Giancoli 7th Edition textbook cover
Giancoli's Physics: Principles with Applications, 7th Edition
28
Quantum Mechanics of Atoms
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28-2: Wave Function, Double-Slit
28-3: Uncertainty Principle
28-6 to 28-8: Quantum Numbers, Exclusion Principle
28-9: X-Rays
28-11: Lasers

Question by Giancoli, Douglas C., Physics: Principles with Applications, 7th Ed., ©2014, Reprinted by permission of Pearson Education Inc., New York.
Problem 2
Q

Pellets of mass 2.0 g are fired in parallel paths with speeds of 120 m/s through a hole 3.0 mm in diameter. How far from the hole must you be to detect a 1.0-cm-diameter spread in the beam of pellets?

A
4.5×1027 m4.5 \times 10^{27} \textrm{ m}
Giancoli 7th Edition, Chapter 28, Problem 2 solution video poster
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COMMENTS
By kbick on Sun, 5/7/2017 - 6:33 PM

Hi! Just wondering when we can use the small angle approximation - you say here that it is when theta is less than zero, but how can that be?

By Mr. Dychko on Sat, 5/13/2017 - 8:03 PM

Hi kbick, this is a good question since the small angle approximation is often a very useful trick to make the algebra much easier. Yes, I know it seems unbelievable that you can just replace sin(x)sin(x) with simply xx, so what you should do to convince yourself that it's acceptable when xx is small (I'm saying xx, but θ\theta, or whatever variable represents the angle in your equation) is this: plot the graph of y=xy=x and y=sin(x)y=sin(x) on the same graph, but there's a catch. Zoom in on the graph so that you can only see xx between, say, -1 and 1. You'll notice the graphs look the same! This is the reason the approximation works, is that within this restricted domain of 1x1-1 \leq x \leq 1, both functions y=xy=x and y=sin(x)y=sin(x) give the same result. When you zoom out and look at larger values of xx, then of course the graphs do not look the same, and you can no longer claim that xsin(x)x \approx sin(x), like you can for small xx.

All the best,
Mr. Dychko

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