Chapter 15 Problems 405
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?
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.
(a) Suppose that at time t = 0, the masses are at their equilibrium positions and their instantaneous velocities are v1 = - v2. Find the position of each mass as a function of time. What is the frequency of the motion?
(b) Suppose that at time t = 0, the masses are at their equilibrium positions and their instantaneous velocities are v1 = v2. Find the position of each mass as a function of time. What is the frequency of the motion?
***15. A cart consists of a body and four wheels on frictionless axles. The body has a mass m. The wheels are uniform disks of mass M and radius R. The cart rolls, without slipping, back and forth on a horizontal plane under the influence of a spring attached to one end of the cart (Figure 15.36). The spring constant is k. Taking into account the moment of inertia of the wheels, find a formula for the frequency of the back-and-forth motion of the cart.
16. Suppose that a particle of mass 0.24 kg acted upon by a spring undergoes simple harmonic motion with the parameters given in Problem 1.
(a) What is the total energy of this motion?
(b) At what time is the kinetic energy zero? At what time is the potential energy zero?
(c) At what time is the kinetic energy equal to the potential energy?
17. A mass of 8.0 kg is attached to a spring and oscillates with an amplitude of 0.25 m and a frequency of 0.60 Hz. What is the energy of the motion?
18. The separation between the equilibrium positions of the two atoms of a hydrogen molecule is 1.0 Å. Using the data given in Example 7, calculate the value of the vibrational energy that corresponds to an amplitude of vibration of 0.5 Å for each atom. Is it valid to treat the motion as a small oscillation if the energy has this value?
*19. A mass of 3.0 kg sliding along a frictionless floor at 2.0 m/s strikes and compresses a spring of constant k = 300 N/m. The spring stops the mass. How far does the mass travel while being slowed by the spring? How long does the mass take to stop?
*20. Two masses m1 and m2 are joined by a spring of spring constant k. Show that the frequency of vibration of these masses along the line connecting them is
(Hint: The center of mass remains at rest.)
*21. Although it is usually a good approximation to neglect the mass of a spring, sometimes this mass must be taken into account. Suppose that a uniform spring has a relaxed length l and a mass m'; a mass m is attached to the end of the spring. The mass m' is uniformly distributed along the spring. Suppose that if the moving end of the spring has a speed v, all other points of the spring have speeds directly proportional to their distance from the fixed end, for instance, a point midway between the moving and the fixed end has a speed 1/2v.
(a) Show that the kinetic energy in the spring is 1/6m¢ v2 and that the kinetic energy of the mass m and the spring is
K = 1/2mv2 + 1/6m¢ v2 = 1/2(m + 1/3m¢ )v2
Consequently, the effective mass of the combination is m + 1/3m¢ .
(c) Suppose that the spring described in Example 1 has a mass of 5 kg. The frequency of oscillation of the 400-kg mass attached to this spring will then be somewhat smaller than calculated in Example 1. How much smaller? Express your answer as a percentage of the answer of Example 1.
22. A mass suspended from a parachute descending at constant velocity can be regarded as a pendulum. What is the frequency of pendulum oscillations of a human body suspended 7 m below a parachute?
23. A "seconds" pendulum is a pendulum that has a period of exactly 2.0 s: - each one-way swing of the pendulum therefore takes exactly 1.0 s. What is the length of the seconds pendulum in Paris (g = 9.809 m/s2), Buenos Aires (g = 9.797 m/s2) and Washington, D.C. (g = 9.801 m/s2)?
24. A grandfather clock controlled by a pendulum of length 0.9932 m keeps good time in New York (g = 9.803 m/s2).
(a) If we take this clock to Austin (g = 9.793 m/s2), how many minutes per day will it fall behind?
(b) In order to adjust the clock, by how many millimeters must we shorten the pendulum?
25. The pendulum of a grandfather clock has a length of 0.994 m. If the clock runs late by 1 minute per day, how much must you shorten the pendulum to make it run on time?
*26. A pendulum hangs from a wall inclined at an angle of 5° with the vertical (see Figure 15.37). Suppose that this pendulum is released at an initial angle of 10° and it bounces off the wall elastically whenever it hits. What is the period of this pendulum?
*27. The pendulum of a pendulum clock consists of a rod of length 0.99 m with a bob of mass 0.40 kg. The pendulum bob swings back and forth along an arc of length 20 cm.
(a) What are the maximum velocity and the maximum acceleration of the pendulum bob along the arc?
(b) What is the force that the pendulum exerts on its support when it is at the midpoint of its swing? At the endpoint? Neglect the mass of the rod in your calculations.
*28. The pendulum of a regulator clock consists of a mass of 120 g at the end of a (massless) wooden stick of length 44 cm.
(a) What is the total energy (kinetic plus potential) of this pendulum when oscillating with an amplitude of 4°?
(b) What is the speed of the mass when at its lowest point?
**29. Galileo claimed to have verified experimentally that a pendulum oscillating with an amplitude as large as 30° has the same period as a pendulum of identical length oscillating with a much smaller amplitude. Suppose that you let two pendulums of length 1.5 m oscillate for 10 min. Initially, the pendulums oscillate in step. If the amplitude of one of them is 30° and the amplitude of the other is 5°, by what fraction of a (one-way) swing will the pendulums be out of step at the end of the 10-min interval? What can you conclude about Galileo's claim?
30. In windup clocks a strong torsional spring is used to store mechanical energy. Suppose that each week a clock requires four full turns of the winding key to keep running. The initial turn requires a torque of 0.30 N • m and the final turn a torque of 0.45 N • m.
(a) What amount of mechanical energy do you store in the spring when winding the clock?
(b) What is the consumption of mechanical power by the clock?
(c) What is the torsional spring constant?
*31. The balance wheel of a watch, such as that shown in Figure 15.25, can be approximately described as a hoop of diameter 1.0 cm and mass 0.60 g. Each of the screws, whose masses are included in the mass given for the hoop, has a mass of 0.020 g. Suppose the watch runs fast by 1.2 minutes per day. To adjust the watch so that it keeps perfect time, by how much must we increase the moment of inertia of the balance wheel? If we want to achieve this increment by moving one of the screws outward in a radial direction, how far must we move the screw?
*32. Show that the potential energy of a torsional pendulum is U = 1/2Kq2. [Hint: Begin with Eq. (13.21) for the work done by the torque.]
33. At the National Bureau of Standards in Washington, D.C., the value of the acceleration of gravity is 9.80095 m/s2. Suppose that at this location a very precise physical pendulum, designed for measurements of the acceleration of gravity, has a period of 2.10356 s. If we take this pendulum to a new location at the U.S. Coast and Geodetic Survey, also in Washington, D.C., it has a period of 2.10354 s. What is the value of the acceleration of gravity at this new location? What is the percentage change of the acceleration between the two locations?
*34. A pendulum consists of a brass rod with a brass cylinder attached to the end (Figure 15.38). The diameter of the rod is 1.00 cm and its length is 90.00 cm; the diameter of the cylinder is 6.00 cm and its length is 20.00 cm. What is the period of this pendulum?
*35. To test that the acceleration of gravity is the same for a piece of iron and a piece of brass, an experimenter takes a pendulum of length 1.800 m with an iron bob and another pendulum of the same length with a brass bob and starts them swinging in unison. After swinging for 12 min, the two pendulums are no more than one-quarter of a (one-way) swing out of step. What is the largest difference between the values of g for iron and for brass consistent with these data? Express your answer as a fractional difference.
*36. Calculate the natural period of the swinging motion of a human leg. Treat the leg as a rigid physical pendulum with axis at the hip joint. Pretend that the mass distribution of the leg can be approximated as two rods joined rigidly end to end. The upper rod (thigh) has a mass of 6.8 kg and a length of 43 cm; the lower rod (shin plus foot) has a mass of 4.1 kg and a length of 46 cm. Using a watch, measure the period of the natural swinging motion of your leg when you are standing on one leg and letting the other dangle freely. Alternatively, measure the period of the swinging motion of your leg when you walk at a normal rate (this approximates the natural swinging motion). Compare with the calculated number.
*37. A hole has been drilled through a meter stick at the 30-cm mark and the meter stick has been hung on a wall by a nail passing through this hole. If the meter stick is given a push so that it swings about the nail, what is the period of the motion?
*38. A physical pendulum has the shape of a disk of radius R. The pendulum swings about an axis perpendicular to the plane of the disk and at distance l from the center of the disk.
(a) Show that the frequency of the oscillations of this pendulum is
(b) For what value of l is this frequency at a maximum?
*39. A physical pendulum consists of a massless rod of length 2l rotating about an axis through its center. A mass m1 is attached at the lower end of the rod and a smaller mass m2 at the upper end (see Figure 15.39). What is the period of this pendulum?
**40. A thin vertical rod of steel is clamped at its lower end. When you push the upper end to one side, bending the rod, the upper end moves (approximately) along an arc of circle9 of radius R and the rod opposes your push with a restoring force F = - kq , where q is the angular displacement and k is a constant. If you attach a mass m to the upper end, what will be the frequency of small oscillations? For what value of m does the rod become unstable, that is, for what value of m is w = 0? Treat the rod as massless in your calculations. (Hint: Think of the rod as an inverted pendulum of length R, with an extra restoring force - kq.)
*41. Suppose that the physical pendulum in Figure 15.19 is a thin rigid rod of mass m suspended at one end. Suppose that this rod has an initial position q = 20° and an initial angular velocity w = 0. Calculate the force F that the support exerts on the pendulum at this initial instant (give horizontal and vertical components).
*42. The door of a house is made of wood of uniform thickness. The door has a mass of 27 kg and measures 1.90 m X 0.91 m. The door is held shut by a torsional spring with k = 30 N • m/radian arranged so that it exerts a torque of 54 N • m when the door is fully open (at right angles to the wall of the house). What angular speed does the door attain if it slams shut from the fully open position? What linear speed does the edge of the door attain?
**43. According to a proposal described in Example 1.5, very fast trains could travel from one city to another in straight subterranean tunnels (see Figure 15.40). For the following calculations, assume that the density of the Earth is constant so that, according to Eq. (9.45), the acceleration of gravity as a function of the radial distance r from the center of the Earth is g = (GM/R3)r.
(a) Show that the component of the acceleration of gravity along the track of the train is
gx = –(GM/R3)x
where x is measured from the midpoint of the track (see Figure 15.40).
(b) Ignoring friction, show that the motion of the train along the track is simple harmonic motion with a period independent of the length of the track,
(c) Starting from rest, how long would a train take to roll freely along its track from San Francisco to Washington, D.C.? What would be its maximum speed (at the midpoint)? Use the numbers calculated in Example 1.5 for the length and depth of the track.
**44. A physical pendulum consists of a long, thin cone suspended at its apex (Figure 15.41). The height of the cone is l. What is the period of this pendulum?
**45. The net gravitational force on a particle placed midway between two equal spherical bodies is zero. However, if the particle is placed some distance away from this equilibrium point, then the gravitational force, is not zero.
(a) Show that if the particle is at a distance x from the equilibrium point in a direction toward one of the bodies, then the force is approximately 4GMmx/r3, where M is the mass of each spherical body, m is the mass of the particle, and 2r is the distance between the spherical bodies. Assume x « r.
(b) Show that if the particle is at a distance x from the equilibrium point in a direction perpendicular to the line connecting the bodies, then the force is approximately – 2GMmx/r3, where the negative sign indicates that the direction of the force is toward the equilibrium point.
(c) What is the frequency of small oscillations of the mass m about the equilibrium point when moving in a direction perpendicular to the line connecting the bodies? Assume that the bodies remain stationary.
**46. The motion of a simple pendulum is given by
(a) Find the tension in the string of this pendulum; assume that q « 1.
(b) The tension is a function of time. At what time is the tension maximum? What is the value of this maximum tension?
***47. The total energy of a body of mass m orbiting the Sun is
where the quantity in brackets represents the kinetic energy written in terms of the radial and the tangential component of the velocity.
(a) Show that in terms of the (constant) angular moment L, the energy can be written
(b) For a circular orbit the radius is constant: r = r0. For a nearly circular orbit, the radius differs from r0 only by a small quantity: r = r0 + x. Show that in terms of the small quantity x, the energy for a nearly circular orbit is approximately
The last term in this expression is a constant (independent of x).
(c) Show that, except for an additive constant, this expression for the energy coincides with the equation for the energy of a harmonic oscillator, provided that we identify
k = 3L2 /mr40 – 2GmMs/r30
Show that this equals k = GmMs/r30.
(d) What is the frequency of small radial oscillations about the circular orbit? Show that this frequency equals the frequency of the circular orbit. Make a sketch of the shape of the orbit that results from the combination of revolution around the circle and small oscillations along the radius.
48. A pendulum of length 1.50 m is set swinging with an initial amplitude of 10°. After 12 min, friction has reduced the amplitude to 4°. What is the value of g for this pendulum?
49. The pendulum of a grandfather clock has a length of 0.994 m and a mass of 1.2 kg.
(a) If the pendulum is set swinging, the friction of the air reduces its amplitude of oscillation by a factor of 2 in 13.0 min. What is the value of g for this pendulum?
(b) If we want to keep this pendulum swinging at a constant amplitude of 8°, we must supply mechanical energy to it at a rate sufficient to make up for the frictional loss. What is the required mechanical power?
*50. If you stand on one leg and let the other dangle freely back and forth starting at an initial amplitude of, say, 20° or 30°, the amplitude will decay to one-half of the initial amplitude after about four swings. Regarding the dangling leg as a damped oscillator, what value of "Q" can you deduce from this?
9 The radius R of the approximating (oscullating) circle is somewhat shorter than the length of the rod.