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 $50Ns$ 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 $10m{s}^{-1}$
• C. Data are insufficient to determine maximum velocity
• D. Amplitude of oscillation is $0.83m$

Answer 1

Answer: (D)

$T=$ period
$=2\pi \sqrt{\frac{m}{K_1+K_2}}=2\pi \sqrt{\frac{10}{360}}=\frac{\pi }{3}s$
The maximum velocity is always at equilibrium position since at any other point there will be a restoring force attempting to slow the mass.
$\therefore V_{\text{max}}=\frac{\text{impulse}}{\text{mass}}=\frac{50}{10}=5m{s}^{-1}$
$A=\text{amplittude}=\frac{V_{\text{max}}}{\omega }=\frac{5}{\frac{\left(2\pi \right)}{\left(\frac{\pi }{3}\right)}}=\frac{5}{6}=\text{0.83 m}$