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Question

Figures given 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 anti-clockwise) are indicated on each figure. Obtain the corresponding simple harmonic motions of the x-projection of the radius vector of the revolving particle P, in each case.
420431_27e18fd4da0b4e99b536d57f4d983b2c.png

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Solution

(a) Time period, t=2s
Amplitude, A=3cm
At time, t=0, the radius vector OP makes an angle π/2 with the positive x-axis, i.e., phase angle ϕ=+π/2
Therefore, the equation of simple harmonic motion for the x-projection of OP, at the time t, is given by the displacement equation:
x=Acos[2πtT+ϕ]
=3cos(2πt2+π2)=3sin(2πt2)
x=3sin(πt)cm

(b) Time Period, t=4s
Amplitude, a=2m
At time t=0, OP makes an angle π with the x-axis, in the anticlockwise direction, Hence, phase angle ϕ=+π
Therefore, the equation of simple harmonic motion for the x-projection of OP, at the time t, is given as:
x=acos[2πtT+ϕ]
=2cos(2πt4+π)
x=2cos(π2t)m

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