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14.12 Plot the corresponding reference circle for each of the following simple harmonic motions. Indicate the initial (t =0) position of the particle, the radius of the circle, and the angular speed of the rotating particle. For simplicity, the sense of rotation may be fixed to be anticlockwise in every case: (x is in cm and t is in s).

(a) X = –2 sin (3t + $π$ /3)

(b) X = cos ( $π$ 6 – t)

(c) X = 3 sin (2 $π$ t + $π$ /4)

(d) X = 2 cos $π$ t

3 Views | Posted 5 months ago

Answered by

Vishal Baghel | Contributor-Level 10

3 months ago

(a) X = -2 sin( 3t + $\frac{π}{3}$ ) = +2cos ( 3t + $\frac{π}{3}$ + $\frac{π}{2})$ = 2 cos (3t + $\frac{5 π}{6}$ )

when we compare this equation with standard SHM equation

x = Acos ( $ω$ t + $φ$ ), then we get

Amplitude A = 2 cm. Phase angle $φ = \frac{5 π}{6}$ = 150 $°$ , angular velocity $ω$ = 3 rad/s

(b) X= cos ( $\frac{π}{6} -$ t) = cos ( $t - \frac{π}{6}$ )

when we compare this equation with standard SHM equation

x = Acos( $ω$ t + $φ$ ), then we get

Amplitude A = 1 cm. Phase angle $φ = - \frac{π}{6}$ = - 30 $°$ , angular velocity $ω$ = 1 rad/s

(c) X = 3sin (2 $π$ t + $\frac{π}{4}$ ) = -3cos $[(2 π t + \frac{π}{4}) + \frac{π}{2}] = - 3 c o s ? (2 π t + \frac{3 π}{4})$

when we compare this equation w

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(a) X = -2 sin( 3t + $\frac{π}{3}$ ) = +2cos ( 3t + $\frac{π}{3}$ + $\frac{π}{2})$ = 2 cos (3t + $\frac{5 π}{6}$ )

when we compare this equation with standard SHM equation

x = Acos ( $ω$ t + $φ$ ), then we get

Amplitude A = 2 cm. Phase angle $φ = \frac{5 π}{6}$ = 150 $°$ , angular velocity $ω$ = 3 rad/s

(b) X= cos ( $\frac{π}{6} -$ t) = cos ( $t - \frac{π}{6}$ )

when we compare this equation with standard SHM equation

x = Acos( $ω$ t + $φ$ ), then we get

Amplitude A = 1 cm. Phase angle $φ = - \frac{π}{6}$ = - 30 $°$ , angular velocity $ω$ = 1 rad/s

(c) X = 3sin (2 $π$ t + $\frac{π}{4}$ ) = -3cos $[(2 π t + \frac{π}{4}) + \frac{π}{2}] = - 3 c o s ? (2 π t + \frac{3 π}{4})$

when we compare this equation with standard SHM equation

x = Acos( $ω$ t + $φ$ ), then we get

Amplitude A = 3 cm. Phase angle $φ = \frac{3 π}{4}$ = 135 $°$ , angular velocity $ω$ = $2 π$ rad/s

(d) X = 2cos $π$ t

when we compare this equation with standard SHM equation

x = Acos( $ω$ t + $φ$ ), then we get

Amplitude A = 2 cm. Phase angle $φ = 0 °$ , angular velocity $ω$ = $π$ rad/s

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