Fall Semester Examination

December 6, 1996

Time: 2 1/2 hours

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Question 1. (12 %) Give short, concise answers in words.

(a) (3 %) What is the uncertainty principle?

(b) (3 %) What are the postulates of special relativity?

(c) (6 %) Define interference and diffraction of waves and why did Feynman say:

"no one has ever been able to define the difference between interference and diffraction

satisfactorily"?

Question 2. (20%) An object of mass 2kg resting on a frictionless horizontal surface is attached to a spring of force constant 600 N/m. A second object of mass 1 kg slides along the surface towards the first object at 6 m/s.

(a) find the amplitude of oscillation if the objects make a perfectly inelastic collision and remain together on the spring. What is the period of oscillation?

(b) Find the amplitude and period if the collision is elastic.

(c) For each type of collision, write an expression for the position x as a function of time for the object attached to the spring, assuming that the collision occurs at time t=0. What is the impulse delivered to the 2kg object in this case?

Question 3. (16%) When a bicycle rider accelerates, she must accelerate her own and her bicycle’s linear motion as well as the angular motion of the wheels. Suppose the cyclist has mass 55 kg; the bicycle (not counting wheels), 8 kg; and both wheels together, 1.8kg. Assume the wheels, each of radius 30 cm, have all their mass concentrated in the (thin) rim.

(a) At 25 km/h, what fraction of the kinetic energy of the rider plus the bicycle is in linear motion and what fraction is in rotational motion?

(b) Suppose the cyclist loses 3 kg on a diet. What is percentage of the original force is required to accelerate the system uniformly from 0 km/h to 25 km/h in 10 s?

(c) Suppose instead of going on a diet, the cyclist replaces her wheels with ones of total mass 1.2 kg. Now what is the percentage of the original force required to accelerate the system uniformly from 0 km/h to 25 km/h in 10 s?

(a) From the spectrum determine how the clarinet essentially functions (i.e.,what type of resonant tube is it: open at both ends, closed at both ends, closed at one end, etc.?). Show quantitative reasoning in your answer (no marks awarded for a "guess"). Draw a simple figure representing the standing waves for the first 2 harmonics in a clarinet (include the location of the reed).

(b) Determine the length of the clarinet in cm.

(space for figure)

Question 5. (16%) Consider a transverse wave that moves to the right on a string in the form of a Gaussian function, namely:

By taking the time derivative compute the transverse velocity of this pulse, and sketch it as a function of (x-vt)/a.

Do the same for acceleration.

We all remember those halcyon days of our youth when we could just sit on a swing and swing as high as we could. How hard we worked. Gee, that’s a good question. How hard did we work? How much work did we actually do? If we ate our bowl of cereal in the morning, how long could we swing for using that cereal as food energy? Well, on the box of Cheerios at my house it says that a serving of 30 g of Cheerios with 250 ml of milk provides 185 food Calories or 720 kJ of energy. Of course, we cannot convert all that chemical energy to mechanical energy with 100% efficiency. Let’s assume that we can convert 20% to mechanical energy. So now how should we figure out how much energy it takes to swing? One approach would be to start by calculating how much energy there is in the motion of the child on the swing. You could pick a mass for the child and the swing and estimate how high the child swings etc.

Now once you know how much energy there is when the child swings to a certain height, you could figure out how much energy the child on the swing loses due to friction. Think back and remember how quickly or slowly you used to slow down if, after you started swinging, you just sat motionless on the swing and let friction slow you down. (Of course, not all swings slow down at the same rate. Just estimate a reasonable value). Now, if you have determined how much energy the swing loses to friction in a certain amount of time, then you know that that is the amount of energy (approximately!) that you have to supply over a given time to keep it going. If you now know how much energy it takes to keep the swing going for a period of time, you should be able to answer the question : "How long can a child swing after eating a bowl of Cheerios?"

The second interesting question for you to answer is: "How do children swing on a swing?" I mean if they don’t touch the ground how do they keep the swing swinging ? Now this is a tougher question, and you probably won’t get a clean answer that you are happy with or supremely confident about. But I would like you to try to analyze the motion of the child on the swing using the physical principles that you know to gain insight to the question. You might want to look at the forces when the child is at different positions on the swing or maybe you want to look at potential energy for different positions of the child, e.g. legs straight out or bent back, head leaning forwards or backwards, etc.

END OF EXAM