angular velocities 1 and 2. They are brought into contact face to face coinciding the axis of rotation. The expression for loss of energy during this process is:

Law of conservation of angular moment -

$\vec{\tau }= \frac{\vec{dL}}{{dt}}$

- wherein

If net torque is zero

i.e. $\frac{\vec{dL}}{{dt}}= 0$

$\vec{L}= constant$

angular momentum is conserved only when external torque is zero .

$Iw_{1}+Iw_{2} = 2Iw \Rightarrow w=\frac{w_{1}+w_{2}}{2}$

$\left ( K.E \right )_{i} = \frac{1}{2}Iw_{1}^{2}+\frac{1}{2}w_{2}^{2}$

$\left (K.E \right )_{f} = \frac{1}{2}\times 2Iw^{2} = I \left ( \frac{w_{1+w_{2}}}{2} \right )^{2}$

Loss in K.E = $\frac{1}{4}I\left ( w_{1}-w_{2} \right )^{2}$

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