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).
Understand that tutors won't do your work for you.
You need to indicate exactly what you have done to solve each problem and where you're running into trouble.
To answer the first question, we need to determine the magnitude of the gravitational force on an object at a distance r from the center of the Earth. We can use the equation for gravitational force:
F = G * (m1 * m2) / r^2
In this case, m1 represents the mass of the Earth, and m2 represents the mass of the object. Since we are neglecting the amount of mass drilled out and the rotation of the Earth, we can consider the mass of the Earth to be concentrated within a solid sphere of radius r (where r is less than the radius of the Earth, re).
The mass of the Earth within this solid sphere can be calculated using the equation:
m1 = density * V
where density is the mass density of the Earth (assumed to be uniform), and V is the volume of the solid sphere. The volume of a sphere can be calculated as:
V = (4/3) * π * r^3
Combining these equations along with the expression relating me (mass of the Earth) and G to g (gravitational constant at the surface of Earth) and re (radius of the Earth), we can find the magnitude of the gravitational force as a function of r:
F = (4/3) * π * G * (density * r^3 * m) / r^2
Simplifying this equation further:
F = (4/3) * π * G * density * r * m
Therefore, the magnitude of the gravitational force as a function of distance r can be expressed as:
F = (4/3) * π * G * density * r * m
Moving on to the second question, we need to find the time it would take for the object to reach the other side of the Earth through the drilled hole. This can be calculated using the laws of motion. The equation we'll use is:
d = (1/2) * g * t^2
where d is the distance traveled by the object (equal to 2 * re since it's traveling through the Earth's diameter), g is the acceleration due to gravity (equal to g at the surface of the Earth), and t is the time taken.
Rearranging the equation to solve for t:
t^2 = (2 * d) / g
Substituting the value of d as 2 * re, we have:
t^2 = (2 * 2 * re) / g
Simplifying further:
t^2 = (4 * re) / g
Taking the square root of both sides:
t = sqrt((4 * re) / g)
Therefore, the time it would take for the object to reach the other side of the Earth can be expressed as:
t = sqrt((4 * re) / g)