To solve the problem, we need to analyze the given equation of motion for the particle:
### Given Equation:
\[ z = 4 + 12 \cos(2\pi t + \frac{\pi}{2}) \]
### Step 1: Identify the form of the equation
The equation can be rewritten in the form of simple harmonic motion (SHM):
\[ z - 4 = 12 \cos(2\pi t + \frac{\pi}{2}) \]
### Step 2: Convert cosine to sine
Using the trigonometric identity:
\[ \cos(x + \frac{\pi}{2}) = -\sin(x) \]
we can rewrite the equation as:
\[ z - 4 = -12 \sin(2\pi t) \]
or
\[ z = 4 - 12 \sin(2\pi t) \]
### Step 3: Determine the mean position
The mean position (equilibrium position) of the motion is the constant term in the equation:
\[ \text{Mean position} = 4 \, \text{cm} \]
### Step 4: Determine the amplitude
The amplitude \( A \) of the SHM is the coefficient of the sine function:
\[ A = 12 \, \text{cm} \]
### Step 5: Determine the extreme positions
The extreme positions (maximum and minimum) can be found by evaluating the maximum and minimum values of the sine function, which varies between -1 and 1:
- Maximum position:
\[ z_{\text{max}} = 4 - 12(-1) = 4 + 12 = 16 \, \text{cm} \]
- Minimum position:
\[ z_{\text{min}} = 4 - 12(1) = 4 - 12 = -8 \, \text{cm} \]
### Summary of Findings:
1. The mean position is at \( z = 4 \, \text{cm} \).
2. The amplitude of SHM is \( 12 \, \text{cm} \).
3. The extreme positions are \( z = 16 \, \text{cm} \) and \( z = -8 \, \text{cm} \).
### Conclusion:
Based on the analysis:
- The first option (mean position at \( z = 5 \, \text{cm} \)) is incorrect; it should be \( z = 4 \, \text{cm} \).
- The second option (extreme positions at \( z = -7 \, \text{cm} \) and \( z = 17 \, \text{cm} \)) is incorrect; the correct extreme positions are \( z = -8 \, \text{cm} \) and \( z = 16 \, \text{cm} \).
- The third option (amplitude of SHM is \( 13 \, \text{cm} \)) is incorrect; the amplitude is \( 12 \, \text{cm} \).
- The fourth option (amplitude of SHM is \( 12 \, \text{cm} \)) is correct.
### Correct Alternatives:
- The correct options are the fourth option (amplitude of SHM is \( 12 \, \text{cm} \)).
