text: # 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?
  
  According to the question, the distance traveled 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,
  
  $$\begin{array}{l}\theta = \frac{s}{r}\end{array}$$
  
  Î¸ = 0.08m/0.5 m
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