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The free end of a thread wound on a bobbin of inner radius r and outer radius R is passed round a nail A hammered into the wall (Fig. 11). The thread is pulled at a constant velocity v.

Find the velocity v0 of the centre of the bobbin at the instant when the thread forms an angle a with the vertical, assuming that the bobbin rolls over the horizontal surface without slipping.

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If the thread is pulled as shown in Fig. 142, the bobbin rolls to the right, rotating clockwise about its axis.

For point B, the sum of the projections of velocity v0 of translatory motion and the linear velocity of rotary motion (with an angular velocity ω) on the direction of the thread is equal to v:

Since the bobbin is known to move over the horizontal surface without slipping, the sum of the projections of the corresponding velocities for point C is equal to zero:

Solving the obtained equations, we find that the velocity v0 is

It can be seen that for R sin α = r (which corresponds to the case when points A, B, and C lie on the same straight line), the expression for v0 becomes meaningless. It should also be noted that the obtained expression describes the motion of the bobbin to the right (when point B is above the straight line AC and R sin α > r) as well as to the left (when point B is below the straight line AC and R sin α < r).

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