First slide

Applications of SHM

Question

Time period of a spring mass system is T. If this spring is cut into two parts whose lengths are in the ratio 1:3 and the same mass is attached to the longer part. The new time period will be

Moderate

A

$\sqrt{\frac{3}{2}} T$
B

$\frac{T}{\sqrt{3}}$
C

$\frac{\sqrt{3}}{2} T$
D

$\sqrt{3} T$

Solution

The spring mass system oscillates in SHM,
$time period of mass attached to a spring is T=2π \sqrt{\frac{m}{K}} ---(1) where m=mass of oscillator, k=force constant f o r l e n g t h l o f s p r i n g, r e s t o r i n g f o r c e F = - k l - - - - (2) a f t e r s p r i n g i s c u t i n t o 2 p a r t s i n 1 : 3 r a t i o n o f l e n g t h, f o r s h o r t e r p a r t s p r i n g F = - k_{1} \frac{l}{4} f o r l o n g e r p a r t s p r i n g, F = - k_{2} \frac{3 l}{4} - - - (3) d i v i d e e q u a t i o n (3) b y (2) \Rightarrow k_{2} = \frac{4 k}{3} - - - (4) n e w t i m e p e r i o d o f l o n g e r p a r t s p r i n g i s T^{1} =2 π \sqrt{\frac{m}{\frac{4 k}{3}}} - - - (5) d i v i d e e q u a t i o n (5) b y (1) t h e n T^{1} = \frac{\sqrt{3}}{2} T$

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