Chapter 15 Problems 405
 Problems Section 15.1 1. A particle moves back and forth along the x axis between the points x = 0.20 m and x = –0.20 m. The period of the motion is 1.2 s, and it is simple harmonic. At the time t = 0, the particle is at x = 0 and its velocity is positive. (a) What is the frequency of the motion? The angular frequency? (b) What is the amplitude of the motion? (c) What is the phase constant? (d) At what time will the particle reach the point x = 0.20 m? At what time will it reach the point x = –0.10 m? (e) What is the speed of the particle when it is at x = 0? What is the speed of the particle when it reaches the point x = – 0.10 m? 2. The motion of the piston in an automobile engine is approximately simple harmonic. Suppose that the piston travels back and forth over a distance of 8.50 cm and that the piston has a mass of 1.2 kg. What are its maximum acceleration and maximum speed if the engine is turning over at its highest safe rate of 6000 rev/min? What is the maximum force on the piston? 3. A given point on a guitar string (say, the midpoint of the string) executes simple harmonic motion with a frequency of 440 Hz and an amplitude of 1.2 mm. What is the maximum speed of this motion? The maximum acceleration? 4. A particle moves as follows as a function of time: where distance is measured in meters and time in seconds. (a) What is the amplitude of this simple harmonic motion? The frequency? The angular frequency? The period? (b) At what earliest positive time does the particle reach the equilibrium point? The turning point? (c) At what later times does the particle reach the equilibrium point and the turning point again? *5. Experience shows that from one-third to one-half of the passengers in an airliner can be expected to suffer motion sickness if the airliner bounces up and down with a peak acceleration of 0.4 G and a frequency of about 0.3 Hz. Assume that this up-and-down motion is simple harmonic. What is the amplitude of the motion? *6. The frequency of oscillation of a mass attached to a spring is 3.0 Hz. At time t = 0, the mass has an initial displacement of 0.20 m and an initial velocity of 4.0 m/s. (a) What is the position of the mass as a function of time? (b) When will the mass first reach a turning point? What will be its acceleration at that time? Section 15.2 7. The body-mass measurement device used aboard Skylab consisted of a chair supported by a spring. The device was calibrated before the space flight by placing a standard mass of 66.91 kg in the chair; with this mass the period of oscillation of the chair was 2.088 s. During the space flight astronaut Lousma sat in the chair; the period of oscillation was then 2.299 s. What was the mass of the astronaut? Ignore the mass of the chair. 8. The body of an automobile of mass 1100 kg is supported by four vertical springs attached to the axles of the wheels. In order to test the suspension, a man pushes down on the body of the automobile and then suddenly releases it. The body rocks up and down with a period of 0.75 s. What is the spring constant of each of the springs? Assume that all the springs are identical and that the compressional force on each spring is the same; also assume that the shock absorbers of the automobile are completely worn out so that they do not affect the oscillation frequency. 9. Deuterium is an isotope of hydrogen. The mass of the deuterium atom is 1.998 times larger than the mass of the hydrogen atom. Given that the frequency of vibration of the H2 molecule is 1.31X1014 Hz (see Example 7), calculate the frequency of vibration of the D2 molecule. Assume the "spring" connecting the atoms is the same in H2 and D2. 10. Calculate the frequency of vibration of the HD molecule consisting of one atom of hydrogen and one of deuterium. See Problem 9 for necessary data. *11. A mass m = 2.5 kg hangs from the ceiling by a spring with k = 90 N/m. Initially, the spring is in its unstretched configuration and the mass is held at rest by your hand. If, at time t = 0, you release the mass, what will be its position as a function of time? *12. The wheel of a sports car is suspended below the body of the car by a vertical spring with a spring constant 1.1X104 N/m. The mass of the wheel is 14 kg and the diameter of the wheel is 61 cm. (a) What is the frequency of up-and-down oscillations of the wheel? Regard the wheel as a mass on one end of a spring and regard the body of the car as a fixed support for the other end of the spring. (b) Suppose that the wheel is slightly out of round, having a bump on one side. As the wheel rolls on the street it receives a periodic push each time the bump comes in contact with the street. At what speed of the translational motion of the car will the frequency of this push coincide with the natural frequency of the up-and-down oscillations of the wheel? What will happen to the car at this speed? (Note: This problem is not quite realistic because the elasticity of the tire also contributes a restoring force to the up-and-down motion of the wheel.) *13. A mass m slides on a frictionless plane inclined at an angle q with the horizontal. The mass is attached to a spring, parallel to the plane (Figure 15.34); the spring constant is k. How much is the spring stretched at equilibrium? What is the frequency of the oscillations of this mass up and down the plane? **14. Two identical masses slide with one-dimensional motion on a frictionless plane under the influence of three identical springs attached as shown in Figure 15.35. The magnitude of each mass is m and the spring constant of each spring is k.