Nine review problems: Force, energy, motion, frequency, amplitude, harmonic motion, oscillation, speed, tuning fork, beat frequency

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1. An archer pulls his bow string back 0.4 m by exerting a force on the string that increases uniformly from zero to 230 N. What is the equivalent force constant of the bow? How much work does the archer do in pulling the bow?

2. A ball dropped from a height of h makes a perfectly elastic collision with the ground. Assuming no energy is lost due to air resistance, show that the ensuing motion is periodic and determine its period.

3. The position of a particle is given by the expression x=4cos(3?t+?), where x is in meters and t is in seconds. Determine the frequency and period of the motion, the amplitude of the motion, the phase constant, and the position of the particle after first 0.25 seconds of motion. Derive an expressions for particle velocity and acceleration.

4. A piston in a gasoline engine is in simple harmonic motion. If the extremes of its displacement away from its center point are ±5.0 cm, derive the equation of this motion when engine is running at the rate of 3600 rev/min.

5. After a thrilling plunge, bungee-jumpers bounce freely on the bungee cord through many cycles. You can make a pest by figuring out the mass of each person, using a proportion which you set up by solving this problem: An object of mass M is oscillating freely on a vertical spring with a period T. The object of unknown mass m on the same spring oscillates with a period t. Determine the force constant (spring stiffness k) and the unknown mass.

6. A stone is dropped from the top of a cliff. The splash it makes when striking the water below is heard 3.5 s later. How high is the cliff? (Assume speed of sound equal to 340 m/s)

7. At the Indianapolis 500, you can measure the speed of cars just by listening to the difference in pitch of the engine noise between approaching and receding cars. Suppose the sound of a certain car drops by a full octave (262 Hz) as it goes by on the straightway. How fast is it going?

8. A tuning fork is set into vibration above a vertical open tube filled with water. The water level is allowed to drop slowly. As it does so, the air in the tube above the water level is heard to resonate with the tuning fork when the distance from the tube opening to the water level is 0.125 m and again at 0.395 m. What is the frequency of the tuning fork?

9. In certain ranges of a piano keyboard, more than one string is tuned to the same note to provide extra loudness. For example, the note at 110 Hz has two strings at this frequency. If one string slips from its normal tension of 600 N to 540 N, what beat frequency is heard when the hammer strikes the two strings simultaneously?