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Weight and Mass

In the figure...

Question

In the figure, the blocks have unequal masses $m_{1}$ and $m_{2} (m_{1} > m_{2}) . m_{1}$ has a downward acceleration $a$ . The pulley $P$ has a radius $r$ , and some mass. The string does not slip on the pulley. Then which of the following statement(s) is/are true ?

A

The two sections of the string have unequal tensions.

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B

The two blocks have accelerations of equal magnitude.

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C

The angular acceleration of P is

\frac{a}{r}

.

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D

a < (\frac{m_{1} - m_{2}}{m_{1} + m_{2}}) g

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Solution

The correct options are
A The two sections of the string have unequal tensions.
B The two blocks have accelerations of equal magnitude.
C The angular acceleration of P is $\frac{a}{r}$ .
D $a < (\frac{m_{1} - m_{2}}{m_{1} + m_{2}}) g$
Due to unequal masses the tensions will be unequal. We have the force balance for $m_{1}$ as $T_{1} = m_{1} g - m_{1} a$ and for mass $m_{2}$ as $T_{2} = m_{2} g + m_{2} a$ .
or
$\frac{T_{1}}{T_{2}} = \frac{m_{1} g - m_{1} a}{m_{2} g + m_{2} a}$
As $T_{1} > T_{2}$ we have $m_{1} g - m_{1} a > m_{2} g + m_{2} a$
or
$(m_{1} + m_{2}) a < (m_{1} - m_{2}) g$
or
$a > \frac{m_{2} - m_{1}}{m_{1} + m_{2}} g$
or
$a < \frac{m_{1} - m_{2}}{m_{1} + m_{2}} g$
If rope is not sliding then as per the relation $a = r α$ option C is also correct.

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