### P. LeClair

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pring.
otion,
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ue for
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disk is
mbly is
sition
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is
initial value? (b) What If? How long does it take for the
mechanical energy to drop to half its initial value?
(c) Show that, in general, the fractional rate at which the
amplitude decreases in a damped harmonic oscillator is
PH the
125 fractional
/ LeClair rate at which the mechanical energy
half
decreases.
Spring 2014
Problem Set 6
71. A block of mass m is connected to two springs of force constants kInstructions:
1 and k 2 as shown in Figures P15.71a and P15.71b.
In each case,
the block
moves on
a frictionless
it credit.
1. Answer
all questions
below.
Show yourtable
work after
for full
is displaced
from
equilibrium
released.
Show
that
2. All
problems
are due and
by the
end of the
day on
14 in
April 2014
the two cases the block exhibits simple harmonic motion
3. Late penalties will not be incurred until after spring break
with periods
4. You may collaborate, but everyone must turn in their own work.
√
k 2) h above the surface of a planet of mass M and radius R.
1. An object of mass m is droppedm(k
from
a height
1\$
(a)
T ! 2#
Assume the planet has no atmosphere so
k 1kthat
2 friction can be ignored.
√
(a) What is the speed of the mass justmbefore it strikes the surface of the planet? Do not assume that h is
(b)
T ! 2#
small compared with R.
k1 \$ k2
√
(b) Show that the expression from (a) reduces to v = 2gh for h R.
(c) How long does it take for the object to fall to the surface for an arbitrary value of h? Use any means
k1
k2
necessary to evaluate the integral required. Bonus points (10%) for code submissions.
m
2. A block of mass m is connected to two springs of force constants k1 and k2 as shown below. The block
moves on a frictionless table after it is displaced from equilibrium and released. Determine the period of
simple harmonic motion. (Hint:
(a) what is the total force on the block if it is displaced by an amount x?
k1
k2
m
3. A mass m is connected (b)
to two springs in series as shown below. What is the period of simple harmonic
Figure
motion if the mass is displacedP15.71
from equilibrium. Hint: what must be true of the displacement of each spring
if the total displacement is ∆x?
72. A lobsterman’skbuoy is a solid
k wooden cylinder of radius r
and mass M. It is weighted at one end so that it floats upright in calm sea water, having density '. A passing shark
tugs on the slack rope mooring the buoy to a lobster trap,
pulling the buoy down a distance x from its equilibrium
position and releasing it. Show that the buoy will execute
simple
harmonic
if and
thelength
resistive
effects atofone
the
4. A horizontal
plankmotion
of mass m
L is pivoted
end. The plank’s other end is supported by
water
are
neglected,
and
determine
the
period
of
the
a spring of force constant k. The moment of inertia of the plank about the pivot is I = 13 mL2 . The plank
oscillations.
is displaced by a small angle θ from horizontal equilibrium and released. Find the angular frequency ω of
m
73. simpleConsider
bob on(Hint:
a light
stiff rod,
forming
a simple
harmonic amotion.
consider
the torques
about
the pivot point.)
pendulum of length L ! 1.20 m. It is displaced from the
vertical by an angle "max and then released. Predict the
subsequent angular positions if "max is small or if it is large.
Proceed as follows: Set up and carry out a numerical
method to integrate the equation of motion for the simple
pendulum:
d 2"
g
! % sin "
2
dt
L
the mass is 5.00 kg and the spring has a force constant of
100 N/m.
note that the mass of a segment of the
is dm " (m/!)dx. Find (a) the kinetic energy o
system when the block has a speed v and (b) the p
of oscillation.
Pivot
v
θ
dx
k
x
M
Figure P15.61
P15.66
5. Assume the earth to be a solid sphere of uniform density. A hole is drilled throughFigure
the earth,
passing
through its center, and a ball is dropped into the hole. Neglect friction.
62. Review problem. A particle of mass 4.00 kg is attached to
67.
A ball of mass m is connected to two rubber ba
a spring with
a force constant
of the
100ball
N/m.
It is oscillating
(a) Calculate
the time for
to return
to the release point. Hint: what sort of motion results?
length L, each under tension T, as in Figure P15.6
on a horizontal
frictionless
surface
an time
amplitude
(b) Compare
the result
of partwith
a to the
requiredoffor the ball to complete a circular orbit of radius RE
ball is displaced by a small distance y perpendicular
2.00 m. Aabout
6.00-kg
theobject
earth is dropped vertically on top of the
length of the rubber bands. Assuming that the t
4.00-kg object as it passes through its equilibrium point.
does not
showonthat
(a) the restoring
6. A satellite
in a circular
orbitmuch
of radius
r. the
The area A enclosed
by thechange,
orbit depends
r2 because
The two objects
stick istogether.
(a)Earth
By how
does
is \$ (2T/L)y and (b) the system exhibits simple har
2
= πr
Determine system
how thechange
following
amplitudeA of
the. vibrating
as properties
a result ofdepend
the on r: (a) period, (b) kinetic energy, (c) angular
motion with an angular frequency # " √2T/mL .
and (d)
speed.
collision?momentum,
(b) By how
much
does the period change?
(c) By how
much does the energy change? (d) Account for
7. Here are some functions:
the change in energy.
y
2
A
63. A simple pendulum
length
of 2.23 m and a mass fof2 (x, t) =
f1 (x, t)with
= Aea−b(x−vt)
2
b (x − vt) + 1
6.74 kg is given an initial speed of 2.06 m/s at its equilibL
L
2
3
−b
bx
+vt
(
)
rium position.f3 (x,
Assume
t) = Ae it undergoes simple harmonic
f4 (x, t) = A sin (bx) cos (bvt)
motion, and determine its (a) period, (b) total energy, and
Figure P15.67
(c) maximum
angular displacement.
(a) Which ones satisfy the wave equation? Justify your answer with explicit calculations.
64. Review problem.
One end
of a light
spring the
withwave
force
con- write down the corresponding functions g(x, t)
(b) For those
functions
that satisfy
equation,
stant 100 representing
N/m is attached
vertical
wall. A
light string
a waveto
of athe
same shape
traveling
in theisopposite
68.direction.
When a block of mass M, connected to the en
tied to the other end of the horizontal spring. The string
spring of mass ms " 7.40 g and force constant k, is s
8. The
equation for
driven damped
oscillator
changes from
horizontal
to avertical
as it passes
over isa solid
simple harmonic motion, the period of its motion is
pulley of diameter
4.00
cm.
The
pulley
is
free
to
turn
on
a
d2 x
dx
q
+The
2γωo vertical
+ ωo2 xsection
= E(t)
fixed smooth axle.
of the string sup2
M ' (m s(1)
/3)
dt
dt
m
%
"
2
&
ports a 200-g object. The string does not slip at its contact
k
with the (a)
pulley.
Find
frequency
of oscillation
of the
Explain
thethe
significance
of each
term.
object if the
(b) E250
g,
A two-part experiment is conducted with the u
iωt
(b) mass
Let Eof=the
Eo epulley
and isx (a)
= xonegligible,
ei(ωt−α) where
o and xo are real quantities. Substitute into the above
and (c) 750
g.
blocks of various masses suspended vertically fro
expression and show that
√
xo = q
qEo /m
2
(2)
2
(ωo2 − ω 2 ) + (2γωωo )
(c) Derive an expression for the phase lag α, and sketch it as a function of ω, indicating ωo on the sketch.
9. (a) A diatomic molecule has only one mode of vibration, and we may treat it as a pair of masses connected by a spring (figure (a) below). Find the vibrational frequency, assuming that the masses of A and B
are different. Call them ma and mb , and let the spring have constant k.
(b) A diatomic molecule adsorbed on a solid surface (figure (b) below) has more possible modes of vibration.
Presuming the two springs and masses to be equivalent this time, find their frequencies.
Figure 1: From http: // prb. aps. org/ abstract/ PRB/ v19/ i10/ p5355_ 1 .
10. Energetics of diatomic systems An approximate expression for the potential energy of two ions as a
function of their separation is
PE = −
b
ke2
+ 9
r
r
(3)
The first term is the Coulomb interaction representing the electrical attraction of the two ions, while the
second term is introduced to account for the repulsive effect of the two ions at small distances. (a) Find
b as a function of the equilibrium spacing ro . (What must be true at equilibrium?) (b) For KCl, with an
equilibrium spacing of ro = 0.279 nm, calculate the frequency of small oscillations about r = ro . Hint: do a
Taylor expansion of the potential energy to make it look like a harmonic oscillator for small r = ro .
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