First slide

Applications of SHM

Question

A particle at the end of a spring executes simple harmonic motion with a period $t_{1}$ , while the corresponding period for another spring is $t_{2}$ . If the period of oscillation with the two springs in series is T, then

Moderate

A

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

$T^{2} = t_{1}^{2} + t_{2}^{2}$
C

$T^{- 1} = t_{1}^{- 1} + t_{2}^{- 1}$
D

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

Solution

$t_{1} = 2 π \sqrt{\frac{m}{k_{1}}}$ and $t_{2} = 2 π \sqrt{\frac{m}{k_{2}}}$
In series, effective spring constant, $k = \frac{k_{1} k_{2}}{k_{1} + k_{2}}$
$∴$ Time period, $T = 2 π \sqrt{\frac{m (k_{1} + k_{2})}{k_{1} k_{2}}}$ …..(ii)
Now, $t_{1}^{2} + t_{2}^{2} = 4 π^{2} m (\frac{1}{k_{1}} + \frac{1}{k_{2}}) = \frac{4 π^{2} m (k_{1} + k_{2})}{k_{1} k_{2}}$
$t_{1}^{2} + t_{2}^{2} = T^{2} .$ [Using equation (ii)]

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