The displacement of a particle varies according to the relation x = 3 sin 100t + 8 `cos^(2)` 50t. Which of the following is/are correct about this motion.

Options

A. the motion of the particle is not S.H.M.
B. the amplitude of the S.H.M. of the particle is 5 units
C. the amplitude of the resultant S.H.M. is `sqrt(73)`units
D. the maximum displacement of the particle from the origin is 9 units.

Answer

To analyze the motion of the particle described by the displacement equation \( x = 3 \sin(100t) + 8 \cos^2(50t) \), we will follow these steps:

### Step 1: Rewrite the displacement equation
The given displacement equation is:
\[ x = 3 \sin(100t) + 8 \cos^2(50t) \]
We can use the trigonometric identity for \( \cos^2(\theta) \):
\[ \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \]
Applying this identity to \( \cos^2(50t) \):
\[ \cos^2(50t) = \frac{1 + \cos(100t)}{2} \]
Substituting this back into the equation gives:
\[ x = 3 \sin(100t) + 8 \left(\frac{1 + \cos(100t)}{2}\right) \]
This simplifies to:
\[ x = 3 \sin(100t) + 4(1 + \cos(100t)) = 3 \sin(100t) + 4 + 4 \cos(100t) \]

### Step 2: Combine sine and cosine terms
Now we can combine the sine and cosine terms:
\[ x = 4 + 3 \sin(100t) + 4 \cos(100t) \]
To combine \( 3 \sin(100t) + 4 \cos(100t) \), we can express it in the form \( R \sin(100t + \phi) \), where:
- \( R = \sqrt{A^2 + B^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \)
- \( \tan(\phi) = \frac{B}{A} = \frac{4}{3} \)

Thus, we can rewrite:
\[ x = 4 + 5 \sin(100t + \phi) \]

### Step 3: Analyze the motion
1. **Determine if the motion is SHM**: The equation \( x = 4 + 5 \sin(100t + \phi) \) represents simple harmonic motion (SHM) because it can be expressed in the standard form of SHM \( x = A \sin(\omega t + \phi) + C \), where \( C \) is a constant that shifts the equilibrium position.

2. **Find the amplitude**: The amplitude \( A \) is given by the coefficient of the sine term, which is \( 5 \).

3. **Maximum displacement from the origin**: The maximum displacement occurs when \( \sin(100t + \phi) = 1 \):
\[ x_{\text{max}} = 4 + 5 = 9 \]

### Summary of Results
- The motion of the particle **is SHM**.
- The **amplitude** of the SHM is **5 units**.
- The **maximum displacement** from the origin is **9 units**.