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).
a) ((m*g)/r_e)*r
b)pi*sqrt(r_e/g)
(a) To find the magnitude of the gravitational force on an object of mass m located inside the earth at a distance r from the center, we can use the concept of gravitational force within a sphere. The mass within a solid sphere of radius r is proportional to the volume of the sphere.
Let's derive an expression for the mass within the solid sphere of radius r. The density of the earth is considered to be uniform, so the mass within a sphere of radius r is equal to the density multiplied by the volume of the sphere:
Mass within the sphere of radius r = Density × Volume of the sphere
= Density × (4/3)πr³
Considering the total mass of the earth as a uniform sphere with radius re, the mass of the entire earth is:
Mass of the earth = Density × Volume of the earth
= Density × (4/3)πre³
Now, let's find the proportion of the mass within the solid sphere of radius r with respect to the mass of the entire earth:
Mass within the sphere of radius r / Mass of the earth = (Density × (4/3)πr³) / (Density × (4/3)πre³)
Canceling the density and the constant factors, we get:
Mass within the sphere of radius r / Mass of the earth = r³ / re³
Since the gravitational force is directly proportional to the mass, the gravitational force on the object at a distance r from the center of the earth is also proportional to r³ / re³. Let's now introduce the acceleration due to gravity on the surface of the earth, g. At the surface of the earth, the gravitational force is simply mg, where m is the mass of the object. Therefore, we can relate g, re, and the gravitational force at a distance r as follows:
g / (r_e)² = (r³ / (r_e)³)
Multiplying both sides of the equation by m:
g × m / (r_e)² = (r³ / (r_e)³) × m
Now, we have an expression for the gravitational force as a function of the distance r from the center of the earth:
Gravitational force = (g × m / (r_e)²) × (r³ / (r_e)³)
(b) The time it takes for an object to fall to the other side of the earth can be determined using the equations of motion. Assuming there is no air resistance and neglecting the rotation of the earth, the object undergoes free fall under gravity.
The distance the object needs to travel is twice the radius of the earth, since it starts at one end of the tunnel and reaches the other end. Therefore, the distance is 2re.
Using the equation of motion for free fall, we have:
Distance = (1/2) × acceleration × time²
Substituting the distance with 2re and the acceleration with g, we get:
2re = (1/2) × g × time²
Simplifying the equation, we have:
time² = (4re / g)
Finally, taking the square root of both sides, we get:
time = √(4re / g)
So, the time it would take for the object to reach the other side of the earth is given by the square root of 4 times the radius of the earth divided by the acceleration due to gravity on the surface of the earth.