Answer:
the requested values are:
- Period T: 4 seconds
- Amplitude A : 6 inches
- Maximum height: 12 inches
- Minimum height: 6 inches
- Equation of the Midline: \(y = 9\) inches
Explanation:
To find the period, amplitude, maximum, minimum, and equation of the midline for the motion of the weight on the spring, we can use the given information:
1. **Period**:
The period T is the time taken for one complete cycle of motion. Since the weight takes 4 seconds to complete one full cycle (from its highest position to its lowest and back to its resting position), the period is T = 4 seconds.
2. **Amplitude**:
The amplitude is half the difference between the maximum and minimum positions. Given that the difference between the lowest and highest points is 12 inches, the amplitude is [tex]\(A = \frac{12}{2} = 6\) inches.[/tex]
3. **Maximum**:
The maximum height occurs when the weight is at its highest point. Since the weight was initially 6 inches high, and the amplitude was 6 inches, the maximum height is [tex]\(6 + 6 = 12\) inches.[/tex]
4. **Minimum**:
The minimum height occurs when the weight is at its lowest point. Since the difference between the lowest and highest points is 12 inches, and the amplitude is 6 inches, the minimum height is [tex]\(12 - 6 = 6\)[/tex] inches.
5. **Equation of the Midline**:
The midline represents the average position of the weight during its oscillation. It is located halfway between the maximum and minimum positions. Since the maximum height is 12 inches and the minimum height is 6 inches, the midline is [tex]\(6 + \frac{1}{2}(12 - 6) = 9\)[/tex]inches.
Therefore, the requested values are:
- Period (\(T\)): 4 seconds
- Amplitude (\(A\)): 6 inches
- Maximum height: 12 inches
- Minimum height: 6 inches
- Equation of the Midline: \(y = 9\) inches
