Question

A block with mass M attached to a horizontal spring with force constant k is moving with simple harmonic motion having amplitude $A_1.$ At the instant when the block passes through its equilibrium position a lump of putty with mass m is dropped vertically on the block from a very small height and sticks to it.
(a) Find the new amplitude and period
(b) Repeat part (a) for the case in which the putty is dropped on the block when it is at one end of its path.

Answer 1

$K{A}_1^2=M{V}^2....\left(1\right)$
$\left(M+m\right)V\left(V_1\to \text{new velocity system}\right)$
$MV=\left(M+M\right)V_1$
$V_1\left(\frac{mv}{M+M}\right)$
$K.E=\left(m+M\right)\frac{V_1^2}{2}......\left(2\right)$
$\frac{K{x}^2}{2}=\left(m+M\right)V_1^2/2$
$x={\left(\frac{M}{m}+M\right)}^{1/2}{A}_1$
time period $=T=2\pi {\left(\frac{M+m}{K}\right)}^{1/2}$