Answer:
The answers to the questions are;
1. The amount of energy required to move a mass m from the center of the Earth to the surface is 0.5·m·g·R
2. The amount of energy required to move the mass from the surface of the planet to a very large distance away m·g·R.
3. The speed of an object of mass m, dropped into this hole, when it reaches the center of the planet is 9682.41783 m/s.
Explanation:
We note that the Work done W by the force F on the mass to move a small distance is given by
F×dr
The sum of such work to move the body to a required location is
W =
F = mg' = m
We integrate from 0 to R (the center to the Earth surface)
Therefore W =
Which gives W =
2. To find the work done we have to integrate from the surface to infinity as follows W = =
The energy required to move the object to a large distance is equal to twice the energy reqired to move the object to the surface.
3 We note that the acceleration due to gravity at the surface is g and reduces to zero at the center of the Earth
v² = u² + 2·g·s
Radius of the Earth = 6371 km
From surface to half radius we have
v₁² = 2×9.81×6371/2×1000 = 62499460.04
v₁ = 7905.66 m/s
From the half the radius of the earth to the Earth center =
v₂² = 7905.66² + 2×9.81/2×6371/2×1000 = 93749215.04
v₂ = 9682.41783 m/s
The speed of an object of mass m, dropped into this hole, when it reaches the center of the planet. is 9682.41783 m/s