The displacement of a particle is given by ` x = 3 sin ( 5 pi t) + 4 cos ( 5 pi t)`. The amplitude of particle is

Options

A. `3`
B. `4`
C. `5`
D. `7`

Explanation

To find the amplitude of the particle given the displacement equation \( x = 3 \sin(5 \pi t) + 4 \cos(5 \pi t) \), we can follow these steps:

### Step 1: Identify the components
The displacement equation consists of two components:
- \( x_1 = 3 \sin(5 \pi t) \)
- \( x_2 = 4 \cos(5 \pi t) \)

### Step 2: Convert the cosine term to sine
We can express the cosine term in terms of sine to facilitate the addition:
\[ x_2 = 4 \cos(5 \pi t) = 4 \sin\left(5 \pi t + \frac{\pi}{2}\right) \]
This shows that \( x_2 \) is a sine function with a phase shift of \( \frac{\pi}{2} \).

### Step 3: Use the principle of superposition
Since both components have the same frequency, we can use the principle of superposition. The resultant displacement can be treated as a vector addition of the two amplitudes.

### Step 4: Calculate the resultant amplitude
The resultant amplitude \( A \) can be calculated using the formula:
\[ A = \sqrt{A_1^2 + A_2^2 + 2 A_1 A_2 \cos(\phi)} \]
where:
- \( A_1 = 3 \) (amplitude of the sine component)
- \( A_2 = 4 \) (amplitude of the cosine component)
- \( \phi = 90^\circ \) (phase difference between sine and cosine)

Since \( \cos(90^\circ) = 0 \), the formula simplifies to:
\[ A = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 5: Conclusion
Thus, the amplitude of the particle is \( 5 \).