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Chapter 12: Q. 10 (page 330)
What is the rotational kinetic energy of the earth? Assume the earth is a uniform sphere. Data for the earth can be found inside the back cover of the book.
The rotational kinetic energy of the earth is
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Evaluate the cross products and . (part )
During most of its lifetime, a star maintains an equilibrium size in which the inward force of gravity on each atom is balanced by an outward pressure force due to the heat of the nuclear reactions in the core. But after all the hydrogen “fuel” is consumed by nuclear fusion, the pressure force drops and the star undergoes a gravitational collapse
until it becomes a neutron star. In a neutron star, the electrons and protons of the atoms are squeezed together by gravity until they fuse into neutrons. Neutron stars spin very rapidly and emit intense pulses of radio and light waves, one pulse per rotation. These “pulsing
stars” were discovered in the 1960s and are called pulsars.
a. A star with the mass M = 2.0 x 1030 kg and size R =7.0 x 108 m of our sun rotates once every 30 days. After undergoing gravitational collapse, the star forms a pulsar that is observed by astronomers to emit radio pulses every 0.10 s. By treating the neutron star as a solid sphere, deduce its radius.
b. What is the speed of a point on the equator of the neutron star? Your answers will be somewhat too large because a star cannot be accurately modeled as a solid sphere. Even so, you will be able to show that a star, whose mass is 106larger than the earth’s, can be compressed by gravitational forces to a size smaller than a typical state in the United States!
How fast, in rpm, would a diameter bowling ball have to spin to have an angular momentum of ?
Flywheels are large, massive wheels used to store energy. They can be spun up slowly, then the wheel’s energy can be released quickly to accomplish a task that demands high power. An industrial flywheel has a 1.5 m diameter and a mass of 250 kg. Its maximum angular velocity is 1200 rpm.
a. A motor spins up the flywheel with a constant torque of 50 N m. How long does it take the flywheel to reach top speed?
b. How much energy is stored in the flywheel?
c. The flywheel is disconnected from the motor and connected to a machine to which it will deliver energy. Half the energy stored in the flywheel is delivered in 2.0 s. What is the average power delivered to the machine?
d. How much torque does the flywheel exert on the machine?
A rod of length L and mass M has a nonuniform mass distribution. The linear mass density (mass per length) is λ= cx2, where x is measured from the center of the rod and c is a constant.
a. What are the units of c?
b. Find an expression for c in terms of L and M.
c. Find an expression in terms of L and M for the moment of inertia of the rod for rotation about an axis through the center.
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