Question

The system shown in the figure can move on a smooth surface. They are initially compressed by 6 cm and then released:

• A. The system performS, SHM with time period $\frac{\pi }{10}$ s
• B. The block of mass 3 kg perform SHM with amplitude 4 cm
• C. The block of mass 6 kg will have maximum momentum of 2.40 kg m/s
• D. The time periods of two blocks are in the ratio of $1:\sqrt{2}$

Answer 1

Answer: (A), (B), (C)

The correct options are
A The system performS, SHM with time period $\frac{\pi }{10}$ s
B The block of mass 3 kg perform SHM with amplitude 4 cm
C The block of mass 6 kg will have maximum momentum of 2.40 kg m/s

$\mu =$Reduced mass
$=\frac{m_1{m}_2}{m_1+m_2}=2$ kg
$T=2\pi \sqrt{\frac{\mu }{k}}=2\pi \sqrt{\frac{2}{800}}=\frac{\pi }{10}s=T_3=T_6$

Assumed $T_3,T_6$ are time periods for mass 3 kg and 6 kg respectively.
$A\propto \frac{1}{m}$
$\therefore$ $\frac{A_3}{A_6}=\frac{6}{3}=\frac{2}{1}$
$\therefore$ $A_3=4$ cm and $A_6=2$ cm
${\left(P_6\right)}_{max}=m_6{\left(v_6\right)}_{max}$
$m_6\left(\omega A_6\right)$
$=\left(6\right)\left(\frac{2\pi }{\pi /10}\right)(2\times {10}^{-2})$
$=2.4$ kg/m