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  4. An SHM is represented by x=5√ 2(sin 2π t+cos 2π t) The amplitude of the SHM is

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Q. An SHM is represented by
$ \, \, \, \, x=5\sqrt 2(sin 2\pi t+cos 2\pi t)$
The amplitude of the SHM is

MHT CETMHT CET 2004Oscillations

Solution:

Here, $ \, \, \, \, x=5\sqrt 2(sin 2\pi t+cos 2\pi t)$
$\Rightarrow \, \, \, \, \, \, x=5\sqrt 2sin 2\pi t+5 \sqrt 2 cos 2\pi t \, \, \, \, \, $ ... (i)
The standard equation of simple harmonic motion is given by
$ \, \, \, \, \, x=A_1 sin \omega t +A_2 cos \omega t \, \, \, \, \, \, \, \, \, $... (ii)
Now, comparing Eqs. (i) and (ii), we obtain
$ \, \, \, \, \, \, A_1 =5\sqrt 2 \, and \, A_2 =5\sqrt 2$
So, the resuitant amplitude of the motion
$ \, \, \, \, \, \, A=\sqrt{A_1^2 +A_2^2} =\sqrt{(2\sqrt 2)^2+(5\sqrt 2)^2}$
$ \, \, \, \, \, \, \, \, \, \, \, =\sqrt{50+50}$
$ \, \, \, \, \, \, \, \, \, \, \, =10 cm$