Question

A block of mass M is pulled along a horizontal frictionless surface by a rope of mass \[{M {\left/
 {\vphantom {M 2}} \right.
} 2}\]. If a force \[2Mg\] is applied at one end of the rope, the force which the rope exerts on the block is –

Answer 1

Hint:Based on the concept of Newton’s second law of motion, we can say that force exerted by the rope on the block is equal to the product of acceleration and total mass of rope and block system.

Complete step by step answer:
Given:
The mass of the block is M.
The mass of rope is \[{M {\left/
 {\vphantom {M 2}} \right.
} 2}\].
Force applied on one end of the rope is \[P = 2Mg\].
The block will start moving under the action of force P with acceleration, and the value of that acceleration is given by:
\[a = \dfrac{{{\rm{total external force}}}}{{{\rm{total mass}}}}\]……(1)
We know that the value of total external force is given by force P, and the total mass is equal to the summation of the mass of the block and rope.
Total mass \[= M + {M {\left/
 {\vphantom {M 2}} \right.
} 2}\\
 \Rightarrow\dfrac{{3M}}{2}
\]
Substitute \[\dfrac{{3M}}{2}\] for total mass and \[2Mg\] for total external force in equation (1).
 \[
a = \dfrac{{2Mg}}{{\dfrac{{3M}}{2}}}\\
\Rightarrow a = \dfrac{{4g}}{3}
\]
From the concept of Newton’s second law, we can write the expression for the total force exerted by the rope on the block as below.
\[F = m \times a\]
Here, m is the total mass of the system.
Let us substitute \[\dfrac{{3M}}{2}\] for m and \[\dfrac{{4g}}{3}\] for a in the above expression.
\[
F = \dfrac{{3M}}{2} \times \dfrac{{4g}}{3}\\
\therefore F = 2Mg
\]

Therefore, we can say that the force exerted by the rope on the block is equal to the force exerted by the block on the rope, and its value is \[2Mg\].

Note: Alternate method: Based on the concept of Newton’s third law of motion, we can say that the force applied by the rope on the block is equal to the force exerted by the block on the rope.Moreover,friction is the force that resists motion when the surface of one object comes in contact with the surface of another. The mechanical advantage of a machine is reduced by friction, or in other words, the ratio of output to input is reduced because of friction.