Law Of Conservation Of Angular Momentum - Practice Questions & MCQ

• Analogy Between Translatory Motion and Rotational Motion for common terms

• - From $\vec{L}=I \vec{\omega}$ we get $\frac{d \vec{L}}{d t}=I \frac{d \vec{\omega}}{d t}=I \vec{\alpha}=\vec{\tau}$ i.e. the rate of change of angular momentum is equal to the net torque acting on the particle.

  This is Rotational analogue of Newton's second law
  - Angular impulse $=\vec{J}=\int \vec{\tau} d t=\Delta \vec{L}$

  $
  \text { or, } \vec{J}=I\left(\vec{w}_f-\vec{w}_i\right)
  $

  i.e., Angular impulse is equal to change in angular momentum
  - As

  $
  \vec{\tau}=\frac{d \vec{L}}{d t}
  $

  
  So if the net external torque on a particle is zero then for that particle

  $
  \begin{aligned}
  & \frac{d \vec{L}}{d t}=0 \Rightarrow \vec{L}=\text { constant } \\
  & \Rightarrow L_i=L_f
  \end{aligned}
  $

  
  Similarly in case of system consists of $n$ particles
  If the net external torque on a system is zero then for that system

  $
  \begin{gathered}
  \frac{d \vec{L}}{d t}=0 \Rightarrow \vec{L}=\text { constant } \\
  \text { Or, } \vec{L}_{n e t}=\vec{L}_1+\vec{L}_2 \ldots \ldots+\vec{L}_n=\text { constant }
  \end{gathered}
  $
   

I.e Angular momentum of a system  remains constant if resultant torque acting on it zero.

This is known as the law of conservation of angular momentum.

• - For a system if $\vec{\tau}_{\text {net }}=0$ then its

  $
  \begin{aligned}
  & \vec{L}=I \vec{\omega}=\text { Constant } \\
  & \text { Or, } I \propto \frac{1}{\omega}
  \end{aligned}
  $
   

Example-In a circus during performance an  acrobat try to bring the arms and legs closer to body to increase spin speed. On bringing the arms and legs closer to body, his moment of inertia I decreases. Hence  increases.