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Oscillations
figures 14 29 correspond to two circular motions...

Question:
Figures 14.29 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.

Answer:

( 1 ) For time period, T = 4 s

Amplitude, A = 3 cm

At time, t = 0, the radius vector OB makes an angle π/2 with the positive x-axis, i.e., Phase angel Φ = + π/2

Therfore, the equation of simple harmonic motion for the x-projection of OB, at time t is:

x = A cos [ 2πt/T + Φ ]

= 3 cos [2πt/4 + π/2 ]

= -3sin ( πt/2 )

= -3sin ( πt/2 ) cm

( 2 ) Time period, T = 8 s

Amplitude, A = 2 m

At time t = 0, OB makes an angle π with the x-axis, in the anticlockwise direction. Thus, phase angle, Φ = + π

Therefore, the equation of simple harmonic motion for the x-projection of OB, at time t is:

x = A cos [ 2πt/T + Φ ]

= 2 cos [ 2πt/8 + π ]

= -2 cos ( πt/4 )

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