Chapter 8, Problem 68GP

To determine

To Find: The length of rope that unwinds from the spool and the distance by which the spool’s center of mass move.

Answer to Problem 68GP

The length of the rope that unwinds from the spool is $\frac{\mathrm{L}}{2}$ and the distance by which the spool’s center of mass move is $\frac{\mathrm{L}}{2}$ .

Explanation of Solution

Given:

Distance moved by the person along with the rope is L .

Formula used:

Write the expression for the distance moved.

  $\mathrm{d}=\mathrm{v}\times \mathrm{t}$

Here, $\mathrm{v}$ is the velocity of the motion of man.

  $\mathrm{t}$ is the time taken.

Write the expression for the velocity at the center of mass.

  ${\mathrm{v}}_{\mathrm{c}}=\mathrm{r}\times \mathrm{\omega }$

  ${\mathrm{v}}_{\mathrm{c}}$ is the velocity of center of mass

  $\mathrm{\omega }$ is the linear velocity of the spool

Calculation:

Write the expression for the velocity of the rope at the top of the spool.

  $\begin{array}{c}\mathrm{v}={\mathrm{v}}_{\mathrm{c}}+\mathrm{r}\times \mathrm{\omega }\\ \mathrm{v}={\mathrm{v}}_{\mathrm{c}}+{\mathrm{v}}_{\mathrm{c}}\\ \mathrm{v}=2{\mathrm{v}}_{\mathrm{c}}\end{array}$

This can be written as:

  ${\mathrm{v}}_{\mathrm{c}}=\frac{\mathrm{v}}{2}$

Based on the above relation, as velocity is half at the center of the mass. Thus for the same time, the distance moved by the center of mass of spool is $\frac{\mathrm{L}}{2}$ and similarly the distance moved by the rope on unwinding is $\frac{\mathrm{L}}{2}$ .

Conclusion:  The length of the rope that unwinds from the spool is $\frac{\mathrm{L}}{2}$ and the distance by which the spool’s center of mass move is $\frac{\mathrm{L}}{2}$ .