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
To answer part (a), we need to determine the magnitude of the gravitational force as a function of the distance r from the center of the earth.
We know that the gravitational force acting on an object is given by the formula:
F = G * (M * m) / r^2
where F is the gravitational force, G is the universal gravitational constant, M is the mass of the object attracting the other object, m is the mass of the object being attracted, and r is the distance between the centers of the two objects.
In this case, we are considering an object located inside the earth, so we can assume that the mass M is the mass of the earth within a solid sphere of radius r.
Now, let's derive an expression for the mass of the earth within a solid sphere of radius r:
The earth's mass density is assumed to be uniform, so the mass contained within a sphere of radius r (M_r) is proportional to the volume of the sphere:
M_r = d * V_r
where d is the mass density and V_r is the volume of the sphere.
We know that the volume of a sphere is given by:
V_r = (4/3) * π * r^3
Substituting this back into the previous equation, we get:
M_r = d * (4/3) * π * r^3 ----(1)
To find an expression relating the mass of the earth (M_e) and the universal gravitational constant (G) to the surface gravity (g) and the radius of the earth (r_e), we need to relate the mass contained within the entire earth to its surface gravity.
The surface gravity is given by:
g = G * M_e / r_e^2
Rearranging this formula to solve for M_e, we have:
M_e = (g * r_e^2) / G ----(2)
Now, we can substitute equation (1) into equation (2) to get an expression relating G, M_e, and g to r_e:
M_r = (g * r_e^2 * d * (4/3) * π * r^3) / G
The gravitational force on the object located inside the earth at a distance r from the center will be given by:
F = G * (M_r * m) / r^2
Substituting equation (1) into this equation, we have:
F = G * (d * (4/3) * π * r^3 * m) / r^2
Simplifying this expression, we get:
F = (4/3) * π * G * d * r * m
Therefore, the magnitude of the gravitational force as a function of the distance r from the center of the earth is (4/3) * π * G * d * r * m.
Now let's move on to part (b).
To determine the time it would take for the object to reach the other side of the earth, we can use the equation of motion for vertical motion under gravity:
h = (1/2) * g * t^2
where h is the distance traveled (which is equal to twice the radius of the earth), g is the acceleration due to gravity, and t is the time taken.
In this case, h is equal to 2 * r_e and g is the surface gravity.
Solving the equation for t, we get:
t = sqrt((2 * h) / g)
Substituting the value for h and g, we have:
t = sqrt((2 * 2 * r_e) / g) = sqrt((4 * r_e) / g)
Therefore, the time it would take for the object to reach the other side of the earth is sqrt((4 * r_e) / g).