(a) Suppose that at time t = 0, the masses
are at their equilibrium positions and their instantaneous velocities
are v_{1} = 
v_{2}. 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 v_{1} = v_{2}.
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 backandforth motion of the cart.
Section 15.3
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 m_{1 }and
m_{2} 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¢ v^{2}
and that the kinetic energy of the mass m and the spring
is
K = 1/2mv^{2}
+ 1/6m¢ v^{2}
= 1/2(m + 1/3m¢ )v^{2}
Consequently, the effective mass of the combination
is m + 1/3m¢ .
(b) Show that the frequency of oscillation is
(c) Suppose that the spring described in Example
1 has a mass of 5 kg. The frequency of oscillation of the 400kg
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.
Section 15.4
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 oneway
swing of the pendulum therefore takes exactly 1.0 s. What is the
length of the seconds pendulum in Paris (g = 9.809 m/s^{2}),
Buenos Aires (g = 9.797 m/s^{2})
and Washington, D.C. (g = 9.801 m/s^{2})?
24. A grandfather clock controlled by a pendulum
of length 0.9932 m keeps good time in New York (g = 9.803
m/s^{2}).
(a) If we take this clock to Austin (g
= 9.793 m/s^{2}), 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 (oneway) swing will the
pendulums be out of step at the end of the 10min interval? What
can you conclude about Galileo's claim?
Section 15.5
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 = 1/2Kq^{2}.
[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/s^{2}.
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 onequarter of a (oneway) 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 30cm 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 m_{1} is attached at
the lower end of the rod and a smaller mass m_{2}
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 circle^{9}
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/R^{3})r.
(a) Show that the component of the acceleration
of gravity along the track of the train is
g_{x }= –(GM/R^{3})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/r^{3},
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/r^{3}, 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 = r_{0}. For a nearly
circular orbit, the radius differs from r_{0}
only by a small quantity: r = r_{0}
+ 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 = 3L^{2 }/mr^{4}_{0}
– 2GmM_{s}/r^{3}_{0}
Show that this equals k = GmM_{s}/r^{3}_{0.}
(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.
Section 15.6
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 onehalf 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.
