Download General Physics I Homework Set 5

314
CHAPTER 9 Rotation of Rigid Bodies
ping betw
9.68. The pulley in Fig. 9.36 has Figure 9..36 Problem 9.86.
University
of Technology der has m
of
radius 0.160 mDalian
and moment
disk with
inertia 0.480 kg · m 2• The rope
from rest
does rot slip on the pulley rim.
the speed
Use energy methods to calculate
312
CHAPTER 9 Rotation of Rigid Bodies
9.9]. A t
the speed of the 4.00-kg block
4
just before it strikes the floor.
·00 kg
hole ofra
then punc
You hang
thin hoop
with9.67.
9.59. A 1hin uniform rod of mass M and length L is bent at 9.87.
its cenFigurea 9.33
Problem
with the h
radius
ter so that the two segments are now perpendicular to each
othe&R over a nail at the rim of
Ball
A disk of radius 25.0 cm is free toFind
turn
about an axis perpendicular to it
through
perpendic
theplane
hoop. You displace it to the
its moment of inertia about an axis perpendicular to its
I
its center. It has a very thin but
strong
string
wrapped
itsside
rim,
and
inertia of
(within
the plane of the hoop)
and passing
through
(a) the
point where around
the two segments
meet
angle f3 from its equi1.00 kg _
through
of inertia
line connecting its
two ends.
(b) the
the string is attached to a ball and
that
is midpoint
pulledof the
tangentially
away
from
theanrim
of the ho
librium position and let it go.
of the disk (see the figure). The*Section
pull increases
in magnitude
and produces
9.6 Moment-of-Inertia
Calculations
angular speed
What is its an
Disk when it returns to its equilibrium posi- 9.94. A p
tion?
[Hinr: Use Eq. (9.18).]
and radiu
acceleration of the ball that obeys
the
equation
a(t)in =
At,
t is in
seconds
*9.60.
Using
the information
Table
9.2 where
and the parallel-axis
the9.68. M
A passenger bus in Zurich, Switzerland, derived its mctive from the
of
inertia
of
the
slender
rod
with
mass
orem,
find
the
moment
and A is a constant. The cylinder starts from rest, and at the end ofpower
the third
and length L shown
in Fig. 9.23 about an axis through 0, at anfrom the eneQ!Y stored in a large flywheel. The wheel was sphere is
2
second, the ball’s acceleration isarbitrary
1.80 m/s
distance. h from one end. Compare your resultbrought
to that up to speed periodieally, when the bus stopped at a station, the pivot
9.68.
Whenwhich
a toycould
car isthen
rapidly
scooted to
across
the floor, it
stores
theorem
by an electric
motor,
be attached
the electric
found by integration in Example 9.11 (Section 9.6).
(a) Find A.
in a flywheel.
has mass
and
ignored, /
power
The flywheel
was a The
solidcar
cylinder
with0.180
masskg,
1000
kgits flywheel
*9.61. Use Eq. (9.20) to calculate the moment of inertia of
a uni-lines.energy
2
has 1.80
moment
of top
inertia
4.00 speed
X
kg ·3000
m • The
car is 15.0
is 1%em
of M
and diameter
m; its
angular
was
rev/min.
form, solid disk with mass M and radius R for an axis perpendicu(b) Express the angular acceleration
of the disk as a function of time.
Anspeed,
advertisement
claims
that eneQ!Y
the car of
canthetravel
a scale
at th
angular
what is the
kinetic
fly- at axis
lar to the plane of the disk and passing through its center. (a) At this long.
ofaverage
up to 700
km/h
( 440
mi/h).
The scale
speed
9.95.
Per
If the
required
to operate
the speed
bus isis the
wheel?
(b) speed
*9.62.has
Use Eq.
(9.20) to
moment
inertia of
a slen(c) How much time after the disk
begun
tocalculate
turnthe
does
it of
reach
an
angular
speed
ofpower
15.0
rad/s?
toylong
car multiplied
by thebetween
ratio ofstops?
the length of an actual
car
is a thin,
X 104of
W,the
how
could it operate
at one
der, uniform rod with mass M and length L about an axis1.86
the length
of
the
toy. radius
Assume
aFigure
length9.37
of 3.0 m for a real
car.
(d) Through what angle has the
turned
just
(Hint:
See
Section
2.6.)
xy-plane
9.89. Two to
metal
disks,
one
with
end,disk
perpendicular
to the
rod. as it reaches 15.0 rad/s?
of 700 km/h,Problem9.89.
what is the actual translational
(a) For a scale 1speed
within or
R1 =
2.50cmandmassM
= 0.80kgaod
*9.63. A slender rod with length L has a mass per unit length
that
speedradius
of theRcar?
Ifem
all the
in the
about
the
the kg
other
with
5.00
andkinetic energy that is initially
0, according
to and
varies
left end,
where
2 = (b)
Two metal disks, one with radius
Rwith
2.50 from
cm the
and
mass
Mx1= =
0.80
the
other
with
1 =distance
the translational kinetic energy of an
theaxis
toy, th
converted
mass
M2 =flywheel
1.60 kg,isare
welded to
together
dmfdx = 'YX, where 'Y has units of kg/m2 • (a) Calculate the
total
radius R2 = 5.00 cm and mass Mmass
1.60 kg,terms
areof'Y
welded
together and mounted
on
a frictionless
2 =oftherodin
how
much
energy is
originally
(c) What
massinielem
R2
and mounted
on
a frictionless
axis
throughstored in the flywheel?
andL. (b) Use Eq. (9.20) to calculate
of What
the flywheel was needed to store
tial center
angular
velocity
common
(Ftg.
9.37). (a)
This the
is e
axis through their common center
(see the
figure).
the moment
of inertia
of the rod for an axis at the left end, their
perpenenergy
in part (b)?
is (a)
theto
total amount
momentofof
inertiacalculated
of the two
does not
dicular to the rod. Use the expression you derived in part
9.69. A classicwrapped
1957 Chevrolet
starts
Ma
mass
around Corvette of mass 1240 kg
disks?to(b)Alightstringis
express
I in rerms
M and
L How
does your result compare
(a) What is the total moment of
inertia
of ofthe
two
disks?
rest and
speeds
with a constant tangential acceleration
of
the smaller
disk,
and aup1.50-kg
cular-axis
that for a uniform rod? Explain this comparison. (c) Repeatthe
partedge
(b) offrom
2
(b) A light string is wrapped for
around
edge
ofthethe
smaller
disk,for
and
a3.00
1.50-kg
block
is
m/sfrom
on the
a circular
test
track of radius 60.0 m. Treat theis
car
block
is suspended
free end
ofthe
inas
thea p
an axis atthe
the right
end of
rod. How
do the results
parts
particle.
its angular
angular
If the
blockat
is(a)released
from
rest atacceleration? (b) Whatis its may
use
andthe
(c) compare?
Explain
this result.
suspended from the free end(b)of
string.
If the
block is releasedstring.
from
rest
aWhatis
distance
(c) What is its radial acceleration
at this
speed
s afterit
a distance of
2.006.00
m above
the starts?
floor, what
axis
theor
of 2.00 m above the floor, what is its speed just before it strikesisthe
floor?
time?
(d) Sketch
a view
L, th
side the
its speed
just before
it strikes
thefrom
floor?above showing the circular track,
Problems
car, calculation
the velocity
vector,
and
vectors
(c) Repeat
the
part (b),
thisthe acceleration component sheet
that
(c) Repeat the calculation of part
(b), this time with the string wrapped
around
the ofedge
of
6.00string
s afterwrapped
the car starts.
9.64. Sketch a wheel lying in the plane of your paper and time
rotating
What are the magnitudes of the
usetotal
the in
with the
around(e)
the
the larger disk. In which case
is the final speed of the block theedge
greatest?
Explain
whyforthethe car at 1.50
counrerclockwise. Choose a poinr on the rim and draw a vector
1 acceleration
and
net force
this kg
time? (f) What angle
ofthe
larger disk. In
which
case is
9.96. do
A th
this is so.
from the center of the wheel to that point. (a) What is the direction
total
acceleration
andExplain
net forcewhy
make
at
the total
final speed the
of the
block
the greatest?
thiswith
is so.the car's velocity
of ru? (b) Show that the velocity of the point is = 9.90.
ru X 1.
time? aod mass combination described in Example through t
In thethis
cylinder
Calculations of the moments of inertia
for
three
shapes.
Challenge
Problems
315
(c) Show that the radial acceleration of the point is 9.9 (Section
=
(Hint:
9.70.
designing
Figure
9.34rubProblem
9.70.Us
of ideal
9.4),Engineers
suppose theare
falling
mass m ais made
ru X V = ru X ( ru X 1) (see Exercise 9.28).
by whicheneQ!Y
a falling
masswhen
m the mass hits the *9.97. A
ber, so thatsystem
no mechanical
is lost
31 worlcing
(a) A thin, uniform rod Crab
is bent
into
a tosquare
of
side
a. Ifwith
total
ismoment
Misenergy
, originally
9.65.
are
a project
NASA
toCalculate
imparts
kinetic
to a rotating
ground.
(a)mass
If the cylinder
not
rotating
and
the mass m increases
*9."110.
the
of inerf"JgUre
9AO
Challenge
Nebula
at aTrip
rate
ofMars.
about 5You
X 11)
W, length
abouton10"
times
thethe
launch
a rocket
to energy.
Mars,
with
the rocket
blasting
from
earth
uniform
drum
to h which
it ground,
is 9.100.
is
released
from
rest
at
a height
above
the
to what height where a i
tiaperpendicular
of a uniform
solid
cone
an Problem
rate at which
the sun
radiates
The Crab
Nebula
obtainsoff
its
find the moment of inertia
about
an
axis
through
the
center
and
to about
when
and Mars
are aligned
a straight
line from will
the sun.
attached
by thin,
very
wire back up from the ofDrum
the cyl
this
mass
rebound
if
it bounces
straight
axis
through
its
center
(Fig.
9.40).light
eneigy from
the earth
rotational
kinetic
energy along
of a rapidly
spinning
the plane of the square.
(Hint:
Use
parallel-axis
theorem.)
Ifat
Mars
is nowthe
60" shead
earth in
its orbit
when
wrapped
rim
of the
the answer to part (a) is of M and
floor?
(b) Explain,
terms
ofthe
CUCQ!Y,
The
cone
has
massinMaround
and altitode
h. why
neutron star
its cente&
This
objectof
rotates
once
everyaround
0.0331the
s, sun,
(Fig.
9.34).
There is no
should
you
launch
rocket?
(Note:
Alleach
the
planets
orbitless
the than
sun
greater or
h. drum
and thisRperiod
bythe
4.22
X w-u
s for
second
of
The
radius
of its circular
base is R.
(b) A cylinder with radius
andis increasing
mass
M
has
density
that
increases
linearly
with
in the (a)
same
direction,
1 year
on Mars
in 9.18,
the axle
of mass is released of inertia
In
the
system shown
in Fig.
a 12.0-kg
9.101.and
On
aappreciable
compact
discfriction
(CD),
music
time that elapses.
If the
rate at which
energy
is lostisby1.9
theearth-years,
neu- 9.91.
distance r from the cylinder
axis,
ρ = atαr,
where
α is a positive constant.
assume
for
both planets.)
theCalculate
and
everything
starts cylinder of diame- mass and
rest and
falls,
causing
uniform
is coded
in
a drum,
pattern
of the
tiny
pits 10.0-kg
tocircular
the rate orbits
which
energy
is released by the neb- from
tron star is equal
rest.
This
system
A roller
inabout
a of
printing
press
turns
anofangle
li(t) em
being
toaits
tum
about
aspirals
frictionless
through its center. How density?
ter 30.0
the moment of inertia
cylinder
a
longitudinal
axis
through
center
arranged
infrom
track
that
out- isaxle
the9.68.
moment
of inertia
the neutron
star.
(b)through
Theories
ula, of
find the
2
2
3
li(t)
'Yt - star
{Jt ,inwhere
'Y =Nebula
3.20rad/s
fJwill
= thetested
on
earth,
but
it isAs
to be
used
far
mass
have
give
the cylinder 250 J of makes qu
ward
toward
the rim
oftothedescend
disc.
supernovaegiven
predictbythat
the =neutron
the Crab
has a and
in terms of M and R.
3
rad/s
Calculate
the angular
velocitystar
of as
thea roller
as aeneQ!Y?
on inside
Mars,a where
the the
acceleration
• (a)
kinetic
9.96. Neu
the disc
spins
CD player,
mass about0.500
1.4 times
that
of the
sun. Modeling
the neutron
2
(c) Calculate the moment
of
inertia
of
a
uniform
solid
cone
about
an
axis
through
its
function
time. (b) its
Calculate
the
angular acceleration
roller
due9.38,
toatgravity
is 3.71
m/sFigure
• In the
nova Rem
9.91.
In scanned
Fig.
cylinder
9.38 Problem 9.92.
track
is
a the
constant
linear
radius in
kilometers.
(c) What isof the
solid uniform
sphere,ofcalculate
tests,
as acone
function
of
(c) What
is
the
maximum
positive
angular
m is set
without
friction
ula is a
pulley
speed
ofradius
v earth
=tum
1.25
m/s.
Because
theto 15.0 kg
the linear
speed
of a point
ontime.
the
equator
neutron
star? Comcenter (see the figure).
The
has
mass
Mof the
and
altitude
h. and
The
ofwhen
its
at (d)
what
value oft
it occur?star is uni- radius
and
allowed
through 5.00 m, it gives 250.0 J ofabout
kinetic
velocity,
about stationary
horizontal
axles
10 l
of the
track
variestoasfall
it
spirals
speed ofand
light.
Assume
thatdoes
the neutron
Axis
circular base is R. pare to the *9.67.
to thecenters.
drum.of
(a)
If the system is operated on Mars, about
through65
energy
disk
ofradius
25.0 em
free
to turnof
about
an axle that
perpenpass through
their
A the
outward,
the
angular
speed
form and calculateAits
density.
Compare
toisthe
density
ordinary
dicular
toand
it through
its cente&
has very
1hin (about
but strong
string
what
distance
would
the 15.0-0
mass
to
fall to
give the
same
earth
(Ftg
is
wrapped
around
the
light
rope
disc
must
change
as the
CD
played
(See
Exercise
9.22.)
Let's
kg/m')on
to theedge
density
of an
atomic
nucleus
You connect a light stringrock
to (3000
a point
the
of
a Ituniform
vertical
disk
with
radius
Risenergy
and
mass
Mhave
. The
disk
its rim,that
anda the
stringstar
is attached
to a ballcylinder,
that
is angular
around
of kinetic
toCylinder
the
drum?v (b)
How The
fast would
amount
of a the
star
passes
over
the pulley,
see
what
acceleratim
is required
to keep
constant.
Justify the
statement
neutron
is essentially
1017 kgfrri').wrapped
is free to rotate without afriction
about
a stationary
through
center.
Initially,
the
is gained
at
15.0-kg
mass
rim of the disk axis
(Fig. 9.33).
The
be
just
the
250.0exp
J
pulled
tangentially
away from thehorizontal
has aof
3.00-kg
box
suspended
and
large atomic
nucleus.
equation
aits
spiral
is r(
6) moving
= To + on
{36,Mars
where
ro as
isdisk
the drum
radius
of nova
in magnitude
and produces
acceleration
offrom
the
ball
energy?
pullthe
increases
its free
is
no
1054 A.D.
the
spiral
atofend.
6kinetic
= There
0 and
fJwith
is aslipconstant
On a CD,horizontal
To is the inner
rest with the string connection at
highest
point
on theandisk.
You
pull
the
string
a constant
A is aof the
9.71.
A vacuum
is looped
over aofshaft
that obeys theequationa(t) =At, wheretisin seconds and
radius
spiral
track. If cleaner
we take belt
the rotation
direction
the of radius
force F~ until the wheel has
madeconstant.
exactly
one-quarter
about
ato horizontal
axis
through
its
center,
and
Challenge
Problems
The cylinder
starts from revolution
rest, and at the end
of the
wheel
of radius
em.
The
of the
0.45 emfJ and
CDthird
be positive,
musta be
positive
so that2.00
r increases
as arrangement
the disc
2
shaft, and
second, the
is 1.80
A. (b) turns
Express
similar
that of
the chain
and sprockets in
• (a) Find
then you let go.
and 6belt,
increases.
(a) wheel
Wbenisthe
disc to
rotates
through
a small
9.99. The moment
of ball's
inertiaacceleration
of a sphere
with m/s
uniform
density
2 Satellite
the through
angular acceleration
the 2disk
as a function
of time. (c)
How
Fig.
9.14. The
motor
turnsthe
thetrack
shaftisatds60.0
angle
d8, the
distance
scanned
along
= r rev/sand
d8. Usingthe moving
about an axis
its center isofiMR
= 0.400MR
•
Z θ2
2 • anthe
belt turns the
much
the disk
has begun
to turn
it reach
angular
in turn
is connected
by another
shaft to
above expression
forwheel,
T( 6), which
integrate
ds to
find the total
disshowtime
thatafter
the earth's
moment
ofinertia
is does
0.3308MR
observations
speed
ofsuggest
15.0done
rad/s?
(d)
what
has the
turned
the roller
that
beats
theasdirt
out of theofrug
(a) Use W =
τz dθ toGeophysical
find the
work
byThrough
the
string.
s scanned
along
the
track
a function
thebeing
total vacuumed.
angle 6 Assume
data
the earth
consists
of angle
five main
re-disktance
that the
the disc
belt has
doesn't
slip on
just ascore
it reaches
(Hint:
wheel. (a) What
rotated.
(b)either
Since the
the shaft
trackor
is the
scanned
θ1
gions: the inner
( T = 015.0
to Trad/s?
= 1220
Ian)See
of Section
average 2.6.)
density through which
Z P2
12,900kg/m3, the outer core (T = 1220km toT= 3480km) of at a constant linear speed v, the distance s found in part (a) is
3
to vt. Use this to find 6 as a function of time. There will be
density
kg/m
, the lower mantle ( T = 3480 Ian to
(b) Use W =
F~ · d~` average
to find
the10,900
work
done
by the string. Do youequal
obtain
the same result as in part (a)?
two solutions for 6; choose the positive one, and explain why this
T = 5700 Ian) of average density 4900 kg/m3, the upper mantle
P1
3
is the solution to choose. (c) Use your expression for 6(t) to find
( T = 5700 Ian to T = 6350 Ian) of average density 3600 kgfm ,
(c) Find the final angular
speed
of the
disk.( r = 6350 km to T = 6370 Ian) of the angular velocity w, and the angular acceleratim a, as funcand the
outer crust
and oceans
average density 2400 kgfm3• (a) Show that the moment of inertia tions of time. Is a, constant? (d) On a CD, the inner radius of the
(d) Find the maximum about
tangential
acceleration
of a point on the disk.
a diameter of a uniform spherical shell of inner radius R" track is 25.0 mm, the track radius increases by 1.55 ,_m per revolution, and the playing time is 74.0 min. Find the values of r 0 and
outerradiusR.,
and
density
pis
I=
p(Pm/15)(Ri(Hint: on
(e) Find the maximum radial (centripetal) acceleration of R.'}.
a point
the disk.
Form the shell by superposition of a sphere of density p and a {J, and find the total number of revolutions made during the playing
Using your results from parts (c) and (d), make
of density -p.) (b) Check the given data by using
A lawn roller in the formsmaller
of a sphere
thin-walled,
hollow cylinder with mass Mtime.
is (e)
pulled
horizontally with2 a constant
them to calculate the mass of the earth. (c) Use the given data to graphs of w, (in rad/s) versus t and a, (m rad/s ) versus t between
2
horizontal force F applied
by
a
handle
attached
to
the
axle.
If
it
rolls
without
slipping,
find the acceleration
t
=
0
andt
=
74.0min.
calculate the earth's moment of inertia in terms of MR •
School of Physics and Optoelectronic Technology
General Physics I
Homework Set 5
5-1
w-s
5-2
5-3
5-4
5-5
j
v
and the friction force.
Problem 6 is on the next page.
5-6
moment of inertia about the rotation axis. A neutron star with
angular speed ru0 = 70.4 rad/s underwent such a glitch in October
1975 that increased its angular speed to ru = ru0 + 11ru, where
11ru/ruo = 2.01 X 10-6• 1f the radius of the neutron star before the
2
glitch was 11 km, by how much did its radius decrease in the starquake? Assume that the neutron star is a uniform sphere.
10.91. A 500.0-g bird is flying Figure 10.63 Problem 10.91.
horizontally at 2.25 rnfs, not
A 500.0-g bird is flying horizontally at 2.25 m/s, not paying much attention,
when it
paying much attention, when it
suddenly flies into a stationary vertical bar, hitting it 25.0 cm below suddenly
the top
flies(see
into athe
stationary
25.0 em
bar, hitting
figure). The bar is uniform, 0.750 m long, has a mass of 1.50 kg, andvertical
is hinged
at itits
below the top (Fig. 10.63). The
base. The collision stuns the bird so that it just drops to the groundbarafterward
(but
is uniform, 0.750
m long,
has a mass of 1.50 kg, and is
soon recovers to fly happily away).
hinged at its base. The collision
the bird so that it just - - - H i n g e - .
(a) What is the angular velocity of the bar just after it is hit by thestons
bird?
drops to the ground afterward
(but soon recovers to fly hap(b) What is the angular velocity of the bar just as it reaches the ground?
pily away). What is the angular velocity of the bar (a) just afterit is
hit by the bird, and (b) just as it reaches the ground?
10.92. A small block with mass 0.250 kg is attached to a string
passing through a hole in a frictionless, horizontal surface (see
Fig. 10.48). The block is originally revolving in a circle with a
radius of 0.800 m about the hole with a tangential speed of
4.00 rnfs. The string is then pulled slowly from below, shortening
the radius of the circle in which the block revolves. The breaking
strength of the string is 30.0 N. What is the radius of the circle
when the string breaks?
10.93. A horizontal plywood disk with mass 7.00 kg and diameter
1.00 m pivots on frictionless bearings about a vertical axis through
its center. You attach a circnlar model-railroad track of negligible
25.oT
·-fl·-
show that
is given b
10.90. A
turntable m
The runne
The turnta
velocity o
the turntab
rotation is
if the runn
the runner
10.97. Rec
earth-moo
ing away
angular m
earth to th
of the moo
E and the
increasing
10.98. Ce
less, horiz
0.800 kg,
the hat (Fi
ter of mas
ing perpe
J = f.:F d
What mus
the bat beg
mass and
two motio
the collis