
Concept explainers
(a)
The escape velocity on the earth.

Answer to Problem 9C
Escape velocity on earth is approximately
Explanation of Solution
Given:
Mass of the earth =
Radius of earth =
Universal gravitational constant =
Formula used:
The escape velocity is given by,
Calculation:
Substitute all given values in the above equation,
Conclusion:
Escape velocity of object from the surface of earth is
(b)
To prove:
The expression of Schwarzschild radius with the help of Newtonian mechanics is given by

Explanation of Solution
Given:
Mass of sphere =
Radius of sphere =
Mass of object = m.
Calculation:
By conservation of mechanical energy,
Loss of potential energy = gain in kinetic energy
Conclusion:
Thus, we have derived Schwarzschild radius using the concept of Newtonian mechanics.
(c)
The Schwarzschild radius of earth when escape velocity is the speed of light for any object. Also, compute the density of Earth and compare it with the current density of Earth.

Answer to Problem 9C
Schwarzschild radius of earth is
Explanation of Solution
Given:
Mass of the earth =
Current radius of earth =
Universal gravitational constant =
Speed of light =
Formula used:
Schwarzschild radius is defined as,
Calculation:
Substitute all given values in the above equation,
Now, density is defined as
Density =
Since, earth is in spherical so, volume of earth would be
So, density is defined as
Substituting the given values,
This is current density which is extremely very high to its initial value.
Conclusion:
Thus, Schwarzschild radius is
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