First slide

Applications of SHM

Question

A mass M is suspended from a spring of negligible mass. The spring is pulled a little and then released so that the mass executes SHM of time period T. If the mass is increased by ‘m’ the time period becomes $\frac{5 T}{3}$ , then the ratio of $\frac{m}{M}$ is

Moderate

A

$\frac{3}{5}$
B

$\frac{25}{9}$
C

$\frac{16}{9}$
D

$\frac{5}{3}$

Solution

As time period $T=2π \sqrt{\frac{M}{K}}$ ------(1)
when mass m is added to M
$T^{1} =2π \sqrt{\frac{M+m}{K}}$ substitute value of new time period $T^{1} = \frac{5 T}{3}$
$\frac{5T}{3} = 2π \sqrt{\frac{M+m}{K}}$ ---------(2)
Dividing equation(1) by equation (2), we get
$\frac{3}{5} = \sqrt{\frac{M}{M + m}}$
squaring on both sides
$\frac{9}{25} = \frac{M}{M+m}$
cross multiply
$9M+9m=25M$
$16M=9m$
$\frac{m}{M} = \frac{16}{9}$

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