Whatis the displacement equation of S.H.M. with an amplitude 2m , if 120 oscillations are performed during one minute and initial phase is `60^(@)` [ Consider displacement time equation of the form `y = A sin( omega t + phi)]` ?

Options

A. `2 sin (4pi t + (pi)/(3))`
B. ` 2 sin ( 4 pi t )`
C. ` 2 sin ( 2pit (pi)/(3))`
D. ` 2 sin ( pit + (pi)/(3))`

Answer

To find the displacement equation of Simple Harmonic Motion (S.H.M.) given the amplitude, frequency, and initial phase, we can follow these steps:

### Step 1: Identify the amplitude (A)
The amplitude is given as 2 meters. Therefore, we have:
\[ A = 2 \, \text{m} \]

### Step 2: Calculate the frequency (f)
We know that 120 oscillations are performed in 1 minute (60 seconds). To find the frequency, we can use the formula:
\[ f = \frac{\text{Number of oscillations}}{\text{Time in seconds}} \]
Substituting the values:
\[ f = \frac{120}{60} = 2 \, \text{Hz} \]

### Step 3: Calculate the angular frequency (ω)
The angular frequency (ω) is related to the frequency (f) by the formula:
\[ \omega = 2 \pi f \]
Substituting the frequency we calculated:
\[ \omega = 2 \pi \times 2 = 4 \pi \, \text{rad/s} \]

### Step 4: Convert the initial phase (φ) to radians
The initial phase is given as 60 degrees. We need to convert this to radians:
\[ \phi = 60^\circ = \frac{60 \times \pi}{180} = \frac{\pi}{3} \, \text{radians} \]

### Step 5: Write the displacement equation
The general form of the displacement equation for S.H.M. is:
\[ y = A \sin(\omega t + \phi) \]
Substituting the values we found:
\[ y = 2 \sin(4 \pi t + \frac{\pi}{3}) \]

### Final Displacement Equation
Thus, the displacement equation of S.H.M. is:
\[ y = 2 \sin(4 \pi t + \frac{\pi}{3}) \]

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