Question

The time period of a
mass suspended
from spring is T. If
the spring is cut
into four equal
parts and the same
mass is suspended
from one of the
parts, then the new
time period will be
​

Answer 1

Initially ,the mass M is suspended from a spring of spring constant k

The time period of oscillation is given by the formula ,

$T = 2\pi \sqrt{ \frac{m}{k} }$

Then the spring is cut into 4 equal parts

When a spring is cut into n equal parts spring constant of each part is equal to

k ' = n k

here , n = 4

k ' = 4 k

or

When n no of springs are connected in series ,

$\frac{1}{k ' } = \frac{n}{k }$

here, n =4

$k ' = 4k$

New time period

$T ' = 2\pi \sqrt{ \frac{m}{4k} }$

$T ' = \frac{2\pi}{2} \sqrt{ \frac{m}{k} }$

$\boxed{T ' = \frac{T}{2} }$