The displacement of a particle performing S.H.M. is, x = 3 sin (314t) + 4 cos (314t), where x and t are in CGS units. Then the amplitude of the S.H.M. is,

Options

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

Explanation

To find the amplitude of the simple harmonic motion (S.H.M.) given the displacement equation \( x = 3 \sin(314t) + 4 \cos(314t) \), we can follow these steps:

### Step 1: Identify the coefficients
The given equation can be rewritten in the form:
- \( A_1 = 3 \) (coefficient of \( \sin(314t) \))
- \( A_2 = 4 \) (coefficient of \( \cos(314t) \))

### Step 2: Use the formula for resultant amplitude
The resultant amplitude \( A \) when two waves are combined can be calculated using the formula:
\[ A = \sqrt{A_1^2 + A_2^2} \]
This formula is applicable because the sine and cosine functions are orthogonal (i.e., they are at a phase difference of \( \frac{\pi}{2} \)).

### Step 3: Substitute the values into the formula
Now, substitute the values of \( A_1 \) and \( A_2 \) into the formula:
\[ A = \sqrt{3^2 + 4^2} \]

### Step 4: Calculate the squares
Calculate \( 3^2 \) and \( 4^2 \):
\[ 3^2 = 9 \]
\[ 4^2 = 16 \]

### Step 5: Add the squares
Now add these values:
\[ A = \sqrt{9 + 16} = \sqrt{25} \]

### Step 6: Calculate the square root
Finally, calculate the square root:
\[ A = 5 \]

### Conclusion
Thus, the amplitude of the S.H.M. is \( 5 \) cm.

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