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State and prove the principle of conservation of angular momentum. Explain the principle of conservation of angular momentum with examples. |
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Answer» Solution : LAW of conservation of angular momentum: When no external torque is acting on a body then the angular momentum of that rotating body is constant. `I_1 omega_1 = I_2 omega_2 ( " when " tau = 0)` Explanation : Here `l_1 " and " I_2`are moment ofinertia of rotating bodies and `omega_1 " and "omega_2` are their initial and final angular velocities. If external torque `tau = 0 " then " (dvecL)/(dt) = tau = 0` `therefore` Change in angular momentum is also zero. `rArr dvecL_2 - dvecL_1 =0` i.e., ` I_2omega_2 - I_1 omega_1 = 0 " or " I_1omega_1 = I_2 omega_2` Example - 1 :A boy stands over the centres of a horizontal platform which is rotating| freely with a speed `omega_1 (n_1 " revolutions/sec ".)`about a vertical axis passing through the centre of the platform and straight up) through the boy. He holds two bricks in each of his hands CLOSE to his body. The combined moment of inertia of the system is say `I_1`. LET the boy stretches his arms to hold the masses far away from his body. In this position the moment of inertia increases to `I_2`and let `omega_2`is his angular speed. Here `omega_2 lt omega_1` , because moment of inertia increases. `because I_1omega_1 = I_2omega_2 rArr I_1 (2pi)/(T_1) = I_2(2pi)(T_2) rArr I_1 n_1 = I_2n_2` Example – 2: An athlete diving off a high| spring board can BRING his legs and hands close to the body and performs Somersault about a horizontal axis passing through his body in the air before reaching the water below it. During the fall his angular momentum remains constant. |
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