Question

A block having a mass $m$ is resting on a smooth horizontal surface. One end of a uniform rope having a mass $3m$ has been fixed to the block, which is pulled by applying force $F$ at the other end in the horizontal direction. What will be the tension in the middle of the rope?
$\begin{align}
  & A.\dfrac{8}{6}F \\
 & B.\dfrac{1}{7}F \\
 & C.\dfrac{1}{8}F \\
 & D.\dfrac{7}{8}F \\
\end{align}$

Answer 1

Hint: The acceleration can be found by taking the ratio of the force acting or the tension in the wire to the total mass of the body. The mass of the mid rope means the half mass of the rope and the mass of the block. The mass of the full rope means the mass of the block and the mass of the full rope. The acceleration of mid rope and full rope will be equal. Compare them and find the tension. This will help you in answering this question.

Complete answer:
The acceleration can be found by taking the ratio of the force acting or the tension in the wire to the total mass of the body. That is here the acceleration at the mid rope can be found as,
$a=\dfrac{T}{m+\dfrac{m}{6}}=\dfrac{6T}{7m}$
Where $T$ be the tension in the rope and $m$ be the mass. The mass of the mid rope means the half mass of the rope and the mass of the block.
For a full rope we can write that,
$a=\dfrac{F}{m+\dfrac{m}{3}}=\dfrac{3F}{4m}$
Here the mass of the full rope means the mass of the block and the mass of the full rope.
These two accelerations will be similar. That is we can write that,
$\dfrac{6T}{7m}=\dfrac{3F}{4m}$
Therefore the tension can be formulated as,
$T=\dfrac{7F}{8}$
The tension of the rope has been calculated.

The correct answer has been mentioned as option D.

Note:
Acceleration is defined as the rate of variation of the velocity of a body with respect to the time taken. The velocity of the body can be defined as the rate of variation of the displacement of the body with respect to the time taken. The acceleration can be found in units of metre per Second Square.