First slide

Applications of SHM

Question

A particle of mass ‘ $m$ ’ is fixed to one end of light spring having force constant ‘ $k$ ’and unstretched length ‘ $l$ ’. The other end is fixed. The system is given an angular speed $ω$ about the fixed end of the spring such that it rotates in a circle in gravity free space. Then the stretch in the spring is

Moderate

A

$\frac{m l ω^{2}}{k + m ω^{2}}$
B

$\frac{m l ω^{2}}{k - m ω^{2}}$
C

$\frac{m l ω^{2}}{k - ω m}$
D

$\frac{m l ω^{2}}{k + m ω}$

Solution

$r e s t o r i n g f o r c e o f a s p r i n g i s F = - k x, - - = (1) h e r e k i s f o r c e c o n s \tan t, x i s d i s p l a c e m e n t c e n t r i f u g a l f o r c e a c t i n g o n t h e m a s s a t t a c h e d t o s p r i n g i s F = m (l + x) w^{2}, - - - (2) h e r e m i s m a s s a t t a c h e d t o t h e s p r i n g a n d l i s u n s t r e t c h e d l e n g t h o f s p r i n g e q u a t i n g a b o v e t w o e q u a t i o n s 1 a n d 2 k x = m l ω^{2} + m x ω^{2}$
$x (k - m ω^{2}) = m l ω^{2} x = \frac{m l ω^{2}}{k - m ω^{2}}$

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