91Ó°ÊÓ

In order to find in which case the velocity would be greater or smaller, let’s discuss about the distribution of ,ass, because the four ball are moved inward closer to the drum.

The rotational moment of inertial is given by,

$\mathit{l}\mathbf{=}\mathit{m}{\mathit{d}}^{\mathbf{2}}$

It is known that the formula for rotational is $l=m{d}^2$and because the distance is reduced between the balls and the axis of rotation, rotational inertial of the system is going to reduced too. Therefore, $l_1>l_2$.

Draw the diagram for given situation.

Write the Newton’s equations for translational and rotational motion.

For case (1),

$I_1{\mathrm{Î\pm }}_1={\mathrm{Ï„}}_1=T_{1}R$

For case (2),

$I_2{\mathrm{Î\pm }}_2={\mathrm{Ï„}}_2=T_{2}R$

Express$T_1$and$T_2$from Newton’s equation for translational motion, and also note that $m_1=m_2=m$,

For case (1),

$$\begin{gathered} m{a}_1=mg-T_1 \\ {T}_1=mg-m{a}_1 \end{gathered}$$

For case (2),

$$\begin{gathered} m{a}_2=mg-T_2 \\ {T}_2=mg-m{a}_2 \end{gathered}$$

It is known that$\mathrm{Î\pm }=\frac{\mathrm{Î\pm }}{R}$then,

For case (1),

$$\begin{gathered} l_1\frac{a_1}{R}=\left(mg-m{a}_1\right)R \\ =mgR-m{a}_{1}R \end{gathered}$$

For case (2),

$$\begin{gathered} l_2\frac{a_2}{R}=\left(mg-m{a}_2\right)R \\ =mgR-m{a}_{2}R \end{gathered}$$

Divide both sides with R.

$$\begin{gathered} l_1\frac{a_1}{R^2}=\left(mg-m{a}_1\right) \\ {a}_1\left(l_1\frac{a_1}{R^2}\right)=mg \end{gathered}$$

And,

$$\begin{gathered} l_2\frac{a_2}{R^2}=\left(mg-m{a}_2\right) \\ {a}_2\left(l_2\frac{a_2}{R^2}\right)=mg \end{gathered}$$

Final express the above equations for and .

$a_1=\frac{mg}{{\displaystyle \frac{I_1}{R^2}}+m}$

And,

$a_2=\frac{mg}{{\displaystyle \frac{I_2}{R^2}+m}}$

Because, $l_1>l_2$, the denominator in first case is going to be greater than denominator in second case. Depending on that, it will be that ${\mathrm{Î\pm }}_1>{\mathrm{Î\pm }}_2$.

Therefore, for some distance d, because the box in second case have greater linear acceleration, so ,it will have greater velocity too.