Two particles A and B are performing SHM along x and y-axis respectively with equal amplitude and frequency of `2 cm` and `1 Hz` respectively. Equilibrium positions of the particles A and B are at the coordinates `[3 cm, 0]` and `(0, 4 cm)` respectively. At `t = 0 ,B` is at its equilibrium position and moving towards the origin, while A is nearest to the origin and moving away from the origin-
Equation of motion of particle B can be written as-

Explanation

To derive the equation of motion for particle B, we will follow these steps:

### Step 1: Understand the parameters of SHM
Given:
- Amplitude (A) = 2 cm
- Frequency (f) = 1 Hz
- Equilibrium position of particle B = (0, 4 cm)

### Step 2: Determine the angular frequency (ω)
The angular frequency (ω) can be calculated using the formula:
\[ \omega = 2\pi f \]
Substituting the given frequency:
\[ \omega = 2\pi \times 1 = 2\pi \text{ rad/s} \]

### Step 3: Write the general equation of motion for SHM
The general equation of motion for a particle performing SHM is given by:
\[ y(t) = A \sin(\omega t + \phi) \]
Where:
- \(y(t)\) is the displacement from the equilibrium position,
- \(A\) is the amplitude,
- \(\omega\) is the angular frequency,
- \(\phi\) is the phase constant.

### Step 4: Determine the phase constant (φ)
At \(t = 0\), particle B is at its equilibrium position (y = 4 cm) and moving towards the origin. This means that the sine function should start at 0 and be negative (since it is moving downwards). Therefore, we can set:
\[ \phi = 0 \]
Thus, the equation simplifies to:
\[ y(t) = A \sin(\omega t) \]

### Step 5: Substitute the values into the equation
Substituting the values of amplitude and angular frequency:
\[ y(t) = 2 \sin(2\pi t) \]

### Step 6: Adjust for the equilibrium position
Since the equilibrium position of particle B is at (0, 4 cm), we need to adjust the equation to reflect this:
\[ y(t) = 4 - 2 \sin(2\pi t) \]
This indicates that when \(y = 4\), the particle is at its equilibrium position, and as it moves downwards, the sine function will decrease.

### Final Equation of Motion for Particle B
Thus, the equation of motion for particle B can be written as:
\[ y(t) = 4 - 2 \sin(2\pi t) \]

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