Seed Question: What would your acceleration due to gravity be if you were released out in space a distance r from the center of the Earth? Show your work, starting from Newt’s Law of Universal Gravity.
Exploration: Is there a way to find the acceleration due to gravity simply in highly symmetrical cases without having to integrate? Yes. But first we need to talk about the gravitational field. The gravitational field is simply the acceleration due to gravity.
We also need the idea of flux. The flux is the product of the penetrating gravitational field and the area A.
What is the gravitational flux through a horizontal piece of paper of area A in this room? Sketch it!
Now tilt the paper so one edge is vertical. What is the flux now?
How would you find the flux at some intermediate angle? Define the area vector A to be perpendicular to the plane of the sheet. We can’t make the area vector in the plane of the sheet because that would allow for an infinite number of directions.
The flux is defined to be: Φ= ag∙A
Note that this dot product ensures that we are talking about penetration of the gravitational field.
Now tilt one axis of the paper at an angle q from the horizontal. Sketch it! Find the flux.
Gauss came up with an elegant way to find the gravitational acceleration. Look at the Big Idea. Apply Gauss’ Law to the Seed Question. Do your results agree?
Apply Gauss’ Law inside and outside a spherical shell. Do your results agree with yesterday’s PJ entry findings?
Big Idea: Gauss’ Law of Gravity for use in cases of high symmetry:
ag∙dA=-4πGmenclosed
The circle on the integral means that you must integrate over a closed area. The direction of dA is perpendicular to the plane of dA and points out of the enclosed volume.
The Flux Φ= ag∙dA
Discussion: ?