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Physics 204A
Problem set 13, due 5/4/15
1. Pine has density ⇢p = 0.373⇥103 kg/m3 . The density of seawater is ⇢w = 1.03⇥103 kg/m3 .
Physics 204A
Problem set 15, due 12/14/12
What fraction of a pine log floating in seawater is below the surface?
2. A cube a3 is suspended in liquid of density ⇢. The top and bottom faces of the cube are
1. horizontal.
Resonance is
one
humans
can runbetween
so much
efficiently
swim: the
in
Use
thereason
di↵erence
in pressure
themore
top and
bottomthan
faceswetocan
calculate
addition
to
the
pendulum
resonance
of
our
legs,
there
is
a
mass/spring
resonance
in
which
buoyant force on the cube, and show that this buoyant force is equal to the weight of the
the tendons
of our
feet act as springs. Stand up and bounce gently on your toes. From
water
displaced
bylegs
theand
cube.
your natural resonance frequency, estimate the e↵ective spring constant of the spring formed
3. A
has radius
r =ankles,
12.5 mm
and
mass
m =your
67 g.equations,
What would
the any
apparent
weight
bysteel
yourball
Achilles
tendons,
and
feet.
Show
and be
state
assumptions
of
this
ball
under
water?
made.
4.
rubber
hungvaries
fromwith
a spring,
which
stretches
a distance
x as a result. by
Then
the
2. A
The
waterchicken
level in is
a bay
a period
of 12
hours, and
can be approximated
simple
rubber
chicken
(still
suspended
from
the
spring)
is
submerged
in
water,
and
the
spring
stretch
harmonic motion. The di↵erence between the highest and lowest observed water level is 1.2
is
be only
initial stretch
What
is theand
density
of levels,
the rubber
chicken?
m.measured
At t = 0,tothe
water10%
levelof isthe
midway
betweenx.the
highest
lowest
and is
rising.
Hint:
m
=
⇢
V.
rc
rc
Write an equation for the water level y(t).
5.
3. A
A u-shaped
u-shaped tube,
tube, with
with uniform
uniform internal
internal radius
radius r,
r, is
is partially
partially filled
filled with
with aa liquid
liquid of
of density
density ⇢.
⇢.
(Figure
1)
The
total
length
of
liquid-filled
tube
is
L.
If
the
liquid
level
is
somehow
disturbed
(Figure 1) The total length of liquid-filled tube is L. If the liquid level is somehow disturbed
from
from its
its equilibrium
equilibrium position,
position, it
it will
will oscillate.
oscillate. Show
Show that
that the
the oscillation
oscillation is
is simple
simple harmonic,
harmonic,
and
and find
find the
the angular
angular frequency
frequency !
! of
of the
the oscillation.
oscillation. (Assume
(Assume that
that any
any viscous
viscous damping
damping forces
forces
are
negligible.)
are negligible.)
x
Figure 1: Fluid-filled tube
4. A simple pendulum of length L and mass m is attached to a horizontal spring with spring
6.
constant k as shown in figure 2. Show that for small oscillations, the motion of the mass is
approximately simple harmonic, and determine the angular frequency ! of the oscillation.
7. A uniform disk is mounted on a low-friction bearing at its center. A spring is attached to
L
the edge of the disk, so that the spring is in equilibrium when it is tangental to the edge of
the disk, as shown in figure 3. Show that for small oscillations of the disk (“small” meaning
that sin ✓ ⇡ ✓ and cos ✓ ⇡ 1) the motion of the
k disk is simple harmonic, and find the angular
frequency !. The rotational inertia ofm a disk about the center is I = 12 M R2 .
Figure 2: Simple pendulum
1 with an additional spring
approximately simple harmonic, and determine the angular frequency ! of the oscillation.
Figure 1: Fluid-filled tube
4. A simple pendulum of length L and mass m is attached to a horizontal spring with spring
constant k as shown in figure 2. Show that for small oscillations, the motion of the mass is
5. You are standing at the end of a diving board. Your mass is (for this problem) m = 65 kg.
(The mass of the board is negligible.) The end of the board oscillates with a period T = 1.2
L contact with the board?
seconds. At what amplitude will you lose
6. A uniform disk is mounted on a low-friction bearing at its center. A spring is attached to
the edge of the disk, so that the spring is inkequilibrium when it is tangental to the edge of
the disk, as shown in figure 3. Show m
that for small oscillations of the disk (“small” meaning
that sin ✓ ⇡ ✓ and cos ✓ ⇡ 1) the motion of the disk is simple harmonic, and find the angular
frequency !. The rotational inertia of a disk about the center is I = 12 M R2 .
Figure
Figure 2:
2: Simple
Simple pendulum
pendulum with
with an
an additional
additional spring
spring
approximately simple harmonic, and determine the angular frequency ! of the oscillation.
k
1
R
M
Figure 3: Disk with tangental spring at the edge
q
7.
small peg
thea equilibrium
line, located
as
8. A simple
simple pendulum
pendulumhas
is aa point
massaton
massless string,
and Ts half-way
= 2⇡ upLg .the
A string
physical
shown
in figure
4. by
The
pendulum
hasalength
L when it’s to the left of the string, and length
pendulum
formed
a small
mass on
qlightweight string is approximately a simple pendulum,
L
when
it’s
to
the
right.
I
2
but
not quite; its period is T = 2⇡
. Is the period of a physical pendulum longer or
p
mgL
shorter than the period of a simple pendulum with the same length? Assume L is measured
to the center of mass, and justify your answer.
L/2
9. You wish to build your own pendulum clock. The first thing you will need is a pendulum with
L
a period of 2.00 seconds. Assuming that a simple pendulum is an appropriate approximation,
what should be the length of your pendulum?
10. You’re still trying to build a pendulum clock. You don’t have a simple pendulum available,
since point masses are actually quite
m to massless strings, so you’re using a
m difficult to attach
uniform rod instead. You still need the period to be 2.00 seconds. What should be the mass
and length of the bar, which pivots at one end?
Figure 4: Pendulum with a peg at the middle, shown in two positions.
(a) Is this motion simple harmonic? Approximately simple harmonic?
(b) What is the period of the motion?
8. Bubba (mass m = 100 kg) goes bungie-jumping. After the jump, he oscillates at the end of
the bungie cord with an amplitude A = 11 m and period T = 6.8 s. What is his maximum
velocity during this oscillation?
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