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Chapter 10: Problem 18

During startup, a power plant's turbine accelerates from rest at \(0.52 \mathrm{rad} / \mathrm{s}^{2} .\) (a) How long does it take to reach its 3600 -rpm operating speed? (b) How many revolutions does it make during this time?

Short Answer

Expert verified
a) It takes 725 seconds to reach its 3600-rpm operating speed. b) During this time, the turbine makes 21748 revolutions.

Step by step solution

01

Convert rotational speed to rad/s

Initially, the turbine's rotational speed is given in rpm (revolutions per minute) which needs to be converted into rad/s. Use this conversion formula: \( \omega = rpm * \frac{2\pi}{60} \). Substituting the given values, \( \omega = 3600 * \frac{2 \pi}{60} = 377 \, rad/s \)
02

Find time to reach operating speed

The time required to reach this speed can be found using the equation \( t = \frac{\omega}{\alpha} \) where \( \alpha \) is the angular acceleration and \( \omega \) is the final angular speed. Substituting the given and found values, it gives \( t = \frac{377}{0.52} = 725 \, sec \)
03

Find number of revolutions made

Since 1 revolution is equal to \( 2\pi \) rad, to calculate the number of revolutions, the angular displacement should be found first. For uniform angular acceleration, it is given by \( \theta = \alpha * t^2 / 2 \). Substituting the known values gives \( \theta = 0.52 * 725^2 / 2 = 136662 \, rad \). Now convert it to revolutions by dividing it by \( 2\pi \), which gives Number of revolutions = \( \frac{136662}{2\pi} = 21748 \, revolutions \)

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Most popular questions from this chapter

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