Transcript Slide 1

1/21 do now
A particle of mass m slides down a fixed, frictionless sphere of
radius R starting from rest at the top. In terms of m, g, R, and
θ, determine each of the following for the particle while it is
sliding on the sphere.
1. The kinetic energy of the particle
2. The centripetal acceleration of the mass
3. The tangential acceleration of the mass
4. Determine the value of θ at which the particle leaves the
sphere.
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley
Assignment
• Regent book chapter 8 questions are due
Monday
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley
Chapter 10
Dynamics of
Rotational Motion
PowerPoint® Lectures for
University Physics, Twelfth Edition
– Hugh D. Young and Roger A. Freedman
Lectures by James Pazun
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley
Goals for Chapter 10
• To examine examples of torque
• To see how torques cause rotational dynamics (just
as linear forces cause linear accelerations)
• To examine the combination of translation and
rotation
• To calculate the work done by a torque
• To study angular momentum and its conservation
• To relate rotational dynamics and angular
momentum
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley
10.1 Torque
The quantitative measure of the
tendency of a force to change
a body’s rotational motion is
called torque;
Fa applies a torque about point
O to the wrench.
Fb applies a greater torque
about O,
and Fc applies zero torque
about O.
Lines of force and calculations of torques
• The tendency of a force F to cause a rotation about O depends
on
– the magnitude F
– the perpendicular distance l1 between point O and the
line of force. We call the distance l1 the lever arm of
force F1about O.
• We define the torque of the force
F with respect to O as the product
F∙l. We use the Greek letter τ
(tau) for torque.
τ = F∙l
F: force
l: the perpendicular distance between
point O and the line of force
CAUTION: Torque is always
measured about a point
The units of torque
• The SI unit of torque is the Newton-meter.
• Torque is not work or energy, and torque
should be expressed in Newton-meters,
not joules.
If φ is angle between force F and distance r
τ = F∙(r∙sin)
r∙sin - perpendicular distance
τ = r∙(F∙sin)
F∙sin - perpendicular force
The direction of torque
Torque is a vector quantity
Magnitude:
  rF sin 
 
  r F

The direction of torque is
perpendicular to both r and F,
which is directed along the axis of
rotation, with a sense given by the
right-hand rule.
counterclockwise torques are
positive and clockwise torques are
negative.
A dot ● means pointing out of the screen
A cross × means pointing into the screen
Example 10.1 applying a torque
A weekend plumber, unable to loosen a pipe fitting, slips a piece of
scrap pipe over his wrench handle. He than applies his full weight of
900 N to the end of the cheater by standing on it. The distance from
the center of the fitting to the point where the weight acts is 0.80 m,
and the wrench handle and cheater make an angle of 19o with the
horizontal. Find the magnitude and direction of the torque he applies
about the center of the pipe fitting.
Test Your Understanding 10.1
The figure shows a force P being applied to one end of a lever of
length L. What is the magnitude of the torque of this force about
point A
i. PLsinθ
ii. PLcosθ
iii. PLtanθ
Example
•
1.
2.
3.
4.
5.
Torque is the rotational analogue of
kinetic energy
linear momentum
acceleration
force
mass
Example
• A force F of magnitude F making an angle θ with the x axis is
applied to a particle located along axis of rotation A, at
Cartesian coordinates (0,0) in the figure. The vector F lies in
the xy plane, and the four axes of rotation A, B, C, and D all lie
perpendicular to the xy plane.
1. Express the torque about axis A
at Cartesian coordinates (0,0).
2. Express the torque about axis B
in terms of F, θ, φ, π, and/or other
given coordinate data.
b
Example
•
A square metal plate 0.180 m on each side is pivoted about an
axis through point at its center and perpendicular to the plate.
Calculate the net torque about this axis due to the three forces
shown in the figure if the magnitudes of the forces are F1
= 21.0 N , F2 = 17.0 N, and F3 = 14.9 N . The plate and all
forces are in the plane of the page. Take positive torques to be
counterclockwise.
1/23 do now
• Rank the design scenarios (A through C) on the basis of
the tension in the supporting cable from largest to
smallest. In scenarios A, and C, the cable is attached
halfway between the midpoint and end of the pole. In B,
the cable is attached to the end of the pole.
60o
sign
A
45o
30o
sign
sign
B
B, C, A
C
10.2 Torque and Angular Acceleration for a Rigid Body
• For particle 1, Newton’s
second law for the tangential
component is:
F1, tan  m1a1, tan
a1, tan  r1 z
F1, tanr1  m1r1  z
2
 1z  I1 z
For every particle in the body, we add
all the torques:

 ( mi ri ) z
2
iz

z
 I z
τ = Iα is just like F = ma
4 things to note in Eq.

z
 I z
1. The equation is valid only for rigid bodies.
2. Since we used atan = r∙αz, αz must be measured in rad/s2.
3. Since all the internal torques add to zero, so the sum ∑τ
in Eq. ∑τ = Iα includes only the torques of the external
forces.
4. Often, an important external force acting on body is its
weight. We assume that all the weight is concentrated at
the center of mass of the body to get the correct torque
(about any specified axis).
Example 10.2 An unwinding cable I
A cable is wrapped several times around a uniform solid
cylinder that can rotate about its axis. The cylinder has
diameter 0.120 m and mass 50 kg. The cable is pulled with
a force of 9.0 N. Assume that the cable unwinds without
stretching or slipping, what is its acceleration?
Example 10.3 An unwinding cable II
Consider the situation on
the diagram, find the
acceleration of the block
of mass m.
Test Your Understanding 10.2
•
The figure shows a glider of mass m1 that can slide without friction
on horizontal air tract. It is attached to an object of mass m2 by a
massless string. The pulley has radius R and moment of inertia I
about it axis of rotation. When released, the hanging object
accelerates downward, the glider accelerates to the right, and the
string turns the pulley without slipping or stretching. Rank the
magnitudes of the following forces that acting during the motion, in
order from largest to smallest magnitude.
1. The tension force (magnitude T1) in the horizontal part of the string;
2. The tension force (magnitude T2) in the vertical part of the string;
3. The weight m2g of the hanging object.
Example
•
A light rigid massless rod with masses attached to its
ends is pivoted about a horizontal axis as shown
above. When released from rest in a horizontal
orientation, what is the magnitude of the angular
acceleration of the rod?
l
3Mo
2l
Pivot point
Mo
1/24 do now
• Find the magnitude of the angular
acceleration α of the swing bar.
Assignments
• Due Monday 1/27
– Regents book questions 8A-8E
– All classwork packets
1/27 do now
•
1.
2.
3.
4.
5.
If Anya decides to make the star twice as massive, and
not change the length of any crossbar or the location of
any object, what does she have to do with the mass of
the smiley face to keep the mobile in perfect balance?
Note that she may have to change masses of other
objects to keep the entire structure balanced.
make it eight times more massive
make it four times more massive
make it twice as massive
Nothing
impossible to tell
Assignments
• Due today:
– Regents book questions 8A-8E
– The two class work packets
• Due Friday:
– Additional practice packet:
#1,4,8,13,19,22,26,27,28,29
10.3 A rigid body in motion about a moving axis
• When a rigid body rotate about
a moving axis, the motion of
the body is combined
translation and rotation. We
need to combine:
– Translational motion of
the center of mass
– Rotation about an axis
through the center of
mass.
Combined Translation and Rotation: Energy Relationships
• The kinetic energy of a rigid body that has
both translational and rotational motions is the
sum of a part ½ Mvcm2 associated with motion
of the center of mass and a part ½ Icmω2
associated with rotation about an axis through
the center of mass.
1
1
2
2
K  Mv cm  I cm
2
2
Rolling without slipping
vcm = Rω
(condition for rolling without slipping)
If a rigid body changes height as it moves, we must also
consider gravitational potential energy. U = Mgycm
Example 10.4 Speed of a primitive yo-yo
A primitive yo-yo is made by
wrapping a string several times
around a solid cylinder with mass
M and radius R. You hold the end
of the string stationary while
releasing the cylinder with no
initial motion. The string unwinds
but does not slip or stretch as the
cylinder drops and rotates. Use
energy considerations to find the
speed vcm of the center of mass of
the solid cylinder after it has
dropped a distance h.
Example 10.5 Race of the rolling bodies
In a physics lecture demonstrations, an instructor “races”
various round rigid bodies by releasing them from rest at
the top of an inclined plane. What shape should a boby have
to reach the bottom of the incline first?
The smaller the moment of inertia the body has, the faster the
body is moving at the bottom because they have less of their
kinetic energy tied up in rotation and have more available for
translation.
The order of finish is as follows:
Solid sphere
Solid cylinder,
Thin-walled hollow sphere
Thin-walled hollow cylinder
Combined translation and rotation: Dynamics
When a rigid body with total mass M moves, its motion can be
described by combining translational motion and rotational motion


In translational motion:
 Fext  Macm
The rotational motion about the center of mass:

z
 I cmaz
Note: when we learned this equation, we assumed that the axis of
rotation was stationary. But in fact, this equation is valid even
when the axis of rotation moves, provided the following two
conditions are met:
1. The axis through the center of mass must be an axis of symmetry.
2. The axis must not change direction.
Example 10.6 Acceleration of a primitive yo-yo
For the primitive
yo-yo in Example
10.4, find the
downward
acceleration of the
cylinder and the
tension in the string.
Example 10.7 Acceleration of a rolling sphere
A solid bowling ball rolls without slipping down the return ramp at
the side of the alley. The ramp is inclined at an angle β to the
horizontal. What are the ball’s acceleration and the magnitude of the
friction force on the ball? Treat the ball as a uniform solid sphere,
ignoring the finger holes.
Rolling Friction
Test Your Understanding 10.3
•
Suppose the solid cylinder used as a yo-yo in
example 10.6 is replaced by a hollow cylinder
of the same mass and radius
1. Will the acceleration of the yo-yo
a. Increase
b. Decrease,
c. Remain the same?
2. Will the string tension
a. Increase,
b. Decrease,
c. Remain the same?
example
•
•
•
Two uniform identical solid spherical balls each of mass M,
radius r and moment of inertial about its center 2/5MR2, are
released from rest from the same height h above the
horizontal ground Ball A falls straight down, while ball B
rolls down the distance x along the inclined plane without
slipping.
If the velocity of ball A as it hits the ground is VA, what is
the velocity VB of ball B as it reaches the ground?
In terms of acceleration due to earth’s gravity g, determine
the acceleration of ball B along the inclined plane.
A
B
h
x
30o
horizontal
Example
• While exploring a castle, Dan spotted by a dragon who chases him
down a hallway. Dan runs into a room and attempts to swing the
heavy door shut before the dragon gets to him. The door is initially
perpendicular to the wall, so it must be turned through 90o to close it.
The door is 3.00 m tall and 1.25 m wide, and it weighs 750 N. You
can ignore the friction at the hinges. If Dan applies a force of 220 N at
the edge of the door and perpendicular to it, how much time does it
take him to close the door?
If Dan doubles the force, how long
would it take to close the door.
10.4 Work and Power in Rotational Motion
The work dW done by the force Ftan
while a point on the rim moves a
distance ds is dW = Ftan∙ds. If dθ is
measured in radians, then ds = R∙dθ
dW  Ftan  R  d
dW   z  d
2
W    z  d
1
If τz is constant
W   z  (2  1 )   z  
Work done by a constant torque
W   z  (2  1 )   z  
The work done by a constant torque is the product of torque and the
angular displacement. If torque is expressed in Newton-meters and
angular displacement in radian, the work is in joules. Only the
tangent component of force does work, other components do no work.
When a torque does work on a rotating rigid body, the kinetic
energy changes by an amount equal to the work done.
Wtot = ½ I∙ω22 – ½ I∙ω12
Power is the rate of doing work
dW  z d
d
P

z
  z
dt
dt
dt
When a torque acts on a body the rotates with angular
velocity ωz, its power is the product of τz and ωz. This is
the analog of the relationship P = F∙v
Example 10.8 Engine power and torque
The power output of an automobile engine is advertised
to be 200 hp at 6000 rpm. What is the corresponding
torque?
Example 10.9 Calculating power from torque
An electric motor exerts a constant torque of 10 N∙m on a grinding
stone mounted on its shaft. The moment of inertia of the grinding
stone about the shaft is 2.0 kg∙m2. If the system starts from rest, find
the work done by the motor in 8.0 seconds and the kinetic energy at
the end of this time. What was the average power delivered by the
motor?
Test Your Understanding 10.4
•
1.
2.
3.
You apply equal torques to two different cylinders,
one of which has a moment of inertial twice as large
as the other cylinder. Each cylinder is initially at rest
after one complete rotation, which cylinder has the
greater kinetic energy?
The cylinder with the larger moment of inertia;
The cylinder with the smaller moment of inertia;
Both cylinders have the same kinetic energy.
example
• A thin, uniform, 19.0 kg post, 1.95 m long, is held
vertically using a cable and is attached to a 5.00 kg
mass and a pivot at its bottom end as shown. The string
attached to the 5.00 kg mass passes over a massless,
frictionless pulley and pulls perpendicular to the post.
Suddenly the cable breaks. Find the angular acceleration
of the post about the pivot just after the cable breaks.
10.5 angular Momentum
• angular momentum of a particle is a vector quantity
denoted as L.
   

L  r  p  r  mv
The value of L depends on the choice of origin O, since it involves
the particle’s position vector relative to O.
The unit of angular momentum are kg∙m2/s.
L is perpendicular to the xy-plane. It direction is determined by
the right-hand rule.
The magnitude of L is L = (mvsinΦ)r or L = mv(rsinΦ)
Perpendicular
component of p
or Perpendicular
component of r
• When a net force F acts on a particle, its velocity and
momentum changes:

dp 
 Fnet
dt
• Similarly, when the torque of the net force acting on a
particle, its angular velocity and angular momentum
changes:


dL  
 r  Fnet   net
dt
The derivation of
If we take the time derivative of
the rule for the derivative of a product:
, using
Angular Momentum of a Rigid Body
In a rigid body rotating around an axis, each
particle moves in a circle centered at the
origin, and at each instant its velocity vi is
perpendicular to its position vector ri. Hence
Φ = 90o for every particle. A particle with
mass mi at a distance ri from O has a speed
vi = riω. The magnitude of its angular
momentum is:
Li = (mivisinΦ)ri = (miriω)ri = miri2ω
The total angular momentum of the slice of the
body lying in the xy-plane is the sum ∑Li of the
angular momenta Li of the particles:


2
L   Li  ( mi ri )  I
Example
•
A particle of mass m moves with a constant speed v
along the dashed line y = a. When the x-coordinate of
the particle is xo, what is the magnitude of the angular
momentum of the particle with respect to the origin of
the system?
y
(0, a)
v
m
xo
x
Example 10.10 Angular momentum and torque
A turbine fan in a jet engine has a moment of inertia of 2.5 kg∙m2 about
its axis of rotation. As the turbine is starting up, its angular velocity as a
function of time is
3 2
z  (40rad / s )t
a. Find the fan’s angular momentum as a function of time, and find its
value at time t = 3.0 s
b. Find the net torque acting on the fan as a function of time, and find
the torque at time t = 3.0 s.
Example
•
A uniform rod of mass M and length L has a moment of
inertia about one end I = ML2/3. It is released from rest
in horizontal direction about the fixed axis
perpendicular to the paper as shown below.
Axis
P
cm
A
1. What is the linear velocity of the center of mass, cm,
when the rod is in the vertical position?
2. What is the angular momentum of the rod about the
axis of rotation at one end?
Test Your Understanding 10.5
•
A ball is attached to one end of a piece
of string. You hold the other end of the
string and whirl the ball in a circle around
your hand.
1. If the ball moves at a constant speed, is
its linear momentum p constant? Why or
why not?
2. Is its angular momentum L constant?
why or why not?
10.6 conservation of Angular Momentum
• Like conservation of energy and of linear momentum, the
principle of conservation of angular momentum is a universal
conservation law, valid at all scales from atomic and nuclear
systems to the motions of galaxies.
• The principle follows directly from equation: ∑τ = dL/dt
• If ∑τ = 0, then dL/dt = 0, and L is constant.
When the net external torque acting on a system is zero, the
total angular momentum of the system is constant (conserved).
• A circus acrobat, a diver, and an ice skater pirouetting on the
toe of one skate all take advantage of this principle.
I1ω1z = I2ω2z
When a system has several parts, the internal forces that the parts
exert on each other cause changes in the angular momenta of the
parts, but the total angular momentum doesn’t change. The total
angular momentum of the system is constant.
Example 10.11 Anyone can be a ballerina
An acrobatic physics professor stands at the center of a turntable,
holding his arms extended horizontally with a 5.0 kg dumbbell in each
hand. He is set rotating about a vertical axis, making one revolution in
2.0 s. Find the prof’s new angular velocity if he pulls the dumbbells in to
his stomach. His moment of inertia without the dumbbells is 3.0 kgm2
when his arms are out-stretched, dropping to 2.2 kgm2 when his hands
are at his stomach. The dumbbells are 1.0 m from the axis initially and
0.20 m from it at the end. Treat the dumbbells as particles.
Example 10.12 A rotational “collision” I
The figure bellow shows two
disks: A is an engine flywheel,
and B is a clutch plate attached to
a transmission shaft. Their
moments of inertia are IA and IB;
initially, they are rotating with
constant angular speeds ωA and
ωB, respectively. We then push
the disks together with forces
acting along the axis, so as not to
apply any torque on either disk.
The disks rub against each other
and eventually reach a common
final angular speed ω. Derive an
expression for ω.
Example 10.12 A rotational “collision” II
In Example 10.12, suppose flywheel A has a mass of 2.0 kg, a radius
of 0.20 m, and an initial angular speed of 50 rad/s and that clutch
plate B has a mass of 4.0 kg, a radius of 0.10 m, and an initial angular
speed of 200. rad/s. Find the common final angular speed ω after the
disks are pushed into contact. What happens to the kinetic energy
during this process?
Example 10.4 Angular momentum in a crime bust
A door 1.00 m wide, of mass 15 kg, is hinged at one side so that it can
rotate without friction about a vertical axis. It is unlatched. A police
officer fires a bullet with a mass of 10 g and a speed of 400 m/s into the
exact center of the door, in a direction perpendicular to the plane of the
door. Find the angular speed of the door just after the bullet embeds itself
in the door. Is kinetic energy conserved?
Test Your Understanding 10.6
•
1.
2.
3.
If the polar ice caps were to completely melt due to global
warming, the melted ice would redistribute itself over the
earth. This change would cause the length of the day (the
time needed for the earth to rotate once in its axis) to
Increase
Decrease
Remain the same
(hint: use angular momentum ideas. Assume that the sun,
moon, and planets exert negligibly small torques on the
earth.)
L = Iω
I increases, ω decreases, days is
longer.
Example
•
A uniform rod of mass M and length L has a
moment of inertia about one end I = ML2/3. It is
released from rest in horizontal direction about the
fixed axis perpendicular to the paper as shown
below.
Axis
P
cm
A
1. What is the linear velocity of the center of mass, cm,
when the rod is in the vertical position?
2. What is the angular momentum of the rod about the
axis of rotation at one end?
example
•
Two equal masses, each m, are resting at the ends of
a uniform rod of length 2a and negligible mass. The
system is in equilibrium about the center C of the rod.
A piece of clay of mass m is dropped down on the
mass at the right end, hits it with velocity v as shown
below and sticks to it.
m
v
m
a
a
m
C
•
What is the ratio of the kinetic energy Ef just after the
collision to the kinetic energy Ei just before the
collision, Ef/Ei, of the system?
example
•
A skater is spinning on ice with her arms
outstretched about the vertical axis at an angular
speed of ω. When she brings her arms close to her
body, which of the following statements is correct?
A.
Her angular velocity and angular momentum remain
constant.
Her angular momentum is increased.
Her kinetic energy is increased.
Her kinetic energy is decreased.
The net torque on her about the axis of rotation increases.
B.
C.
D.
E.
example
•
A uniform diving board, 12 meters long and 20 kg in
mass, is hinged at P, which is 5 meters from the edge
of the platform. An 80 kg diver is standing at the other
end of the board.
P
5m
0
hinge
platform
1.
2.
What will be the force exerted by the hinge on the
board?
What will be the normal force on the board at the edge
of the platform?
Example
M1
L
M2
P
• Masses M1 and M2 are separated by a distance L. what is the
distance of the center of mass of the system at P from M1 as
shown above?
• What is the moment of inertial of the system about the center
of mass at P?
example
M
axis
P
M
h
L
rod
•
•
A mass M slides down a smooth surface from height h
and collides inelastically with the lower end of a rod
that is free to rotate about a fixed axis at P as shown
below. The mass of the rod is also M, the length is l,
and the moment of inertial about P is ml2/3.
What is the angular velocity of the rod about the axis P
jut after the mass sticks to it?
Example
• A solid, uniform cylinder with mass 8.45 kg and diameter
11.0 cm is spinning with angular velocity 230 rpm on a
thin, frictionless axle that passes along the cylinder axis.
You design a simple friction-brake to stop the cylinder by
pressing the brake against the outer rim with a normal
force. The coefficient of kinetic friction between the brake
and rim is 0.334. What must the applied normal force be
to bring the cylinder to rest after it has turned through 5.15
revolutions?
Example
•
A wheel with a weight of 395 N comes off a moving truck
and rolls without slipping along a highway. At the bottom
of a hill it is rotating at an angular velocity of 26.8 rad/s.
The radius of the wheel is 0.652 m and its moment of
inertia about its rotation axis is 0.800 MR2 . Friction does
work on the wheel as it rolls up the hill to a stop, at a
height of above the bottom of the hill; this work has a
magnitude of 3520 J. Calculate h. Use 9.81 m/s2 for
the acceleration due to gravity.
Example
• The radius of the pulley is r and mass M is initially at height h.
The system is initially at rest and is then released at time t = 0.
Assume M>m.
• Assuming the pulley to be massless and frictionless, what is the
angular acceleration of the pulley while M is falling?
r
M
m
h
Example
•
An object of moment of inertial I is initially
at rest when torque T begins to act on it as
shown below. After t seconds,
1. what is the angular velocity of the object in
terms of T, I and t?
2. What is the kinetic energy of the object?
T
Moment of
inertial I
Example
• A solid, uniform cylinder with mass 8.45 kg and diameter
11.0 cm is spinning with angular velocity 230 rpm on a
thin, frictionless axle that passes along the cylinder axis.
You design a simple friction-brake to stop the cylinder by
pressing the brake against the outer rim with a normal
force. The coefficient of kinetic friction between the brake
and rim is 0.334. What must the applied normal force be
to bring the cylinder to rest after it has turned through 5.15
revolutions?
Example
•
A hoop is rolling to the right without slipping on a horizontal
floor at a steady 1.8 m/s (Vcm).
1. Find the velocity vector of each of the following points, as
viewed by a person at rest on the ground:
A. The highest point on the hoop
B. The lowest point on the hoop
C. A point on the right side of the hoop, mideway abetween
the top and the bottom
2. find the velocity vector of each of the above points, as
viewed by a person moving along with the same velocity as
the hoop.
Example
•
Find the magnitude of the angular momentum of
the second hand on a clock about an axis
through the center of the clock face. The clock
hand has a length of 15.0 cm and a mass of
6.00 g. Take the second hand to be a slender
rod rotating with constant angular velocity about
one end.
F