A hole is drilled with smooth sides straight through the center of the earth to the other side of the earth. The air is removed from this tube (and the tube doesn't fill up with water, liquid rock or iron from the core). An object is dropped into one end of the tube and just reaches the opposite end. You can assume the earth is of uniform mass density. You can neglect the amount of mass drilled out and the rotation of the earth.
(a) The gravitational force on an object of mass m located inside the earth a distance r<re from the center (re is the radius of the earth) is due only to the mass of the earth that lies within a solid sphere of radius r . What is the magnitude of the gravitational force as a function of the distance r from the center of the earth? Express your answer in terms of the gravitational of the r, m, g, and re (enter r_e for re).
Note: you do not need the mass of the earth me or the universal gravitation constant G to answer this question but you will need to find an expression relating me and G to g and re .
(b) How long would it take for this object to reach the other side of the earth? Express your answer in terms of the gravitational constant at the surface of earth g, m, and re as needed (enter r_e for re).
http://ocw.mit.edu/courses/physics/8-01sc-physics-i-classical-mechanics-fall-2010/conservation-of-energy/mechanical-energy-and-the-simple-harmonic-oscillator/MIT8_01SC_problems15_soln.pdf
http://web.mit.edu/8.01t/www/materials/ProblemSets/Raw/ps11sol.pdf
a) ((m*g)/r_e)*r
b)pi*sqrt(r_e/g)
To answer part (a), we need to find the magnitude of the gravitational force on an object of mass m at a distance r from the center of the earth. We can use the formula for the gravitational force:
F = (G * (mass of object) * (mass of earth))/r^2
However, we don't have the mass of the earth or the universal gravitational constant G. But we can find a relationship between G and g, the acceleration due to gravity at the surface of the earth.
We know that the acceleration due to gravity at the surface of the earth is given by:
g = (G * (mass of earth))/re^2
Where re is the radius of the earth.
Rearranging this equation, we can find an expression for G:
G = (g * re^2) / (mass of earth)
Now we can substitute this expression for G into the formula for the gravitational force:
F = ((g * re^2) / (mass of earth) * m) / r^2
Simplifying this, we get:
F = (g * re^2 * m) / (r^2 * mass of earth)
So, the magnitude of the gravitational force as a function of the distance r from the center of the earth is given by:
F = (g * re^2 * m) / (r^2 * mass of earth)
Now, let's move on to part (b), which asks for the time it would take for the object to reach the other side of the earth.
The equation for the time it takes for an object to fall a certain distance under gravity is given by:
t = sqrt((2 * distance) / g)
In this case, the distance the object needs to fall is twice the radius of the earth (2 * re), as it starts from one end of the tube and needs to reach the other end.
So, substituting the distance into the equation, we get:
t = sqrt((2 * re) / g)
Therefore, the time it would take for the object to reach the other side of the earth is given by:
t = sqrt((2 * re) / g)
Note that this calculation assumes there is no air resistance and that the object is dropped without any initial velocity.