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  4. The displacement equation of a particle is x=3 sin 2t+4 cos 2t, where x is in metre and t in second. The amplitude and maximum velocity will be respectively:

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Q. The displacement equation of a particle is $ x=3\,\sin \,2t+4\,\cos \,2t, $ where $ x $ is in metre and t in second. The amplitude and maximum velocity will be respectively:

Rajasthan PMTRajasthan PMT 2004Motion in a Plane

Solution:

Displacement equation is
$x=3 \sin 2 t+4 \cos 2 t$
Comparing this equation with
$x=a_{1} \sin \omega t+a_{2} \cos \omega t$
We get,
$ a_{1}=3, a_{2}=4, \omega=2 $
$\therefore a=\sqrt{a_{1}^{2}+a_{2}^{2}} $
$=\sqrt{3^{2}+4^{2}}=5$ metre
and maximum velocity
$v_{\max }=a \omega=5 \times 2=10\, m / s$