Question

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.

Answer 1

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).