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Phy 202
Your 'question form' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
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3/2 10 am
I am struggling completing the worksheets from 2/16 and 2/23. Specifically,
How would the result change if the BB was fired into a slightly irregular tile-lined 'box' approximately 5 cm on a side?
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What does this question have to do with the kinetic theory of gases?
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If the box is irregular then the BB will be bouncing around in 3 dimensions. The average force on any given tile will become 1/3 as great as if it was bouncing back and forth between two parallel tiles.
In kinetic theory we started with a particle bouncing back and forth between the two walls, staying along the axis, and found the expression for the average force. Then we invoked a number of particles, which began colliding with one another so that the velocities became distributed equally over 3 directions of space.*@
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Optional problem: If the BB loses 10% of its momentum with each collision, what average force does it exert over the first 10 round trips? General College Physics students can use estimates, as can University Physics students. However University Physics students should consider applying calculus and/or differential equations.
@& You could make the first crude estimate that the BB loses most of its momentum in 10 trips, reducing its average speed over the 10 trips to, say, 50% of the original. This would reduce the frequency of collisions by 50%, and the momentum transferred with each collision by an equal amount, yielding 1/4 the force.
We can make a more detailed estimate.
With each round trip the BB's speed is reduced twice (two collisions per round trip) to .9 * (.9) = .81 of the previous value.
In two round trips it would be reduced to .81 * .81 = .64 of its original value.
In two more trips (4 round trips total) that figure would reduce to .64 * .64 = .40 of the original.
Another round trip would reduce it to .81 * .40 = .32 of the original. So in 5 round trips the momentum, and hence the speed, would be well below 50% of the original.
5 more trips would reduce this to about .1 of the original momentum.
The momentum vs. # of round trips curve therefore decreases at a decreasing rate, approaching zero and reaching about 10% of the original value after 10 round trips.
A reasonable estimate would be that the average momentum is about 1/3 of the original, leading to 1/9 of the average force.
More sophisticated estimates are possible (e.g., graph ave. force, rather than momentum, vs. number of collisions; calculus-literate students could set up an integral).
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and from 2/23
q002. A laser ray directed from the left parallel to the central axis of a lens, but 2 cm from the axis, is refracted by the lens and passes through the axis at a point 10 cm to the right of the lens. A second ray originates from a point 20 cm to the left of the lens, strikes the lens on the central axis and passes through with practically no deflection from its original path.
How far to the right of the lens will this ray intersect the first, and how far will it be from the axis when that occurs?
@& Sketch the picture and see what pairs of similar triangles emerge.
Consult also notes from 3/02 class, where this was done for the general case of an object at distance o and a lens with focal distance f.*@
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`q003. A laser ray enters a cylindrical lens of radius 10 cm, along a path parallel to but 2 cm from the central axis. It is deflected at that point toward a radial line, in such a way that its angle with the radial line is decreased by 50%. Will the ray get to the opposite surface of the mirror before it reaches the central axis? Where will it go then?
How do you determine where they intersect? We drew pictures in class, but I'm not sure how to answer.
@& The radial line from the center of the lens to the point where the ray strikes makes an angle with the central axis which is equal to the angle of the incident ray and the radial line.
Sketch a dotted line that continues the incident ray, without deflection, through the lens.
Look at the angle between that extended line and the radial line.
The refracted ray will split that angle in half.
Moving in the resulting direction, will the ray reach the other side of the cylinder before or after it reaches the central axis?
This can be answered by constructing the appropriate pairs of similar triangles.*@
@& Good questions.
See if my notes help.*@