First slide

Simple hormonic motion

Question

A mass m is suspended separately by two different springs of spring constant $K_{1} and K_{2}$ gives the time period $t_{1} and t_{2}$ respectively. If the same mass m is connected by both springs as shown in the figure, then time period t is given by the relation

Difficult

A

$t = t_{1} + t_{2}$
B

$t = \frac{t_{1} t_{2}}{t_{1} + t_{2}}$
C

$t^{2} = t_{1}^{2} + t_{2}^{2}$
D

$t^{- 2} = t_{1}^{- 2} + t_{2}^{- 2}$

Solution

$t_{1} = 2 π \sqrt{\frac{m}{K_{1}}} and t_{2} = 2 π \sqrt{\frac{m}{K_{2}}}$
Equivalent spring constant for shown combination is $K_{1} + K_{2}$ . So time period t is given by
$t = 2 π \sqrt{\frac{m}{K_{1} + K_{2}}}$
By solving these equations we get $t^{- 2} = t_{1}^{- 2} + t_{2}^{- 2}$

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