If the displacement of a moving point at any time is given by an equation of the form `y (t) = a cos omega t + b sin omega t`, shown that the motion is simple harmonic . If `a = 3 m, b = 4m and omega = 2`: determine the period , amplitude, maximum velocity and maximum acceleration.

Explanation

To solve the problem step by step, we will start by proving that the motion described by the equation \( y(t) = a \cos(\omega t) + b \sin(\omega t) \) is simple harmonic motion (SHM), and then we will calculate the period, amplitude, maximum velocity, and maximum acceleration given \( a = 3 \, \text{m} \), \( b = 4 \, \text{m} \), and \( \omega = 2 \, \text{rad/s} \).

### Step 1: Proving the motion is simple harmonic
The equation \( y(t) = a \cos(\omega t) + b \sin(\omega t) \) can be rewritten in the form of a single sine function using the phasor method.

1. **Convert to a single sine function:**
\[
y(t) = R \sin(\omega t + \phi)
...