First slide

Dynamics of uniform and non uniform circular motion

Question

A particle of mass m is fixed to one end of a 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

$\frac{m l ω^{2}}{k + m ω}$
B

$\frac{m l ω^{2}}{k - ω m}$
C

$\frac{m l ω^{2}}{k + m ω^{2}}$
D

$\frac{m l ω^{2}}{k - m ω^{2}}$

Solution

$\begin{array}{l} L e t s a y i n c r e a s e i n t h e l e n g t h o f t h e s p r i n g = x \\ a n d L (= l + x) b e t h e t o t a l l e n g t h o f t h e s p r i n g d u r i n g m o t i o n . \\ T h e c e n t r i p e t a l f o r c e a c t i n g o n t h e p a r t i c l e i s p r o v i d e d b y t h e s p r i n g . \\ ∴ m ω^{2} L = k x \\ \Rightarrow m ω^{2} (l + x) = k x \\ \Rightarrow (\frac{l}{x} + 1) = \frac{k}{m ω^{2}} \\ \Rightarrow x = \frac{m l ω^{2}}{k - m ω^{2}} \end{array}$

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