Question

A sphere of radius r and mass m has a velocity v0 directed to the left and no angular velocity as it is placed on a belt moving to the right with a constant velocity v1. If after sliding on the belt the sphere is to have no linear velocity relative to the ground as it starts rolling on the belt without sliding. In terms of v1, the velocity v0 is.

Answer 1

Answer:

Velocity of the sphere in terms of v1 is given as

$\frac{9}{5} v_1 = v_o$

Explanation:

Sphere is performing pure rolling on the belts

Here we know that initially belt is moving to right with speed v1 and sphere is moving to left with speed vo

$R\omega - v_1 = v_o$

So after pure rolling is achieved

$v_o = R\omega$

So we have

$v_1 - v_o = -\mu_k g t$

Also we have

$\mu_k mg R = \frac{2}{5}mR^2 \alpha$

$\alpha = \frac{5\mu_k g}{2R}$

so we have

$\omega = \frac{5\mu_k g}{2R} t$

now for pure rolling

$R\omega - v_1 = v_1$

$\frac{5\mu_k g}{2} t = 2v_1$

$\mu_k g t = \frac{4}{5} v_1$

now we have

$v_1 - v_o = -\frac{4}{5}v_1$

$\frac{9}{5} v_1 = v_o$

#Learn

Topic: Rotational motion

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