time t, the final velocity of the object is ‘v’ and the total displacement of the object is ‘s’.

We know that acceleration is defined as the rate of change of velocity. Or we can write,

$a = \frac{\mathrm{d} v}{\mathrm{d} t}$

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=dxdt, we can write,

$a = v \frac{\mathrm{d} v}{\mathrm{d} x}$

v dv=a dx

Upon 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 first equation into the second equation, we get,

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

$2vat – a^{2}t^{2}= 2as$

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

$s = vt – \frac{1}{2} at^{2}$

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

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

1) Neena goes around a circular track that has a diameter of 8.5 m. If she runs around the entire track for a distance of 60 m, what is her angular displacement?

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

Also, the diameter of the curved path, d = 8.5

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

And according to formula for angular displacement,

$\theta = \frac{s}{r}$

θ = 60m /4.25 m

2) Rohit bought a pizza of 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?

According to question, the distance travelled by the fly on the pizza is, s = 80 cm = 0.08 m.

The radius of the pizza is given to be, r = 0.5 m.

Using the formula for angular displacement,

$\theta = \frac{s}{r}$

θ = 0.08m/0.5 m