Q75P - University Physics with Modern Physics

Question

A uniform solid cylinder with mass M and radius 2R rests on a horizontal tabletop. A string is attached by a yoke to a frictionless axle through the center of the cylinder so that the cylinder can rotate about the axle. The string runs over a disk-shaped pulley with mass M and radius R that is mounted on a frictionless axle through its center. A block of mass M is suspended from the free end of the string (Fig. P10.75). The string doesn’t slip over the pulley surface, and the cylinder rolls without slipping on the tabletop. Find the magnitude of the acceleration of the block after the system is released from rest.

                                                                   

Step-by-Step Solution

Answer

$\frac{\mathrm{g}}{3}$

Step 1: Acceleration

The rate of velocity changes with respect to time is known as acceleration.

Step 2: Given Data

$$\begin{gathered} Mass = m \\ radius= 2R \end{gathered}$$

Step 3: Determine the magnitude of the acceleration of the block

$$\begin{gathered} \mathrm{Apply} \sum _{}F=ma \tau =I\alpha and a=r\alpha \Rightarrow \alpha = \frac{\mathrm{a}}{\mathrm{R}} \\ \mathrm{For} \mathrm{hanging} \mathrm{mass}, \\ /F = ma \\ Mg-T_1=Ma..........\left(1\right) \\ For the pully, \\ \tau = I\alpha \\ {T}_{1}R-T_{2}R=\frac{M{R}^2}{2} . \frac{a}{R} \\ {T}_1-T_2=\frac{Ma}{2}..........................\left(2\right) \end{gathered}$$

For the cylinder,

$$\begin{gathered} F=ma \\ {T}_2-f=Ma.......\left(3\right) \end{gathered}$$

$$\begin{gathered} \tau = I\alpha \\ f.2R =\frac{M.{\left(2R\right)}^2}{2}.\frac{a}{2R} \\ f=\frac{Ma}{2}............\left(4\right) \\ From \left(3\right) and \left(4\right), \\ {T}_2=\frac{3Ma}{2}......\left(5\right) \end{gathered}$$

$$\begin{gathered} \mathrm{Substituting} T_{1 }\mathrm{and} T_{2 }in \left(2\right) \mathrm{by} \left(1\right) \mathrm{in} \left(5\right) \\ \left(Mg-Ma\right)-\left(\frac{3Ma}{2}\right)=\frac{Ma}{2} \\ g-a-\frac{3a}{2}=\frac{a}{2} \\ a=\frac{g}{3} \end{gathered}$$

Hence the magnitude of the acceleration of the block is $\frac{g}{3}$