Question

Two springs with negligible masses and force constant of $k_{1}=200 \, \text{N m}^{- 1}$ and $k_{2}=160 \, \text{N m}^{- 1}$ are attached to the block of mass $m = 10 \, \text{kg }$ as shown in the figure. Initially, the block is at rest, at the equilibrium position in which both springs are neither stretched nor compressed. At time $t = 0 , the \, $ sharp impulse of $50 \, N \, s$ is given to the block with a hammer along with the spring.

• A. Period of oscillations for the mass m is $\frac{\pi }{6} \text{ s}$
• B. Maximum velocity of the mass m during its oscillation is $10ms^{- 1}$
• C. Data are insufficient to determine maximum velocity
• D. Amplitude of oscillation is $0.83m$

Answer 1

Answer: (D)

$T =$ period
$= 2 \pi \sqrt{\frac{m}{K_{1} + K_{2}}} = 2 \pi \sqrt{\frac{1 0}{3 6 0}} = \frac{\pi }{3} s$
The maximum velocity is always at equilibrium position since at any other point there will be a restoring force attempting to slow the mass.
$∴ \, \, \, V_{\text{max}} = \frac{\text{impulse}}{\text{mass}} = \frac{5 0}{1 0} = 5 m s^{- 1}$
$A = \text{amplittude} = \frac{V_{\text{max}}}{\omega } = \frac{5}{\frac{\left(2 \pi \right)}{\left(\frac{\pi }{3}\right)}} = \frac{5}{6} = \text{0.83 m}$