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).
(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).
a) ((m*g)/r_e)*r
b)pi*sqrt(r_e/g)
(a) The gravitational force on an object of mass m located inside the earth a distance r<re from the center is due only to the mass of the earth that lies within a solid sphere of radius r.
To find the magnitude of the gravitational force as a function of the distance r from the center of the earth, we can use the formula for gravitational force:
F = (G * m * M) / r^2
Where:
F is the gravitational force,
G is the gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2),
m is the mass of the object,
M is the mass of the earth, and
r is the distance from the center of the earth.
Since we are considering a solid sphere of radius r, the mass of the earth within this sphere can be calculated as:
M = (4/3) * π * ρ * r^3
Where:
ρ is the mass density of the earth.
Substituting this into the formula for gravitational force:
F = (G * m * (4/3) * π * ρ * r^3) / r^2
Simplifying the equation:
F = (4/3) * π * G * m * ρ * r
Now we have the magnitude of the gravitational force as a function of the distance r from the center of the earth expressed in terms of the gravitational constant (G), the mass of the object (m), the mass density of the earth (ρ), and the distance from the center of the earth (r).
(b) To calculate how long it would take for this object to reach the other side of the earth, we can use the formula for the gravitational acceleration near the surface of the earth:
g = (G * M) / r_e^2
Where:
g is the gravitational acceleration,
G is the gravitational constant,
M is the mass of the earth, and
r_e is the radius of the earth.
We can rearrange this equation to solve for the time it takes for the object to reach the other side:
t = √((2 * d) / g)
Where:
t is the time,
d is the diameter of the earth (2 * r_e), and
g is the gravitational acceleration.
Substituting the values into the formula:
t = √((2 * r_e) / ((G * M) / r_e^2))
Simplifying the equation:
t = √(2 * r_e^3 / (G * M))
Now we have the time it would take for the object to reach the other side of the earth expressed in terms of the gravitational constant (G), the mass of the earth (M), and the radius of the earth (r_e).