question_answer 11)

Figures 10 (NCT). 4(a) and (b) correspond to two circular motions. The radius of the circle, the period of revolution, the initial, position, and the sense of revolution (i.e. clockwise or anticlockwise) are indicated on each. Obtain the corresponding equations of simple harmonic motions of the revolving particle P in each case.

Answer:

In Fig. 10 (NCT). 4(a), \[T=2\,s;\,a=-3\,cm;\] At \[t=0\], OP makes an angle \[a=2\cdot 0\] with x-axis, i.e., \[\text{ }\!\!\omega\!\!\text{ =}\sqrt{\frac{\text{k}}{\text{m}}}\text{=}\sqrt{\frac{\text{1200}}{\text{3}}}\text{=0}\cdot 4m{{s}^{-1}}\]\[\text{x = a sin }\!\!\omega\!\!\text{ t, we have x = 2 sin 20 t}\] radian. While moving clockwise, here \[\pi /2\] \[\text{x = a sin}\left( \text{ }\!\!\omega\!\!\text{ t+}\frac{\text{3 }\!\!\pi\!\!\text{ }}{\text{2}} \right)\text{=a cos }\!\!\omega\!\!\text{ t=2 cos 20 t}\] Thus the (-projection of OP at time t will give us the equation of S.H.M., given by \[\text{3 }\!\!\pi\!\!\text{ /2 rad}\text{.}\] or \[x=-3\,\sin \pi t\] In Fig. 10 (NCT). 4 (b), \[T=4s;\,a=2m\] At t = 0, OP makes an angle 31 with the positive direction of x-axis, i.e., \[\pi /2\] =\[\phi =\]. While moving anticlockwise, here \[\pi /2\] Thus the x-projection of OP at time t will give us the equation of S.H.M. as \[\phi =\]