A simple harmonic motion is represented by :
`y=5(sin3pit+sqrt(3)cos3pit)cm` The amplitude and time period of the motion by :

Options

A. `10cm,(2)/(3)s`
B. `10cm,(3)/(2)s`
C. `5cm,(3)/(2)s`
D. `5cm,(2)/(3)s`

Explanation

To find the amplitude and time period of the given simple harmonic motion represented by the equation:

\[ y = 5 \sin(3\pi t) + \sqrt{3} \cos(3\pi t) \]

we can follow these steps:

### Step 1: Rewrite the equation in a standard form
We can express the equation in the form \( A \sin(\omega t + \phi) \). To do this, we will factor out a common coefficient.

### Step 2: Factor out the coefficient
We can factor out 2 from the equation:

\[ y = 5 \sin(3\pi t) + \sqrt{3} \cos(3\pi t) = 2 \left( \frac{5}{2} \sin(3\pi t) + \frac{\sqrt{3}}{2} \cos(3\pi t) \right) \]

### Step 3: Identify sine and cosine components
Now, we can identify the coefficients of sine and cosine:

- \( A = \frac{5}{2} \)
- \( B = \frac{\sqrt{3}}{2} \)

### Step 4: Calculate the amplitude
The amplitude \( A \) of the motion can be calculated using the formula:

\[ \text{Amplitude} = \sqrt{A^2 + B^2} \]

Substituting the values:

\[ \text{Amplitude} = \sqrt{\left(\frac{5}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2} = \sqrt{\frac{25}{4} + \frac{3}{4}} = \sqrt{\frac{28}{4}} = \sqrt{7} \]

### Step 5: Calculate the angular frequency
The angular frequency \( \omega \) is given by the coefficient of \( t \) in the sine and cosine functions, which is \( 3\pi \).

### Step 6: Calculate the time period
The time period \( T \) can be calculated using the formula:

\[ T = \frac{2\pi}{\omega} \]

Substituting the value of \( \omega \):

\[ T = \frac{2\pi}{3\pi} = \frac{2}{3} \text{ seconds} \]

### Final Results
- Amplitude: \( \sqrt{7} \, \text{cm} \)
- Time Period: \( \frac{2}{3} \, \text{seconds} \)