Given that the efficiency of heating your room is roughly 35%, what mass of natural gas is actually necessary to heat your room
1 answer:
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Answer:
Explanation:
1) Notation and info given
represent the density at the center of the planet
represent the densisty at the surface of the planet
r represent the radius
represent the radius of the Earth
2) Solution to the problem
So we can use a model to describe the density as function of the radius
So we can create a linear model in the for y=b+mx, where the intercept b= and the slope would be given by
So then our linear model would be
Since the goal for the problem is find the gravitational acceleration we need to begin finding the total mass of the planet, and for this we can use a finite element and spherical coordinates. The volume for the differential element would be .
And the total mass would be given by the following integral
Replacing dV we have the following result:
We can solve the integrals one by one and the final result would be the following
Simplyfind this last expression we have:
And replacing the values we got:
And now that for any shape the gravitational acceleration is given by: