First slide

Applications of SHM

Question

A particle of mass m is attached to three identical springs A, B and C each of force constant k a shown in figure. If the particle of mass m is pushed slightly against the spring A and released then the time period of oscillations is

Moderate

A

$2 π \sqrt{\frac{2 m}{k}}$
B

$2 π \sqrt{\frac{m}{2 k}}$
C

$2 π \sqrt{\frac{m}{k}}$
D

$2 π \sqrt{\frac{m}{3 k}}$

Solution

When the particle of mass m at O is pushed by y in the direction of A The spring A will be compressed by y while spring B and C will be stretched by $y^{'} = ycos 45 ° .$ So that the total restoring force on the mass m along OA.
$F_{net} = F_{A} + F_{B} \cos 45 ° + F_{C} \cos 45 °$
$= ky + 2 {ky}^{'} \cos 45 ° = ky + 2 k (ycos 45 °) \cos 45 ° = 2 ky$
Also $F_{net} = k^{'} y \Rightarrow k^{'} y = 2 ky \Rightarrow k^{'} = 2 k$
$T = 2 π \sqrt{\frac{m}{k^{'}}} = 2 π \sqrt{\frac{m}{2 k}}$

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