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Answer :
Sure, let's solve the problem step by step:
1. Identify the given information:
- Initial angular velocity ([tex]\( \omega_0 \)[/tex]) = 500 revolutions per minute (rpm)
- Final angular velocity ([tex]\( \omega_f \)[/tex]) = 200 revolutions per minute (rpm)
- Time taken ([tex]\( t \)[/tex]) = 4 seconds
2. Convert the time from seconds to minutes because the angular velocities are given in revolutions per minute.
- Time in minutes = 4 seconds ÷ 60 = [tex]\(\frac{4}{60}\)[/tex] minutes = [tex]\(\frac{1}{15}\)[/tex] minutes
3. Calculate the angular acceleration:
- Angular acceleration ( [tex]\( \alpha \)[/tex] ) is given by the formula:
[tex]\[
\alpha = \frac{\omega_f - \omega_0}{t}
\][/tex]
- Substituting the given values:
[tex]\[
\alpha = \frac{200 \, \text{rpm} - 500 \, \text{rpm}}{ \frac{1}{15} \, \text{minutes} }
\][/tex]
- Simplifying the numerator:
[tex]\[
200 \, \text{rpm} - 500 \, \text{rpm} = -300 \, \text{rpm}
\][/tex]
- Thus:
[tex]\[
\alpha = \frac{-300 \, \text{rpm}}{ \frac{1}{15} \, \text{minutes} } = -300 \, \text{rpm} \times 15 = -4500 \, \text{rpm}^2/\text{min}
\][/tex]
4. Calculate the number of revolutions made by the fan:
- The distance (in terms of revolutions) covered under uniformly decelerating motion can be found using the average velocity:
[tex]\[
\text{Average velocity} = \frac{\omega_0 + \omega_f}{2}
\][/tex]
- Substituting the given values:
[tex]\[
\text{Average velocity} = \frac{500 \, \text{rpm} + 200 \, \text{rpm}}{2} = 350 \, \text{rpm}
\][/tex]
- Now, the total number of revolutions made ([tex]\( \theta \)[/tex]) is the average velocity multiplied by the time in minutes:
[tex]\[
\theta = \text{Average velocity} \times t = 350 \, \text{rpm} \times \frac{1}{15} \, \text{minutes}
\][/tex]
- Simplifying:
[tex]\[
\theta = 350 \, \text{rpm} \times \frac{1}{15} = \frac{350}{15} = 23.\overline{3}\, \text{revolutions}
\][/tex]
Therefore, the angular acceleration is [tex]\(-4500 \, \text{rpm}^2/\text{min}\)[/tex], and the number of revolutions made by the fan in 4 seconds is approximately [tex]\(23.\overline{3}\)[/tex] revolutions.
1. Identify the given information:
- Initial angular velocity ([tex]\( \omega_0 \)[/tex]) = 500 revolutions per minute (rpm)
- Final angular velocity ([tex]\( \omega_f \)[/tex]) = 200 revolutions per minute (rpm)
- Time taken ([tex]\( t \)[/tex]) = 4 seconds
2. Convert the time from seconds to minutes because the angular velocities are given in revolutions per minute.
- Time in minutes = 4 seconds ÷ 60 = [tex]\(\frac{4}{60}\)[/tex] minutes = [tex]\(\frac{1}{15}\)[/tex] minutes
3. Calculate the angular acceleration:
- Angular acceleration ( [tex]\( \alpha \)[/tex] ) is given by the formula:
[tex]\[
\alpha = \frac{\omega_f - \omega_0}{t}
\][/tex]
- Substituting the given values:
[tex]\[
\alpha = \frac{200 \, \text{rpm} - 500 \, \text{rpm}}{ \frac{1}{15} \, \text{minutes} }
\][/tex]
- Simplifying the numerator:
[tex]\[
200 \, \text{rpm} - 500 \, \text{rpm} = -300 \, \text{rpm}
\][/tex]
- Thus:
[tex]\[
\alpha = \frac{-300 \, \text{rpm}}{ \frac{1}{15} \, \text{minutes} } = -300 \, \text{rpm} \times 15 = -4500 \, \text{rpm}^2/\text{min}
\][/tex]
4. Calculate the number of revolutions made by the fan:
- The distance (in terms of revolutions) covered under uniformly decelerating motion can be found using the average velocity:
[tex]\[
\text{Average velocity} = \frac{\omega_0 + \omega_f}{2}
\][/tex]
- Substituting the given values:
[tex]\[
\text{Average velocity} = \frac{500 \, \text{rpm} + 200 \, \text{rpm}}{2} = 350 \, \text{rpm}
\][/tex]
- Now, the total number of revolutions made ([tex]\( \theta \)[/tex]) is the average velocity multiplied by the time in minutes:
[tex]\[
\theta = \text{Average velocity} \times t = 350 \, \text{rpm} \times \frac{1}{15} \, \text{minutes}
\][/tex]
- Simplifying:
[tex]\[
\theta = 350 \, \text{rpm} \times \frac{1}{15} = \frac{350}{15} = 23.\overline{3}\, \text{revolutions}
\][/tex]
Therefore, the angular acceleration is [tex]\(-4500 \, \text{rpm}^2/\text{min}\)[/tex], and the number of revolutions made by the fan in 4 seconds is approximately [tex]\(23.\overline{3}\)[/tex] revolutions.
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