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).

1. x = –2 sin (3t + \(\frac{\pi}{3}\))
2. x = cos (\(\frac{\pi}{6}\) – t)
3. x = 3 sin (2πt + \(\frac{\pi}{4}\))
4. x = 2 cos πt 
   
    

Answer 1

a. \(x\) =\(-2\sin\bigg(3t+\frac{\pi}{3}\bigg)+2\cos\bigg(3t+\frac{\pi}{3}+\frac{\pi}{2}\bigg)\)
= \(2\cos\bigg(3t+\frac{5\pi}{6}\bigg)\)
If this equation is compared with the standard SHM equation \(x\) =\(A\cos\bigg(\frac{2π}{T} t+\phi\bigg)\) , then we get:
Amplitude, \(A\) = \(2 \;cm\)
Phase angle, \(\phi\) = \(\frac{5\pi}{6}\)=\(150º\)
Angular velocity, \(\omega\) = \(\frac{2\pi}{T}\)=\(3 \,rad/sec.\)
The motion of the particle can be plotted as shown in the following figure.

\(x\) =\(\cos\bigg(\frac{\pi}{6}-t\bigg)\)=\(\cos \bigg(t-\frac{π}{6}\bigg)\)
If this equation is compared with the standard SHM equation \(x\) =\(A\cos\bigg(\frac{2\pi}{T} t+\phi\bigg)\), then we get : 
Amplitude, \(A\) =\(1\)
Phase angle, \(\phi\) = \(-\frac{π}{6}\)=\(-30º\)
Angular velocity, \(\omega\) = \(\frac{2\pi}{T}\)=\(1 \;rad/s\)
The motion of the particle can be plotted as shown in the following figure.

\(x\) = \(3\sin\bigg(2\pi t+\frac{\pi}{4}\bigg)\)
=\(-3\cos\bigg[\bigg(2\pi t+\frac{\pi}{4}\bigg)+\frac{\pi}{2}\bigg]\)=-\(3\cos\bigg(2\pi t+\frac{3\pi}{4}\bigg)\)
If this equation is compared with the standard SHM equation \(x\) = \(A\cos\bigg(\frac{2\pi}{T} t+\phi\bigg)\) , then we get:
Amplitude, \(A\) = \(3 \,cm\)
Phase angle, \(\phi\) = \(\frac{3\pi}4\)=\(135º\)
Angular velocity, \(\omega\) =\(\frac{2\pi}{T}\) = \(2\pi \;rad/s\)
The motion of the particle can be plotted as shown in the following figure.

\(x\) = \(2 \cos \pi t\) 
If this equation is compared with the standard SHM equation we get: \(A\cos\bigg(\frac{2\pi}{T} t+\phi\bigg)\) then we get:
Amplitude, \(A\) = \(2 \;cm\) 
Phase angle, \(\phi\) = \(0\) 
Angular velocity, \(\omega\) = \(\pi \; rad/s\) 
The motion of the particle can be plotted as shown in the following figure.