Question

A block of mass '$M$' rests on a piston executing S.H.M. of period one second. The amplitude of oscillations, so that the mass is separated from the piston, is (acceleration due to gravity, $g = 10 \text{ ms}^{-2}, \pi^2 = 10$ )}

• A. 0.25 m
• B. 0.5 m
• C. 1 m
• D. \infty

Hint:

Separation in vertical S.H.M. occurs when the support acceleration reaches $g$. Solve $\omega^2 A = g$.

Answer 1

The Correct Option is A

Step 1: The mass will separate from the piston when the downward acceleration of the piston exceeds the acceleration due to gravity $g$ at the highest point of oscillation.
Step 2: In S.H.M., the maximum acceleration is given by $a_{max} = \omega^2 A$. For separation to occur: \[ \omega^2 A \ge g \]
Step 3: The angular frequency $\omega$ is calculated from the time period $T = 1$ s as: \[ \omega = \frac{2\pi}{T} = \frac{2\pi}{1} = 2\pi \text{ rad/s} \]
Step 4: Substitute $\omega = 2\pi$, $g = 10$, and $\pi^2 = 10$ into the condition for separation at the threshold: \[ (2\pi)^2 A = 10 \Rightarrow 4\pi^2 A = 10 \] \[ 4(10) A = 10 \Rightarrow 40A = 10 \Rightarrow A = 0.25 \text{ m} \]