Concept explainers
Two astronauts (Fig. P10.67), each having a mass of 75.0 kg, are connected by a 10.0-m rope of negligible mass. They are isolated in space, orbiting their center of mass at speeds of 5.00 m/s. Treating the astronauts as particles, calculate
- (a) the magnitude of the
angular momentum of the two-astronaut system and - (b) the rotational energy of the system. By pulling on the rope, one astronaut shortens the distance between them to 5.00 m.
- (c) What is the new angular momentum of the system?
- (d) What are the astronauts’ new speeds?
- (e) What is the new rotational energy of the system?
- (f) How much chemical potential energy in the body of the astronaut was converted to mechanical energy in the system when he shortened the rope?
Figure P10.67 Problems 67 and 68.
(a)
The magnitude of angular momentum of the two astronaut system.
Answer to Problem 67P
The magnitude of angular momentum of the two astronaut system is
Explanation of Solution
Consider the figure given below.
Write the expression for the magnitude of angular momentum.
Here,
From figure
Conclusion:
Substitute,
Therefore, the magnitude of angular momentum of the two astronaut system is
(b)
The rotational kinetic energy of the system.
Answer to Problem 67P
The rotational kinetic energy of the system is
Explanation of Solution
The total rotational kinetic energy is the sum of the kinetic energy of two astronauts.
Write the expression for the rotational kinetic energy.
Here,
Conclusion:
The mass, and speed of two astronauts are same.
Substitute,
Therefore, the rotational kinetic energy of the system is
(c)
The angular momentum when one of the astronaut shortens the distance between them to
Answer to Problem 67P
The angular momentum when one of the astronaut shortens the distance between them to
Explanation of Solution
Even if the distance between the astronauts changed, the tension of the rope not generating ant torque about the center of mass. Since there is no change in torque the angular momentum of two astronaut –rope system will be same as that of the initial case
Since there is no outside torque the angular momentum when one of the astronaut shortens the distance between them to
Conclusion:
Therefore, the angular momentum when one of the astronaut shortens the distance between them to
(d)
The speed of the astronauts after shortening the distance.
Answer to Problem 67P
The speed of the astronauts after shortening the distance is
Explanation of Solution
Use equation (III) to find the new speed of the astronauts. The angular momentum of the system remains same even if he distance between the astronauts changes.
Conclusion:
Substitute,
Therefore, the speed of the astronauts after shortening the distance is
(f)
The new rotational kinetic energy of the system.
Answer to Problem 67P
The new rotational kinetic energy of the system is
Explanation of Solution
The total rotational kinetic energy is the sum of the kinetic energy of two astronauts.
Write the expression for the rotational kinetic energy.
Here,
Conclusion:
The mass, and speed of two astronauts are same.
Substitute,
Therefore, the rotational kinetic energy of the system is
(e)
The amount of chemical; energy converted to mechanical energy.
Answer to Problem 67P
The amount of chemical; energy converted to mechanical energy is
Explanation of Solution
The amount of chemical; energy converted to mechanical energy is equal to the work done by the astronaut. According to work energy theorem the work dine will be the change in rotational kinetic energy.
Write the expression for work done.
Conclusion:
Substitute,
Therefore, the amount of chemical; energy converted to mechanical energy is
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Chapter 10 Solutions
Principles of Physics: A Calculus-Based Text
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