Three blocks of different masses connected with inextensible string are pulled by a force $F$ on a frictionless surface as shown in the figure. The ratio of tensions $T_{1}$ to $T_{2}$ is

Question

Three blocks of different masses connected with inextensible string are pulled by a force F on a frictionless surface as shown in the figure. The ratio of tensions T 1 ​ to T 2 ​ is

TestHub · Accepted Answer

Let the common acceleration of the system be $a$ .

Total mass of the system, $M = 2 kg + 8 kg + 12 kg = 22 kg$ .

The acceleration is given by $a = \frac{F}{22}$ .

The tension $T_{1}$ is responsible for pulling the blocks of mass $2 kg$ and $8 kg$ together.

Equation of motion for this combined mass is:
$T_{1} = (2 + 8) a = 10 a$

The tension $T_{2}$ is responsible for pulling only the block of mass $2 kg$ .

Equation of motion for the $2 kg$ block is:
$T_{2} = 2 a$

The ratio of the tensions is:
$\frac{T _{1}}{T _{2}} = \frac{10 a}{2 a} = \frac{5}{1}$

Therefore, $T_{1} : T_{2} = 5 : 1$ .

Answer: $5 : 1$

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