Question

A load of mass m falls from a height h onto the scale pan hanging from a spring as shown in the figure. If the spring constant is k, mass of scale pan is zero, and the mass does not bounce relative to the pan, then the amplitude of vibration is 

• A. \( \dfrac{mg}{k} \)
• B. \( \dfrac{mg}{k}\sqrt{\dfrac{1+2hk}{mg}} \)
• C. \( \dfrac{mg}{k} + \dfrac{mg}{k}\sqrt{\dfrac{1+2hk}{mg}} \)
• D. (mg)/(k)√((1+2hk)/(mg)) - (mg)/(k)

Hint:

Amplitude is measured from new equilibrium position, not from natural length.

Answer 1

The Correct Option is B

Step 1: Maximum extension x is obtained using energy conservation: mgh + mgx = (1)/(2)kx²
Step 2: Solving quadratic, x = (mg)/(k)(1 + √(1 + (2hk)/(mg)))
Step 3: Amplitude of oscillation is measured from equilibrium extension (mg)/(k): A = x - (mg)/(k) = (mg)/(k)√((1+2hk)/(mg))