Question

Two blocks $A\left(5\text{kg}\right)\text{and B}\left(2\text{kg}\right)$ attached to the ends of a spring of spring constant 1120 N/m are placed on a smooth horizontal plane with spring undeformed. Simultaneously velocities of $3\text{m/s and 10 m/s}$ along the line of the spring in the same direction are imparted to $A$ and $B$ then

• A. the amplitude of oscillation of $5\text{kg}$ block is $0.18\text{m}$
• B. the first maximum compression occurs after start at $\frac{3\pi }{56}\text{s}$
• C. the maximum extension of the spring is $0.50\text{m}$
• D. time period of oscillation is $\frac{\pi }{14}\text{s}$

Answer 1

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

The correct options are
A the amplitude of oscillation of $5\text{kg}$ block is $0.18\text{m}$
B the first maximum compression occurs after start at $\frac{3\pi }{56}\text{s}$
C the maximum extension of the spring is $0.50\text{m}$

By momentum conservation,
$5\times 3+10\times 2=7v$
$v=5m/s$
By energy conservation,
$\frac{1}{2}\times 5\times 3^2+\frac{1}{2}\times 2\times {10}^2=\frac{1}{2}\times 7\times 5^2+\frac{1}{2}\times k\times x^2$
which gives $x=0.25m$

Time period of oscillation, $T=2\pi \sqrt{\frac{\mu }{k}}$
Here, $\mu =\frac{m_1{m}_2}{m_1+m_2}=\frac{10}{7}$

$T=\frac{\pi }{14}$
First compression occurs = $\frac{3T}{4}=\frac{3\pi }{56}s$