First slide

Periodic Motion

Question

A force F₁ when it acts on a body causes it to execute SHM of time period T₁ whereas another force F₂ when actions on the same body causes it to execute SHM of time period T₂.If both forces act on the body simultaneously and in a manner that they are always acting in the same phase, the time period of SHM will be:

Easy

A

$\sqrt{T_{1}^{2} + T_{2}^{2}}$
B

$\frac{\sqrt{T_{1}^{2} + T_{2}^{2}}}{T_{1} T_{2}}$
C

$\sqrt{\frac{T_{1}^{2} T_{2}^{2}}{T_{1}^{2} + T_{2}^{2}}}$
D

$\sqrt{\frac{T_{1}^{2} + T_{2}^{2}}{T_{1} T_{2}}}$

Solution

As
$T \propto \frac{1}{\sqrt{k}} and k_{eff} = k_{1} + k_{2} So, \frac{1}{T^{2}} = \frac{1}{T_{1}^{2}} + \frac{1}{T_{2}^{2}} \Rightarrow T = \frac{T_{1} T_{2}}{\sqrt{T_{1}^{2} + T_{2}^{2}}}$

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