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, number of oscillations, and initial phase, we can follow these steps:

### Step 1: Identify the given parameters
- Amplitude (A) = 2 m
- Number of oscillations = 120 in 1 minute
- Initial phase (φ) = 60° = π/3 radians (since we need to convert degrees to radians)

### Step 2: Convert the time period from minutes to seconds
Since 1 minute = 60 seconds, we have:
- Time (T) = 60 seconds

### Step 3: Calculate the frequency (f)
The frequency (f) is the number of oscillations per second. Given that there are 120 oscillations in 60 seconds:
\[ f = \frac{120 \text{ oscillations}}{60 \text{ seconds}} = 2 \text{ Hz} \]

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

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

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