15 • Oscillatory Motion - ECHSPhysics

(a)
(b)
(c)
(d)
k
k
k
m 1
k
A
m 2
m 1
m 1
53. A large block P executes horizontal simple harmonic
motion as it slides across a frictionless surface with a frequency
f 1.50 Hz. Block B rests on it, as shown in Figure
P15.53, and the coefficient of static friction between the
two is s 0.600. What maximum amplitude of oscillation
can the system have if block B is not to slip?
v
m 2
Figure P15.52
µ µ s
m 1
Figure P15.53 Problems 53 and 54.
54. A large block P executes horizontal simple harmonic motion
as it slides across a frictionless surface with a frequency
f. Block B rests on it, as shown in Figure P15.53,
and the coefficient of static friction between the two is s.
What maximum amplitude of oscillation can the system
have if the upper block is not to slip?
55. The mass of the deuterium molecule (D 2) is twice that of
the hydrogen molecule (H 2). If the vibrational frequency
of H 2 is 1.30 10 14 Hz, what is the vibrational frequency
of D 2? Assume that the “spring constant” of attracting
forces is the same for the two molecules.
B
P
D
v
m 2
Problems 481
56. A solid sphere (radius R) rolls without slipping in a
cylindrical trough (radius 5R) as shown in Figure
P15.56. Show that, for small displacements from equilibrium
perpendicular to the length of the trough, the
sphere executes simple harmonic motion with a period
T 2 √28R/5g.
5R
57. A light, cubical container of volume a 3 is initially filled
with a liquid of mass density . The cube is initially supported
by a light string to form a simple pendulum of
length L i, measured from the center of mass of the filled
container, where L i a. The liquid is allowed to flow
from the bottom of the container at a constant rate
(dM/dt). At any time t, the level of the fluid in the container
is h and the length of the pendulum is L (measured
relative to the instantaneous center of mass). (a) Sketch
the apparatus and label the dimensions a, h, L i, and L.
(b) Find the time rate of change of the period as a function
of time t. (c) Find the period as a function of time.
58. After a thrilling plunge, bungee-jumpers bounce freely on
the bungee cord through many cycles (Fig. P15.22). After
the first few cycles, the cord does not go slack. Your little
brother can make a pest of himself 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. An object of unknown
mass m on the same spring oscillates with a period
T. Determine (a) the spring constant and (b) the unknown
mass.
59. A pendulum of length L and mass M has a spring of force
constant k connected to it at a distance h below its point of
suspension (Fig. P15.59). Find the frequency of vibration
L
R
Figure P15.56
θ
M
h
k
Figure P15.59