Question

A man of mass $60kg$ standing on a light weighing machine kept in a box of mass $30kg$ . The box is hanging from a pulley fixed to the ceiling through a light rope, the other end of which is held by the man himself. If the man manages to keep the box at rest, what is the weight shown by the machine?

Answer 1

You can start the solution by calculating the net upwards and downwards force on the system. Since the net upwards and downwards force is equal in equilibrium (the sum of the weight of the man and the box is equal to two times the tension), you can use this relation to calculate the value of $T$ . Then use the value $T$ to calculate the weight of the man displayed on the machine.

Complete step by step answer:

Consider that the tension in the wire is $T$ . This is the upward force acting on the system, it acts on both the man and the box.

For the man the upwards force is,

$Force = T$

For the box the upward force is

$Force = T$

Net upwards force on the system is

$Forc{e_{Upwards}} = T + T$

$\Rightarrow Forc{e_{Upwards}} = 2T$

Then there is downwards gravitational force mg on the man and the box.

For the man this force is,

$mg = 60g$

For the box downward force is equal to

$mg = 30g$

Net downwards force on the system is

$Forc{e_{Downwards}} = (30 + 60)kg$

$\Rightarrow Forc{e_{Downwards}} = 90g$

We know that the system is in equilibrium so that the downwards force is equal to the upwards force

$Forc{e_{Upwards}} = Forc{e_{Downwards}}$

$\Rightarrow 2T = 90g$

$\Rightarrow T = 45g$

Now consider the net force on man, the difference between the net downwards force and the net upwards force is

${F_{Man}} = {F_{Downwards}} - {F_{Upwards}}$

$\Rightarrow {F_{Man}} = mg - T$

$\Rightarrow {F_{Man}} = 60g - 45g$

$\Rightarrow {F_{Man}} = 15g$

So this is the force that the man applies on the machine.

Let his mass displayed on the machine be $m'$

$\Rightarrow m'g = 15g$

$\Rightarrow m' = 15kg$

Hence, the weight of the man displayed on the machine for the given system is $15kg$.

Note: In the solution, we took the force applied by the man on the machine to calculate the weight displayed on the man. Well, actually the weight of the man is the force that the machine applies on the man but as by the third law of Newton the force is equal to the force applied by the man on the machine.