(a)
The mass of a hypothetical spherical asteroid.
Answer to Problem 11Q
The mass of the hypothetical spherical asteroid is
Explanation of Solution
Given:
The diameter of the asteroid is
The average density of the asteroid is
Concept used:
The density of a body is defined as the total mass of it present in the total volume of the substance in the region of space.
Write the expression for the density of the body.
Rearrange the above expression for
Here,
Write the expression for the volume of the spherical asteroid.
Here,
Substitute
Calculation:
Substitute
Conclusion:
Thus, the mass of the hypothetical spherical asteroid is
(b)
The speed required to escape from the surface of the asteroid.
Answer to Problem 11Q
The escape velocity of the asteroid is
Explanation of Solution
Given:
The mass of the hypothetical spherical asteroid is
The diameter of the asteroid is
Concept used:
According to the law of conservation of mechanical energy, the total mechanical energy of the system is always a constant and therefore, the sum of the kinetic and the potential energy is also a constant, provided only conserved forces are applied.
Write the expression for conservation of mechanical energy as the sum of the kinetic and the potential energy.
Here,
Write the expression for kinetic energy.
Here,
Write the expression for the gravitational potential energy.
Here,
Substitute
Simplify the above expression.
Here,
As a body escapes with velocity, the only force on it is the gravitational force of the massive body and when it goes out of the gravitational pull, the force on it is nearly zero. Therefore, it is considered to be at infinity and final velocity
Substitute
Simplify and rearrange the above expression for
Here,
Substitute
Calculation:
Substitute
Conclusion:
Thus, the escape velocity of the asteroid is
(c)
The condition if an astronaut starts to jog on the surface of the asteroid with a velocity of
Answer to Problem 11Q
The astronaut will escape out from the gravitational field of the asteroid.
Explanation of Solution
Concept used:
The escape velocity of the asteroid is
Conclusion:
Thus, the astronaut will escape out from the gravitational field of the asteroid.
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Chapter 7 Solutions
UNIVERSE (LOOSELEAF):STARS+GALAXIES
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