Question

A ball of mass m is suspended on a weightless elastic thread whose stiffness factor is k. The ball is lifted so that the thread is not stretched and let fall with zero initial velocity. The maximum stretch of the thread in the process of ball’s motion is:
A. $\dfrac{{mg}}{k}$
B. $\dfrac{{2mg}}{k}$
C. $\dfrac{{mg}}{{2k}}$
D. $\dfrac{{mg}}{{4k}}$

Answer 1

As the ball moves downwards, the gravitational potential energy decreases. This decrease in the potential energy is equal to the increase in the elastic potential energy of the thread. Use the formula for gravitational potential energy and elastic potential energy and equate them to determine the elongation in the thread.

Formula used:
The gravitational potential energy of the ball of mass m is,
$U = mgh$
Here, m is the mass, g is the acceleration due to gravity and h is the height above the ground.
The elastic energy stored in spring or elastic thread of spring constant k is,
$U = \dfrac{1}{2}k{x^2}$
Here, k is spring constant and x is the elongation or compression in the spring from the mean position.

Complete step by step answer:
We know that when a spring stretches or compresses, an elastic potential energy gets stored in it. Also, we know when the body moves downward, the gravitational potential energy decreases.
We can observe here, a decrease in the gravitational potential energy from the mean position of the thread is equal to gain in the elastic potential energy of the thread. Therefore, we can write,
$mgx = \dfrac{1}{2}k{x^2}$
Here, x is the elongation in the thread, m is the mass of the ball, g is the acceleration due to gravity and k is the stiffness factor of the thread.
We can rearrange the above equation for x as follows,
$mg = \dfrac{1}{2}kx$
$\Rightarrow x = \dfrac{{2mg}}{k}$

From the above equation we observe that the elongation or stretching in the thread is proportional to the mass of the ball.

So, the correct answer is “Option B”.

Note:
The stiffness factor and spring constant is the same as they both have the same unit. While solving such types of questions, students should understand which energy is being converted into another form of energy. For every motion, there is decrease in one form of energy and increase in another form of energy and if there is no loss in the energy, the decrease in one form of energy is equal to increase in another form of energy.