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Chapter 10: Problem 13

Determine the angular speed, in rad/s, of (a) Earth about its axis; (b) the minute hand of a clock; (c) the hour hand of a clock; and (d) an eggbeater turning at 300 rpm.

Short Answer

Expert verified
The angular speed of (a) the Earth is \(7.27 \times 10^{-5}\) rad/s, (b) the minute hand of a clock is \(0.0001745\) rad/s, (c) the hour hand of a clock is \(1.45 \times 10^{-4}\) rad/s, and (d) an eggbeater turning at 300 rpm is \(10\pi\) rad/s.

Step by step solution

01

Calculate the angular speed of earth

The Earth completes one rotation every 24 hours. Hence the time period \( T \) of Earth is \( 24 \times 60 \times 60 = 86400 \) seconds. Now we apply the formula: \(\omega_{earth} = \frac{2\pi}{T} = \frac{2\pi}{86400} = 7.27 \times 10^{-5}\) rad/s.
02

Calculate the angular speed of a minute hand

A minute hand completes one rotation every 60 minutes (or 1 hour). Hence the time period \( T \) is \( 60 \times 60 = 3600 \) seconds. Now, substitute in the formula: \(\omega_{minute} = \frac{2\pi}{T} = \frac{2\pi}{3600} = 0.0001745\) rad/s.
03

Calculate the angular speed of an hour hand

An hour hand completes one rotation every 12 hours. Hence the time period \( T \) is \( 12 \times 60 \times 60 = 43200 \) seconds. Substituting in our formula gives: \(\omega_{hour} = \frac{2\pi}{T} = \frac{2\pi}{43200} = 1.45 \times 10^{-4}\) rad/s.
04

Calculate the angular speed of an eggbeater

The eggbeater is given as 300 rpm (rotations per minute), that is, \( 300 \times \frac{1}{60} = 5\) rotations per second. Hence, the angular speed of the eggbeater is \( \omega_{eggbeater} = 5 \times 2\pi = 10\pi \) rad/s.

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Most popular questions from this chapter

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?

A circular saw spins at 5800 rpm, and its electronic brake is supposed to stop it in less than 2 s. As a quality-control specialist, you're testing saws with a device that counts the number of blade revolutions. A particular saw turns 75 revolutions while stopping. Does it meet its specs?

You're an astronaut in the first crew of a new space station. The station is shaped like a wheel 22 m in diameter, with essentially all its \(5 \times 10^{5}-\mathrm{kg}\) mass at the rim. When the crew arrives, it will be set rotating at a rate that requires an object at the rim to have radial acceleration \(g\), thereby simulating Earth's surface gravity. This will be accomplished using two small rockets, each with \(100-\mathrm{N}\) thrust, mounted on the station's rim. Your job is to determine how long to fire the rockets and the number of revolutions the station will make during the firing.

A compact disc (CD) player varies the rotation rate of the disc in order to keep the part of the disc from which information is being read moving at a constant linear speed of \(1.30 \mathrm{m} / \mathrm{s}\) Compare the rotation rates of a 12.0 -cm-diameter CD when information is being read (a) from its outer edge and (b) from a point \(3.75 \mathrm{cm}\) from the center. Give your answers in \(\mathrm{rad} / \mathrm{s}\) and rpm.

The lower part of a horse's leg contains essentially no muscle. How does this help the horse to run fast? Explain in terms of rotational inertia.
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