A simple harmonic motion is given by `y = 5(sin3pit + sqrt(3) cos3pit)`. What is the amplitude of motion if `y` is in `m` ?

Options

A. `100 cm`
B. `5 cm`
C. `200 cm`
D. `1000 cm`

Explanation

To find the amplitude of the simple harmonic motion given by the equation \( y = 5(\sin(3\pi t) + \sqrt{3} \cos(3\pi t)) \), we can follow these steps:

### Step 1: Identify the components of the equation
The equation is in the form \( y = A(\sin(\omega t) + B \cos(\omega t)) \), where \( A = 5 \), \( \omega = 3\pi \), and \( B = \sqrt{3} \).

### Step 2: Use the amplitude formula
The amplitude \( A \) of a simple harmonic motion described by \( y = A(\sin(\omega t) + B \cos(\omega t)) \) can be calculated using the formula:
\[ A = \sqrt{A^2 + B^2} \]
Here, we need to find \( A = 5 \) and \( B = \sqrt{3} \).

### Step 3: Substitute the values into the formula
Now we substitute the values into the amplitude formula:
\[ A = \sqrt{5^2 + (\sqrt{3})^2} \]
Calculating this gives:
\[ A = \sqrt{25 + 3} = \sqrt{28} \]

### Step 4: Simplify the expression
We can simplify \( \sqrt{28} \):
\[ A = \sqrt{4 \times 7} = 2\sqrt{7} \]

### Step 5: Convert the amplitude to centimeters
Since the problem states that \( y \) is in meters, we need to convert the amplitude to centimeters:
\[ A = 2\sqrt{7} \text{ meters} = 2\sqrt{7} \times 100 \text{ centimeters} \]

### Step 6: Calculate the numerical value
Calculating \( 2\sqrt{7} \):
\[ \sqrt{7} \approx 2.64575 \Rightarrow 2\sqrt{7} \approx 5.2915 \text{ meters} \]
Now converting to centimeters:
\[ 5.2915 \text{ meters} \approx 529.15 \text{ centimeters} \]

### Conclusion
Thus, the amplitude of the motion is approximately \( 529.15 \) centimeters.

### Final Answer
The amplitude of motion is approximately \( 529.15 \) centimeters.

---