Sunday, August 23, 2009

What's wrong with these physics problems?

"What's wrong with these physics problems?"*

Sometimes authors of introductory physics textbooks illustrate physics concepts or principles with real world examples to which they apply poorly, or not at all. Here are some examples. I may go into more detail about some of them later.

Examples

1) Treating a bow as an ideal spring over the entire distance of its draw.
Clue #1 It is actually hard to approximate an ideal spring for non-small displacements.
Clue #2 F = kx would be a poor design for a bow. In order to store the maximum amount of work in a bow, you want the force to be about as much as the archer is comfortable with applying. That means that you do not want F to be 0 when x is 0.

2) Assuming that an athlete should throw an object at an angle of 45 degrees to the horizontal to get maximum range.

It is true that, if the speed of a projectile is independent of its direction, and you can ignore air friction, and it lands at the same height at which it started, and the speed is small compared to the escape velocity, the range of a projectile is greatest when its initial direction is at 45 degrees to the horizontal.
Clue #1: Usually a human can throw an object faster in the forward direction. This is the main problem.
Clue #2 Sometimes air friction is important.

3) Assuming that air resistance is proportional to speed for, say, a car. If the “Reynold’s number” for the motion of the car is high enough, and for typical speeds of cars it always is high enough, the force of the air resistance will be approximately proportional to the square of the speed of the car.

Water waves are particular offenders. Introductory physics texts often use them as examples, because they are familiar and easy to see. However, water waves are quite complicated mathematically.

4) Drawing water waves as sinusoidal when their amplitude is not small compared to their wavelength.

The solution for the behavior of water waves that gives them as sinusoidal is only valid when the slope of the wave is small everywhere, which is only true when the amplitude is small compared to wavelength.

Apparently, when the amplitude is not small compared to the wavelength, the tops of the waves are pointier than the bottoms.

5) Calling water waves "transverse".
Clue #1: In the solution for a gravity wave of small amplitude in water, the particles move in ellipses. (If the water is deep compared to the wavelength, they move in circles.) So, water waves are neither transverse nor longitudinal. What they are is surface waves.

6) Trying to illustrate sonic booms with the wake of a boat.
Clue #1: The velocity of water waves depends on their wavelength.

7) Using Bernoulli's principle in examples to which it does not apply.

Bernoulli’s principle is conservation of mechanical energy applied to a “piece of fluid”.

Clue #1 If friction (viscosity) is important, the amount of mechanical energy of the piece of fluid won’t actually stay the same.

8) Problems about incandescent light bulbs that assume that their resistance stays the same while the current thru them varies.

Clue #1 If the current thru the wire in the incandescent bulb increases, so will the temperature of the wire. If the temperature of the wire increases, so will the resistance of the wire. Therefore, if you increase the current thru an incandescent bulb you increase the resistance of the bulb.

*This title was inspired by some essays of David Mermin.

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