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Sagot :
Let's solve this problem step-by-step.
### Given:
- The disk took 0.750 seconds to make the second complete revolution.
- The disk started from rest with a constant angular acceleration.
### Assumptions and Variables:
- Let's denote the angular acceleration as [tex]\( \alpha \)[/tex].
- Let the time taken to complete the first revolution be [tex]\( t_1 \)[/tex].
- Let the total time taken to complete the second revolution be [tex]\( t_2 = 0.750 \)[/tex] seconds.
- For a rotating object with constant angular acceleration starting from rest, the angle [tex]\( \theta \)[/tex] covered is given by:
[tex]\[ \theta = \frac{1}{2} \alpha t^2 \][/tex]
- One complete revolution corresponds to [tex]\( 2\pi \)[/tex] radians.
- Therefore, after the first and second revolutions, the total angles covered are [tex]\( 2\pi \)[/tex] and [tex]\( 4\pi \)[/tex] radians, respectively.
### Steps to Solution
#### Part (b): Determine the angular acceleration [tex]\( \alpha \)[/tex]
1. Equation for the second revolution:
[tex]\[ 4\pi = \frac{1}{2} \alpha t_2^2 \][/tex]
2. Plug in the given value for [tex]\( t_2 \)[/tex]:
[tex]\[ 4\pi = \frac{1}{2} \alpha (0.750)^2 \][/tex]
3. Solve for [tex]\( \alpha \)[/tex]:
[tex]\[ \alpha = \frac{4\pi}{0.5 \times (0.750)^2} \][/tex]
[tex]\[ \alpha \approx 44.68 \text{ rad/sec}^2 \][/tex]
#### Part (a): Determine the time to complete the first rotation
1. Equation for the first revolution:
[tex]\[ 2\pi = \frac{1}{2} \alpha t_1^2 \][/tex]
2. Substitute the value of [tex]\( \alpha \)[/tex] found in part (b):
[tex]\[ 2\pi = \frac{1}{2} \times 44.68 \times t_1^2 \][/tex]
3. Solve for [tex]\( t_1 \)[/tex]:
[tex]\[ t_1^2 = \frac{2\pi}{0.5 \times 44.68} \][/tex]
[tex]\[ t_1 \approx 0.530 \text{ sec} \][/tex]
### Summary of Results:
- The time to complete the first revolution ([tex]\( t_1 \)[/tex]): [tex]\( 0.530 \)[/tex] seconds.
- The angular acceleration ([tex]\( \alpha \)[/tex]): [tex]\( 44.68 \)[/tex] rad/sec².
Therefore, the detailed step-by-step solution concludes that the time to complete the first rotation is approximately 0.530 seconds, and the angular acceleration is approximately 44.68 rad/sec².
### Given:
- The disk took 0.750 seconds to make the second complete revolution.
- The disk started from rest with a constant angular acceleration.
### Assumptions and Variables:
- Let's denote the angular acceleration as [tex]\( \alpha \)[/tex].
- Let the time taken to complete the first revolution be [tex]\( t_1 \)[/tex].
- Let the total time taken to complete the second revolution be [tex]\( t_2 = 0.750 \)[/tex] seconds.
- For a rotating object with constant angular acceleration starting from rest, the angle [tex]\( \theta \)[/tex] covered is given by:
[tex]\[ \theta = \frac{1}{2} \alpha t^2 \][/tex]
- One complete revolution corresponds to [tex]\( 2\pi \)[/tex] radians.
- Therefore, after the first and second revolutions, the total angles covered are [tex]\( 2\pi \)[/tex] and [tex]\( 4\pi \)[/tex] radians, respectively.
### Steps to Solution
#### Part (b): Determine the angular acceleration [tex]\( \alpha \)[/tex]
1. Equation for the second revolution:
[tex]\[ 4\pi = \frac{1}{2} \alpha t_2^2 \][/tex]
2. Plug in the given value for [tex]\( t_2 \)[/tex]:
[tex]\[ 4\pi = \frac{1}{2} \alpha (0.750)^2 \][/tex]
3. Solve for [tex]\( \alpha \)[/tex]:
[tex]\[ \alpha = \frac{4\pi}{0.5 \times (0.750)^2} \][/tex]
[tex]\[ \alpha \approx 44.68 \text{ rad/sec}^2 \][/tex]
#### Part (a): Determine the time to complete the first rotation
1. Equation for the first revolution:
[tex]\[ 2\pi = \frac{1}{2} \alpha t_1^2 \][/tex]
2. Substitute the value of [tex]\( \alpha \)[/tex] found in part (b):
[tex]\[ 2\pi = \frac{1}{2} \times 44.68 \times t_1^2 \][/tex]
3. Solve for [tex]\( t_1 \)[/tex]:
[tex]\[ t_1^2 = \frac{2\pi}{0.5 \times 44.68} \][/tex]
[tex]\[ t_1 \approx 0.530 \text{ sec} \][/tex]
### Summary of Results:
- The time to complete the first revolution ([tex]\( t_1 \)[/tex]): [tex]\( 0.530 \)[/tex] seconds.
- The angular acceleration ([tex]\( \alpha \)[/tex]): [tex]\( 44.68 \)[/tex] rad/sec².
Therefore, the detailed step-by-step solution concludes that the time to complete the first rotation is approximately 0.530 seconds, and the angular acceleration is approximately 44.68 rad/sec².
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