A $1\,kg$ mass is attached to a spring of force constant $600\,N / m$ and rests on a smooth horizontal surface with other end of the spring tied to wall as shown in figure. A second mass of $0.5\,kg$ slides along the surface towards the first at $3\,m / s$. If the masses make a perfectly inelastic collision, then find amplitude and time period of oscillation of combined mass.
Medium
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(a)
Applying linear momentum conservation,
$0.5 \times 3=(1+0.5) v \text { or } v =1 m / s$
By conversation of energy,
After collision
$\frac{1}{2}(1+0.5) v ^2=\frac{1}{2} kA ^2$
$\Rightarrow A =\sqrt{\frac{1.5}{ k }} \times v$
$\Rightarrow A =\sqrt{\frac{1.5}{600}} \times 1=\frac{1}{20} m =0.05 m$
$A =5 cm$
Time period of oscillation,
$T =2 \pi \sqrt{\frac{ m _1+ m _2}{ k }}=2 \pi \sqrt{\frac{1.5}{600}}=\frac{2 \pi}{20}=\frac{\pi}{10} s$
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