A block of mass \[m\] on a smooth horizontal surface is connected to a second mass $2m$ by a light cord over a light friction less pulley as shown. (Neglect the mass of the cord and the pulley). A force of magnitude ${F_0}$​ is applied to mass $m$ as shown. Neglect any friction. Find the value of force ​${F_0}$ for which the system will be in equilibrium

A.$mg$
B.$2mg$
C.$3mg$
D.$4mg$

Hint: when there is a mass attached to a rope tension is acted in the opposite direction of gravitational force acting on that mass. For this question find the tension on the rope, by that we will find the value of the required force.

Complete answer:

We have been given a block of mass \[m\] on a smooth horizontal surface that is connected to a second mass $2m$ by a light cord over a light frictionless pulley. Finding the tension acting on the rope.
Tension $T$ is acted in the upward direction which is balancing the downward weight $2mg$
So, $T = 2mg$
From the figure we can see on mass $m$ , tension $2T$ is acting which is balancing the force ${F_0}$
So, ${F_0} = 2T$
Putting the value of tension from above we get,
$ \Rightarrow {F_0} = 2 \times 2mg$
$ \Rightarrow {F_0} = 4mg$

Hence, option D) $4mg$ is the correct option.

Note:
In physics, a tension force is the force created when a rope, string, or cable is stretched under a force. Tension is applied along the length of the rope/cable in the opposite direction of the force applied. Tension is also known by the terms stress, tensity, and tautness.