Halliday, Resnick & Walker Chapter 10 & 11
Rotation, Rolling,
Torque, Angular Momentum
Physics 1A – PHYS1121
Professor Michael Burton
Rotation
10-1 Rotational Variables
!
The motion of rotation
!
The same laws apply
!
But new quantities are needed to express them
o
o
!
Torque
Rotational Inertia (or Moment of Inertia)
A rigid body rotates as a unit, locked together
!
Consider just rotation about a fixed axis
!
This excludes:
o
The Sun, where layers of gas rotate separately
o
A rolling bowling ball, where rotation and translation occur
10-1 Rotational Variables
!
!
The fixed axis is called the axis of rotation
The angular position, !, of a reference line is taken
relative to a fixed direction, the zero angular position
From side
From above
10-1 Rotational Variables
!
Unit is radians (rad): dimensionless
Eq. (10-1)
Eq. (10-2)
!
Do not reset ! to zero after a full rotation
!
Knowledge of !(t) yields the kinematics of the rotation
!
The angular displacement is:
10-1 Rotational Variables
!
!
“Clocks are negative”
i.e. their angular displacement is in the negative
direction!
–
10-1 Rotational Variables
!
Average angular velocity: angular displacement
during a time interval
Eq. (10-5)
!
Instantaneous angular velocity: limit as "t " 0
Eq. (10-6)
!
!
If the body is rigid, these equations hold for all points
on the body
Magnitude of angular velocity = angular speed
10-1 Rotational Variables
!
Average angular acceleration: angular velocity
change during a time interval
Eq. (10-7)
!
Instantaneous angular velocity: limit as "t " 0
Eq. (10-8)
RMM02VD3: Angular Acceleration
10-1 Rotational Variables
!
!
With right-hand rule to determine direction, angular
velocity & acceleration can be written as vectors
If the body rotates around the vector, then the vector
points along the axis of rotation
10-1 Rotational Variables
!
Angular displacements are not vectors, because the order
of rotation matters for rotations around different axes
!
Does not obey the rule for vector addition!
The wheels of a bicycle roll without slipping on a horizontal road. The bicycle is moving due
east at a constant velocity. What is the direction of the angular velocity of the wheels?
a) down
b) west
c) east
d) north
e) south
E
10-2 Rotation with Constant Angular Acceleration
!
The same equations hold as for constant linear
acceleration.
!
!
Simply change linear quantities to angular ones.
Eqns. 10-12 and 10-13 are the basic equations: all
others can be derived from them.
10-3 Relating the Linear
and Angular Variables
!
!
Linear and angular variables are related by r, the
perpendicular distance from the rotational axis
Position (note ! must be in radians):
!
!
Speed:
!
!
! is the angular position
# is the angular velocity (in radians / s)
The Period (in s) is given by:
T=
2"r 2"r 2"
=
=
#r #
v
10-3 Relating the Linear and Angular Variables
!
Tangential acceleration, at :
!
!
$ angular acceleration (radians / s2):
Radial acceleration, ar, can be written in
terms of angular velocity:
An insect rides the rim of a rotating merry-go-round.
If the angular speed, #, of the system (i.e. merry-goround + insect) is constant, does the insect have:
(a) Radial acceleration?
(b) Tangential acceleration?
If # is decreasing, does the insect have:
(c)  Radial acceleration?
(d) Tangential acceleration?
#
10-4 Kinetic Energy of Rotation
!
Kinetic energy, summing over all the particles:
K = % &mivi2
!
But for rotation each point has a different velocity???
!
Write velocity in terms of angular velocity, i.e. vi=#ri
!
I=!miri2 is the rotational inertia, or moment of inertia
!
!
A constant for a rigid object & given rotational axis
Caution: the axis for I must always be specified
10-4 Kinetic Energy of Rotation
!
We write I, the moment of inertia, as:
!
!
The axis must always be specified.
And rewrite the kinetic energy, K, as:
RMM03VD1: Rotational Kinetic Energy
10-5 Calculating the Moment of Inertia
!
Integrating over a continuous body:
Eq. (10-35)
!
For many shapes the Moment of Inertia has been
calculated for commonly used axes
10-5 Calculating the Moment of Inertia
10-5 Calculating the Moment of Inertia
!
Parallel-axis theorem:
!
!
!
For body of mass M, about an axis distance h from
a centre of mass axis, with moment of inertia Icom
Note the axes must be parallel,
and the first must go through the
centre of mass
Does not relate the moment of
inertias for two arbitrary axes
A flat disk, a solid sphere, and a hollow sphere each have the same mass m and
radius r. The three objects are arranged so that an axis of rotation passes through
the centre of each object. The rotation axis is perpendicular to the plane of the flat
disk. Which of the three objects has the largest rotational inertia?
a) The solid sphere and hollow sphere have the same rotational inertia and it
is the largest.
b) The hollow sphere has the largest rotational inertia.
c) The solid sphere has the largest rotational inertia.
d) The flat disk has the largest rotational inertia.
e) The flat disk and hollow sphere have the same rotational inertia and it is
the largest.
10-5 Calculating the Rotational Inertia
Example Calculate the moment of inertia about an axis
through one particle in 2 ways:
o
(a) Summing by particle:
o
(b) Use parallel-axis theorem
Catastrophic release of Rotational Kinetic Energy!
10-6 Torque
!
!
Force needed to rotate an object
depends on
!
(a) the angle of the force and
!
(b) where it is applied
Resolve force into components to
see how it affects rotation
!
Only tangential component
contributes
10-6 Torque
!
Torque takes these factors into account:
!
!
!
!
i.e.
Line extended through the applied force is called the
line of action of the force
Perpendicular distance from the line of action to the
axis is called the moment arm
Unit of torque is the newton-metre, N m
!
Note that 1 J = 1 N m, but torques are never
expressed in joules, torque is not energy
10-6 Torque
!
!
Torque is positive if it would cause a counterclockwise
rotation, otherwise negative
Net torque or resultant torque is the sum of
individual torques
The corner of a rectangular piece of wood is attached to a rod that is free to rotate as shown.
The length of the longer side of the rectangle is 4.0 m, which is twice the length of the shorter
side. Two equal forces are applied to two of the corners with magnitudes of 22 N. What is the
magnitude of the net torque and direction of rotation on the block, if any?
a) 44 N"m, clockwise
b) 44 N"m, counterclockwise
c) 88 N"m, clockwise
4m
2m
d) 88 N"m, counterclockwise
e) zero N"m, no rotation
RMM05VD1: Torque
1-30
10-7 Newton's Second Law for Rotation
!
Rewrite F = ma with rotational variables:
Eq. (10-42)
!
Torque causes
angular acceleration
Figure 10-17
10-8 Work and Rotational Kinetic Energy
!
The rotational work-kinetic energy theorem states:
Eq. (10-52)
!
The work done in a rotation about a fixed axis can be
calculated by:
Eq. (10-53)
!
Which, for a constant torque, reduces to:
Eq. (10-54)
10-8 Work and Rotational Kinetic Energy
!
We can relate work to power with the equation:
Eq. (10-55)
11-1 Rolling as Translation and Rotation Combined
Straight Line
Cycloid
RMM07VD1: Rolling Motion
1 min
11-1 Rolling as Translation and Rotation Combined
!
!
Consider only objects that roll smoothly (no slip)
The centre of mass (c.o.m.) of the object moves in a
straight line parallel to the surface
!
The object rotates around the c.o.m. as it moves
!
The rotational motion is defined by:
#
Eq. (11-1)
Eq. (11-2)
With " =
d#
dt
11-1 Rolling as Translation and Rotation Combined
!
Rolling is a combination of translation and rotation.
!
Result depends on position on the wheel.
Spokes at top blurred as they are
moving faster than those at bottom.
11-1 Rolling as Translation and Rotation Combined
The rear wheel on a bicycle is half the radius of that of
the front wheel.
(a)  When moving is the linear speed on the top of the
rear wheel greater than, less than, or the same as the
top of the front wheel?
i.e. > or = or <
(a)  Is the angular speed of the real wheel greater than,
equal or less than the front wheel?
i.e. > or = or <
11-2 Kinetic Energy of Rolling
!
Combine translational and rotational kinetic energy:
KE of rotation about c.o.m. + KE of translation of c.o.m.
11-2 Forces associated with (smooth) Rolling
!
If a wheel accelerates, its angular speed changes
!
Friction must act to prevent slip, fs (static friction)
Since v com = "R
dv
d"
Then com =
R
dt
dt
i.e. acom = #R
where # is the angular acceleration
Note: if the wheel does slide then the frictional force is the kinetic friction, fk.
The motion is not then smooth and the above equation does not apply.
11-2 Forces associated with smooth Rolling (no slip)
!
For smooth rolling down a ramp:
1.  The gravitational force is vertically down
2.  The normal force is perpendicular to the ramp
3.  The force of friction points up the slope
O  We make use of the angular version of N2L: Torque "net=I#$
11-2 Forces associated with smooth Rolling (no slip)
Thus, applying N2L along direction of ramp (x - direction) :
f s " Mgsin # = Macom,x
Applying rotational form of N2L about an axis through c.o.m.:
Rf s = Icom " (the moment arm for the friction force is R; for other forces it is 0)
Smoothly rolling, so acom,x = "#R
The negative sign because # is positive, but acom,x is negative.
Solve for f s : f s = "Icom
acom,x
R2
And subsitute to yield : acom,x = "
gsin #
1+ Icom / MR 2
11-3 The Yo-Yo
!
!
As a yo-yo moves down a string, it
loses potential energy mgh but gains
rotational and translational kinetic
energy
Yo-yo accelerating down its string is
like a rolling down a ramp:
1. With an angle of 90°
2. Rolling on axle instead of its outer surface
3. Slowed by tension T rather than friction
11-3 The Yo-Yo
So place " = 90° in acom,x = #
to obtain acom = #
gsin "
1+ Icom / MR 2
g
2
1
2 with I com = 2 MR
1+ Icom / MR
Example Calculate the acceleration of the yo-yo:
o
M = 150 grams, R=2 cm, R0 = 3 mm,
o
So Icom = MR2/2 = 3E-5 kg m2
o
Therefore acom = -9.8 m/s2 / (1 + 3E-5 / (0.15 * 0.0032))
= - 0.4 m/s2
11-4 Torque Revisited
!
!
Previously, torque was defined only for a rotating body
and a fixed axis
Now we redefine it for an individual particle that moves
along any path relative to a fixed point
!
The path need not be a circle; torque is now a vector
!
Direction determined with right-hand-rule
Figure 11-10
11-4 Torque Revisited
!
The general equation for torque is:
!
We can also write the magnitude as:
!
Or, using the perpendicular component of force or the
moment arm of F:
11-4 Torque Revisited
The position vector r of a particle points along the positive z-direction.
(i.e. out of the board – remember the right hand rule for xyz and a⌃b)
What is the direction of the force that is causing the torque if the torque is
(a)  Zero
(b)  In the negative x-direction
(c)  In the negative y-direction
11-5 Angular Momentum
!
!
Angular momentum is the angular
counterpart to linear momentum
Direction given by right-hand rule
!
!
!
!
Perpendicular to both r and p
Magnitude
Particle does need not rotate around
O to have angular momentum about it
Unit of angular momentum:
!
kg m2/s, or J s
11-6 Newton's Second Law in Angular Form
!
We rewrite Newton's second law as:
!
i.e. Torque is the rate of change of angular momentum
!
!
The torque and the angular momentum must be
defined with respect to the same point (usually the
origin)
Note the similarity to the linear form:
11-7 Angular Momentum of a Rigid Body
!
!
!
!
We sum the angular momenta of the particles to find
the angular momentum of a system of particles:
The rate of change of the net angular momentum is:
In other words, the net torque is defined by the rate of
change of the net angular momentum:
With
11-7 Angular Momentum of a Rigid Body
11-7 Angular Momentum of a Rigid Body
I=
1
MR 2
2
I = MR 2
I=
2
MR 2
5
11-8 Conservation of Angular Momentum
!
Since we have a new version of Newton's Second
Law, we also have a new conservation law:
Eq. (11-32)
!
The law of conservation of angular momentum
states that, for an isolated system,
(net initial angular momentum) = (net final angular
momentum)
Eq. (11-33)
11-8 Conservation of Angular Momentum
!
!
!
Since these are vector equations, they are equivalent
to the three corresponding scalar equations
This means we can separate axes and write:
If the distribution of mass changes with no external
torque, we have:
11-8 Conservation of Angular Momentum
Examples
!
A student spinning on a stool: rotation speeds up when arms
are brought in, slows down when arms are extended
!
A springboard diver: rotational speed is controlled by tucking
her arms and legs in, which reduces rotational inertia and
increases rotational speed
!
A long jumper: the angular momentum caused by the torque
during the initial jump can be transferred to the rotation of the
arms, by windmilling them, keeping the jumper upright
11-8 Conservation of Angular Momentum
An insect rides the rim of a rotating disk. As it crawls towards the disk’s
centre, do the following increase, decrease or stay the same?
(a)  Moment of Inertia
(b)  Angular Momentum
(c)  Angular Speed
#
10
Summary
Angular Position
!
Angular Displacement
Measured around a rotation
axis, relative to a reference
line:
O  A change in angular position
Eq. (10-4)
Eq. (10-1)
Angular Velocity and
Speed
Angular Acceleration
O  Average and instantaneous
values:
Eq. (10-5)
Eq. (10-6)
O  Average and instantaneous
values:
Eq. (10-7)
Eq. (10-8)
10
Summary
Kinematic Equations
O  Given in Table 10-1 for constant
acceleration
O  Match the linear case
Rotational Kinetic Energy
and Rotational Inertia
Eq. (10-34)
Linear and Angular
Variables Related
O  Linear and angular
displacement, velocity, and
acceleration are related by r
The Parallel-Axis Theorem
O  Relate moment of inertia around
any parallel axis to value around
com axis
Eq. (10-36)
Eq. (10-33)
10
Summary
Torque
O  Force applied at distance from
an axis:
Newton's Second Law in
Angular Form
Eq. (10-39)
O  Moment arm: perpendicular
distance to the rotation axis
Work and Rotational
Kinetic Energy
Eq. (10-53)
Eq. (10-55)
Eq. (10-42)
11
Summary
Rolling Bodies
Torque as a Vector
Eq. (11-2)
Eq. (11-5)
!
Direction given by the righthand rule
Eq. (11-14)
Eq. (11-6)
Angular Momentum of a
Particle
Newton's Second Law in
Angular Form
Eq. (11-23)
Eq. (11-18)
11
Summary
Angular Momentum of a
System of Particles
Angular Momentum of a
Rigid Body
Eq. (11-31)
Eq. (11-26)
Eq. (11-29)
Conservation of Angular
Momentum
Eq. (11-32)
Eq. (11-33)
Precession of a Gyroscope
Eq. (11-46)