First slide

Simple hormonic motion

Question

A particle at the end of a spring executes simple harmonic motion with a period t₁, while the corresponding period for another spring is t₁ 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

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

$T^{â 2} = t_{1}^{â 2} + t_{2}^{â 2}$

Solution

$Here, t_{1} = 2 Ï \sqrt{(\frac{m}{k_{1}})} or t_{1}^{2} = 4 {Ï}^{2} (\frac{m}{k_{1}}) and t_{2} = 2 Ï \sqrt{(\frac{m}{k_{2}})} or t_{2}^{2} = 4 {Ï}^{2} (\frac{m}{k_{2}}) â´ t_{1}^{2} + t_{2}^{2} = 4 {Ï}^{2} m (\frac{1}{k_{1}} + \frac{1}{k_{3}}) When the springs are connected h seri 6 s, then \frac{1}{k} = \frac{1}{k_{1}} + \frac{1}{k_{2}} Now T = 2 Ï \sqrt{(\frac{m}{k})} = 2 Ï \sqrt{[m (\frac{1}{k_{1}} + \frac{1}{k_{2}})]} or T^{2} = 4 {Ï}^{2} m [\frac{1}{k_{1}} + \frac{1}{k_{2}}] T^{2} = 4 {Ï}^{2} m [\frac{1}{k_{1}} + \frac{1}{k_{2}}] Fmm eqs . [1) atrd (2), we ge T^{2} = t_{1}^{2} + t_{2}^{2}$

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