Question

Two masses $3\text{ kg and 5 kg}$ are suspended with the help of a massless inextensible string, as shown in the figure. The whole system is going upwards with an acceleration of $2\text{ m/}{{\text{s}}^{2}}$. The tension ${{\text{T}}_{1}}\text{ and }{{\text{T}}_{2}}$ is respectively: $\left( g=10\text{ m/}{{\text{s}}^{2}} \right)$

A. $96\text{ N and 36 N}$

B. $36\text{ N and 96 N}$

C. $96\text{ N and 96 N}$

D. $36\text{ N and 36 N}$

Answer 1

Hint: The tension in the strings will be in a direction away from the respective bodies that strings are attached to. The force of gravity will be acting downwards, and the acceleration will be acting upwards. Drawing a free body diagram and plotting all the forces acting on a particular body will give an idea about the tension in the strings.

Complete step-by-step answer:
The free-body diagram for the mass of $3kg$ is

So writing down the forces acting on the body of mass 3 kg, we get,

${{T}_{1}}-mg=ma$

$\Rightarrow {{T}_{1}}=m\left( g+a \right)$

Substituting the values of m, a and g, we get,

${{T}_{1}}=\left( 3kg \right)\times \left( 10+2 \right)m{{s}^{-2}}$

$\therefore {{T}_{1}}=36N$ …. equation (1)

So, the value of tension ${{\text{T}}_{\text{1}}}$ is 36 N.

The free-body diagram for the mass of $5kg$ is

So, writing down the forces acting on the mass 5 kg can be written as,

$ {{T}_{2}}-ma={{T}_{1}}+mg $

$ \Rightarrow {{T}_{2}}=ma+mg+{{T}_{1}} $

Substituting the value of ${{T}_{1}}$ from equation (1) and other values into the above equation, we get

${{T}_{2}}=\left( 5kg \right)\times \left( 10+2 \right)m{{s}^{-2}}+36N$

$\Rightarrow {{T}_{2}}=60N+36N$

$\therefore {{T}_{2}}=96N$

So, the tension ${{\text{T}}_{\text{2}}}$ is found to be 96 N.

Therefore, the answer to the question is the option (B).

Note: The tension is a pulling force that is applied by a string, cable or any one-dimensional continuous object on an object to which it is connected. It is a part of the Newtonian pair of forces which acts on the object and the string. The SI unit of tension is the same as that of force which is the newton.

So, a system containing a string and a mass can be in two states,

1) Equilibrium State: In this state, the system is in equilibrium with no external force acting on it. If T is the tension in the string acting upwards and mg is the force due to gravity acting downwards, the sum of these forces will give zero.

2) Accelerated State: If the system containing the string is accelerated upwards or downwards, a net external force will act on the body other than the tension T and the force of gravity. SO the sum of tension and force due to gravity will not be zero in this case, since an external force is acting on the system.