Answer to Question #115773 in Mechanics | Relativity for saruuol

Question

Connect two bodies of mass m1> m2 with a weightless wire placed on the crossbar (block). The crossbar relative to the axis of rotationю The moment of inertia is I. the radius of the bar R. The wire does not shrink or stretch, eitherdoes not slide over the crossbar. The crossbar rotates without friction. Two bodies first2h away from each other, calm down and start moving. A) The law of conservation of energyю Find the velocities of the bodies when the two bodies collide using B) At the momentю Find the angular velocity of the crossbar.

Answer 1

A) The potential energy converts to their kinetic energy and the energy of rotation of the block:

$$(m_1-m_2)gH=\frac{1}{2}m_1v^2+\frac{1}{2}m_2v^2+\frac{1}{2}I\omega^2=\\ \space\\ =\frac{1}{2}\bigg[(m_1+m_2)v^2+I\bigg(\frac{v}{R}\bigg)^2\bigg]=\\ \space\\ =\frac{v^2}{2}\bigg[(m_1+m_2)+\frac{I}{R^2}\bigg],\\ \space\\ v=2\sqrt{\frac{(m_1-m_2)gh}{m_1+m_2+I/R^2}}.$$

B) The angular velocity of the block is

$$\omega=\frac{v}{R}=\frac{2}{R}\sqrt{\frac{(m_1-m_2)gh}{m_1+m_2+I/R^2}}.$$