Question
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 (2πt + π/4)
(d) x = 2 cos πt
Solution
(a)
If this equation is compared with the standard SHM equation,
We get,
If this equation is compared with the standard SHM equation,
We get,
Amplitude, A = 2 cm
Phase angle, Φ = 5π/6 = 150°
Angular velocity = ω = 2π/T = 3rad/sec
The motion of the particle can be plotted as shown in fig. 10(a).
b)
If this equation is compared with the standard SHM equation,
If this equation is compared with the standard Simple Harmonic Motion the equation is,
b)
If this equation is compared with the standard SHM equation,
Amplitude, A = 1
Phase angle, Φ = -π/6 = -30°.
Angular velocity, ω = 2π/T = 1 rad/s.
The motion of the particle can be plotted as shown in fig. 10(b).
c)
c)
If this equation is compared with the standard Simple Harmonic Motion the equation is,
Amplitude, A = 3 cm
Phase angle, Φ = 3π/4 = 135°
Angular velocity, ω = 2π/T = 2 rad/s.
The motion of the particle can be plotted as shown in fig. 10(c).
d)
If this equation is compared with the SHM equation, we get
, we get
d)
If this equation is compared with the SHM equation, we get
, we getAmplitude, A = 2 cm
Phase angle, Φ = 0
Angular velocity, ω = π rad/s.
The motion of the particle can be plotted as shown in fig. 10(d).
