Question

A helicopter of mass $M$ is lowering a truck of mass $m$ onto the deck of a ship.In the first case the helicopter and the truck move downward together (the length of the cable remains constant). The tension in the cable is ${T_1}$ when their downward speed is decreasing at a rate of —10. In the second case when the truck gets close to the deck, the helicopter stops moving downward. While it hovers stationary, it lets out the cable so that the truck is still moving downward. The truck is moving downward with a speed decreasing at the rate of $\dfrac{g}{{10}}$. Tension in the string is now ${T_2}$. What is the ratio of $\dfrac{{{T_1}}}{{{T_2}}}$?
A. $\dfrac{{10}}{{11}}$
B. $\dfrac{9}{{11}}$
C. $1$
D. None of the above

Answer 1

To solve this problem find the equation of motion of the system for the condition and find the tension in the rope. Newton’s law of motion states that the net force acting on a body is equal to the mass times the acceleration of the body.

Formula used:
Newton’s second law of motion is given by,
${F_{net}} = ma$
where,$F$ is the net force acting on the body, $m$ is the mass of the body and $a$ is the acceleration of the body.

Complete step by step answer:
In the first scenario the helicopter is lowering the truck with itself.So, forces acting on the truck is the tension on the string and the gravitational pull of the earth. In this condition the equation of motion of the truck will be,
${T_1} - mg = m\dfrac{g}{{10}}$
where, given ${T_1}$ is the tension in the string, $\dfrac{g}{{10}}$ is the acceleration of both the truck and the helicopter.
$\Rightarrow {T_1} = \dfrac{{mg}}{{10}} + mg$
$\Rightarrow {T_1} = \dfrac{{11mg}}{{10}}$

Now, in the second scenario the chopper increases the rope length but also in this condition the tension on the rope due to the truck is the same irrespective of the length of the length of the rope. Hence, we can write the equation of motion of the truck as,
${T_2} - mg = m\dfrac{g}{{10}}$
$\Rightarrow {T_2} = \dfrac{{11mg}}{{10}}$
So, ratio of the tension in the string will be,
$\dfrac{{{T_1}}}{{{T_2}}} = \dfrac{{\dfrac{{11mg}}{{10}}}}{{\dfrac{{11mg}}{{10}}}}$
$\therefore \dfrac{{{T_1}}}{{{T_2}}} = 1$
Hence, the ratio of the tension in the string is, $\dfrac{{{T_1}}}{{{T_2}}} = 1$.

Hence, option C is correct.

Note: When the length of the cable is being increased the tension due to the force by the truck does not change as the tension force is being provided by the helicopter.Tension in the string does not depend on the mass of the chopper though we can calculate the tension in terms of helicopter’s mass. Then the tension due to the helicopter will be in the opposite direction but will be equal in magnitude. If the tension is not equal for both the truck and the chopper the cable will snap and the truck would free fall.