3. A model of Earth’s interior: Look up the radius and mass of the Earth.
Radius is 6,371 km
Mass is 5.974 x 10^24 kg
Calculate its average density.
p_e = (5.974 x 10^24 kg)/ (4/3)pi r^3 =
(5.974 x 10^24 kg)/ (4/3)pi (6371)^3 = 5.54 g/cm^3
Imagine a planet of the same radius made completely and uniformly out of an incompressible fluid such that its mass density is equal to the average density of the Earth.
Find how the acceleration of gravity depends on the distance from the center of the planet. Sketch a graph showing this dependence.
I know the dependence would be a straight line but i'm not sure how to prove in mathematically
Now, use the equation of hydrostatic equilibrium to find the pressure as a function of distance from the center. What is the pressure at the center of the planet? Express your answer in Atmospheres.
Any help on the last 2 parts would be great! thanks in advance.
good luck
To find how the acceleration of gravity depends on the distance from the center of the planet, imagine a small mass located at a distance r from the center of the planet. The gravitational force acting on this mass can be given by Newton's law of gravitation:
F = (G * m * M) / r^2
Where:
F is the gravitational force
G is the gravitational constant (6.67 x 10^-11 N m^2 / kg^2)
m is the mass of the object
M is the mass of the planet
r is the distance between the object and the center of the planet
The acceleration due to gravity (g) at a distance r from the center of the planet can be calculated by dividing the force by the mass of the object:
g = F / m = (G * M) / r^2
Since we are considering a planet of the same radius made completely and uniformly out of an incompressible fluid such that its mass density is equal to the average density of the Earth (5.54 g/cm^3), we can substitute the average density formula into the equation:
p = M / (4/3)pi r^3
Simplifying, we have:
M = p * (4/3)pi r^3
Substituting this into the gravity equation:
g = (G * p * (4/3)pi r^3) / r^2
g = (4/3)G * p * pi * r
Therefore, the dependence of acceleration of gravity on the distance from the center of the planet is a linear relationship (directly proportional). As the distance from the center increases, the acceleration of gravity also increases.
To sketch a graph showing this dependence, you can plot the acceleration of gravity (g) on the y-axis and the distance from the center of the planet (r) on the x-axis.
Moving on to the next part, let's use the equation of hydrostatic equilibrium to find the pressure as a function of distance from the center. In hydrostatic equilibrium, the pressure at any point within the planet is balanced by the weight of the material above it.
The equation of hydrostatic equilibrium is given by:
dp/dr = -(G * p * m) / r^2
Where:
dp/dr is the rate of change of pressure with respect to distance (change in pressure per unit distance)
G is the gravitational constant
p is the pressure
m is the mass enclosed within a radius r
r is the distance from the center of the planet
Since we are assuming that the planet is made completely and uniformly out of an incompressible fluid, its density (p) is constant. Therefore, dp/dr can be simplified to:
dp/dr = - (G * p * m) / r^2
Integrating both sides of the equation, we can solve for pressure as a function of distance (r):
∫dp = -∫(G * p * m) / r^2 dr
Integrating, we have:
p = - (G * m) / r + C
Where C is the constant of integration.
Now, to find the pressure at the center of the planet (r = 0), we substitute r = 0 into the equation:
p(0) = - (G * m) / 0 + C
Since the denominator becomes zero, the equation is undefined at r = 0. This means that the pressure at the center of the planet is undefined or infinite.
Note: The above calculations assume a simplified model of a planet and make certain assumptions. In reality, the Earth's interior is more complex and includes solid layers, molten layers, and variations in density. However, the simplified model is useful for demonstrating the concepts.