In the approximation that the Earth is a sphere of uniform density, it can be shown that the gravitational force it exerts on a mass m inside the Earth at a distance r from the center is mg(r/R), where R is the radius of the Earth. (Note that at the the surface and at the center, the force reduces to what we would expect.) Suppose that there were a hole drilled along a diameter straight through the Earth, and the air were pumped out of the hole. If an object is released from one end of the hole, find an expression for how long it will take to reach the other side of the Earth.
I don't really know what to do.
I know you can use to potential and kinetic energy equations to find velocity which would help find the period since T = 2*R/V (if V*T=2R)
But, I keep getting the answer wrong, solving it that way. It might be my use of the energy equations. I understand that PE is supposed to be highest at the surface and 0 at the center, since Fofg would be 0 because r from center would be 0. But where do I Go from there?
Do I use the F of gravity inside earth equation given for g in PE equation m*g*h?
What would happen if you fell into a hole that went completely through the center of the Earth and came out on the other side?
Ignoring the rotaion of the earth on its axis:
The problem of what would happen if you fell through a hole passing through the Earth's center was answerd by Galileo himself. Ignoring friction and air resistance, you would fall faster and faster until you reached your greatest speed, about 5 miles per second at the center of the Earth. While the pull of gravity actually gets weaker as you get closer to the center (less mass below you and gravity is a function of mass), the inertia of your falling body, plus the constant pull of the ever decreasing gravity, would cause your speed to continuously increase to the center of the Earth. Now, once past the center, your speed starts to decrease, as an ever increasing portion of the Earth is behind you, and exerting a stronger pull than the portion of the Earth ahead of you. Your velocity would reach zero just as you arrived at the other end of the hole on the surface. Unless you grabbed the edge of the hole, you would fall back down the hole, ultimately reaching your starting point. Under idealistic conditions, you would travel back and forth forever, making a round trip in about 84 minutes.
Ref: Puzzling Questions About the Solar System by Martin Gardner, Dover Publications, Inc., 1997
To solve this problem, we need to analyze the motion of the object as it falls through the hole.
First, let's consider the potential energy of the object at different points along its path. At the surface of the Earth, the potential energy (PE) is given by mgh, where m is the mass of the object, g is the acceleration due to gravity, and h is the height above the surface. As the object falls and reaches the center of the Earth, the height above the surface becomes zero, so the potential energy is also zero.
Next, we can determine the kinetic energy (KE) of the object at the surface and at the center of the Earth. At the surface, the kinetic energy is given by (1/2)mv^2, where v is the velocity of the object. At the center of the Earth, the velocity becomes zero, so the kinetic energy is also zero.
Since the sum of potential energy and kinetic energy is conserved throughout the motion, we can equate the initial potential energy at the surface to the final kinetic energy at the center. Therefore, we have:
mgh = (1/2)mv^2
Simplifying this equation, we find:
gh = (1/2)v^2
Now, let's find an expression for the velocity v. From the given information, we know that the gravitational force exerted on the object at a distance r from the center is given by mg(r/R), where R is the radius of the Earth. Using Newton's law of universal gravitation, we can equate this force to m*g, where g is the acceleration due to gravity at the surface of the Earth:
mg(r/R) = m*g
Simplifying this equation, we find:
(r/R) = 1
Since the radius of the Earth is much larger than the distance r, we can approximate r/R as zero. This implies that the gravitational force inside the Earth is zero, and the object experiences no net force. Therefore, the object will continue to move with a constant velocity of zero.
Based on this analysis, we can conclude that the object will take an infinite amount of time to reach the other side of the Earth through the hole.