The displacement of a particle performing simple harmonic motion is given by, `x=8 "sin" "omega`t + 6 cos `omega`t, where distance is in cm and time is in second. What is the amplitude of motion?

Options

A. (a)10 cm
B. (b)2 cm
C. (c)14 cm
D. (d)3.5 cm

Answer

To find the amplitude of the motion for the given displacement equation of a particle performing simple harmonic motion, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Displacement Equation**:
The displacement of the particle is given by:
\[ x = 8 \sin(\omega t) + 6 \cos(\omega t) \]

2. **Recognize the Form of the Equation**:
The general form of the displacement in simple harmonic motion can be expressed as:
\[ x = A_1 \sin(\omega t) + A_2 \cos(\omega t) \]
where \( A_1 \) and \( A_2 \) are the coefficients of sine and cosine respectively.

3. **Extract the Coefficients**:
From the given equation, we can identify:
- \( A_1 = 8 \) cm (coefficient of \( \sin(\omega t) \))
- \( A_2 = 6 \) cm (coefficient of \( \cos(\omega t) \))

4. **Calculate the Amplitude**:
The amplitude \( A \) of the motion can be calculated using the formula:
\[ A = \sqrt{A_1^2 + A_2^2} \]
Substituting the values of \( A_1 \) and \( A_2 \):
\[ A = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \text{ cm} \]

5. **Conclusion**:
Therefore, the amplitude of the motion is:
\[ A = 10 \text{ cm} \]