The-time-period-T-of-pendulum-of-length-l-is-given-by-T-2-l-g-where-l-amd-g-are-constant-Find-the-approximate-percentage-increase-in-time-T-when-the-length-of-pendulum-increases-by-4-

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Question Number 20849 by j.masanja06@gmail.com last updated on 04/Sep/17

$$Thetimeperiod,Tofpendulumof$$$$lengthlisgivenbyT=2\mathrm{\Pi }\sqrt{\frac{l}{g}}where$$$$lamdgareconstant.Findthe$$$$approximatepercentageincrease$$$$intimeT,whenthelengthof$$$$pendulumincreasesby4\mathrm{\%}$$

Answered by mrW1 last updated on 04/Sep/17

$$\mathrm{T}=2\pi \sqrt{\frac{\mathrm{l}}{\mathrm{g}}}$$$$\frac{\mathrm{dT}}{\mathrm{dl}}=2\pi \frac{1}{\sqrt{\mathrm{g}}}\times \frac{1}{2\sqrt{\mathrm{l}}}=2\pi \sqrt{\frac{\mathrm{l}}{\mathrm{g}}}\times \frac{1}{2\mathrm{l}}=\frac{\mathrm{T}}{2\mathrm{l}}$$$$\frac{\mathrm{dT}}{\mathrm{T}}=\frac{1}{2}\times \frac{\mathrm{dl}}{\mathrm{l}}$$$$\frac{\mathrm{\Delta }\mathrm{T}}{\mathrm{T}}\approx \frac{\mathrm{dT}}{\mathrm{T}}=\frac{1}{2}\times \frac{\mathrm{dl}}{\mathrm{l}}\approx \frac{1}{2}\times \frac{\mathrm{\Delta }\mathrm{l}}{\mathrm{l}}=\frac{1}{2}\times 0.04=0.02$$$$\Rightarrow \mathrm{Increase}\mathrm{in}\mathrm{T}\mathrm{is}\mathrm{nearly}2\mathrm{\%}.$$

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