Question

Molten - wax of mass $m$ drops on a block of mass $M$, which is oscillating on a frictionless table as shown.
Select the incorrect option.

• A. If the collision takes place at extreme position, amplitude does not change
• B. If the collision takes place at mean position, amplitude decreases
• C. If the collision takes place at mean position, time period decreases
• D. If the collision takes place at extreme position, time period increases

Answer 1

The correct option is C If the collision takes place at mean position, time period decreases

Before collision, the time period of oscillation is
$T=2\pi \sqrt{\frac{M}{k}}$
Due to addition of molten-wax, Mass of the body changes
Total mass of the body executing SHM at a particular instant $m^{\prime }=m+M$
Mass is increasing therefore time period will increase.
$\therefore$ Option (c) is the incorrect and option (d) is the correct.

Let us suppose the common velocity of wax and block be $V$.
If the collision takes place at extreme position
By applying law of conservation of linear momentum we can say that, the amplitude of oscillation remains same.
So, option (a) is the correct answer.

If the collision takes place at mean position, by applying law of conservation of linear momentum
$m_1{u}_1+m_2{u}_2=m_1{v}_1+m_2{v}_2$
$M\left(A\omega \right)=(M+m_w)\left(A_1{\omega }_1\right)$
$A_1=\sqrt{\frac{M}{M+m_w}}$
The amplitude of the oscillation decreases.
So, option (b) is the correct.

Hence, option (c) is the required answer.