PHYS 1443-501, Spring 2004

1
PHYS 1443 Section 501Lecture 20
Monday Apr 12, 2004 Dr. Andrew Brandt

• Rotation Review
• Torque
• Moment of Inertia
• Rotational Kinetic Energy
• Torque and Vector Products

2
Announcements

• HW9 on Ch. 10 will be assigned tomorrow and is
  due 4/19. There will be two more homeworks after
• this. 4/265/3. It would help you to do
  them.
• Updated grades will be posted tomorrow

3
Fundamentals of Rotation
Consider a motion of a rigid body an object
that does not change its shape rotating about
the axis protruding out of the slide.
One radian is the angle swept by an arc length
equal to the radius of the arc.
Since the circumference of a circle is 2pr,
4
Rotational Kinematics
The first type of motion we learned about in
linear kinematics was under a constant
acceleration. We will learn about the rotational
motion under constant angular acceleration,
because these are the simplest motions in both
cases.
Just like the case in linear motion, one can
obtain
Angular Speed under constant angular acceleration
Angular displacement under constant angular
acceleration
One can also obtain
5
Example for Rotational Kinematics
A wheel rotates with a constant angular
acceleration of 3.50 rad/s2. If the angular
speed of the wheel is 2.00 rad/s at ti0, a)
through what angle does the wheel rotate in 2.00s?
Using the angular displacement formula in the
previous slide, one gets
What is the angular speed at t2.00s?
Using the angular speed and acceleration
relationship
Find the angle through which the wheel rotates
between t2.00 s and t3.00 s.
6
Rotational Accelerations
How many different types of linear acceleration
do you see in a circular motion and what are they?
Two
Tangential, at, and the radial acceleration, ar.
Since the tangential speed vt is
The magnitude of tangential acceleration at is
Although every particle in the object has the
same angular acceleration, its tangential
acceleration differs proportional to its distance
from the axis of rotation.
What does this relationship tell you?
The radial or centripetal acceleration ar is
What does this tell you?
The father away the particle from the rotation
axis the more radial acceleration it receives.
In other words, it receives more centripetal
force.
Total linear acceleration is
7
Example for Rotational Motion
Audio information on compact discs are
transmitted digitally through the readout system
consisting of laser and lenses. The digital
information on the disc are stored by the pits
and flat areas on the track. Since the speed of
readout system is constant, it reads out the same
number of pits and flats in the same time
interval. In other words, the linear speed is
the same no matter which track is played. a)
Assuming the linear speed is 1.3 m/s, find the
angular speed of the disc in revolutions per
minute when the inner most (r23mm) and outer
most tracks (r58mm) are read.
Using the relationship between angular and
tangential speed
b) The maximum playing time of a standard music
CD is 74 minutes and 33 seconds. How many
revolutions does the disk make during that time?
c) What is the total length of the track past
through the readout mechanism?
d) What is the angular acceleration of the CD
over the 4473s time interval, assuming constant a?
8
Rolling Motion of a Rigid Body
What is a rolling motion?
A more generalized case of a motion where the
rotational axis moves together with the object
A rotational motion about the moving axis
To simplify the discussion, lets make a few
assumptions

1. Limit our discussion on very symmetric objects,
   such as cylinders, spheres, etc
2. The object rolls on a flat surface

Lets consider a cylinder rolling without
slipping on a flat surface
Under what condition does this Pure Rolling
happen?
The total linear distance the CM of the cylinder
moved is
Thus the linear speed of the CM is
Condition for Pure Rolling
9
More Rolling Motion of a Rigid Body
The magnitude of the linear acceleration of the
CM is
As we learned in the rotational motion, all
points in a rigid body moves at the same angular
speed but at a different linear speed.
At any given time the point that comes to P has 0
linear speed while the point at P has twice the
speed of CM
Why??
CM is moving at the same speed at all times.
A rolling motion can be interpreted as the sum of
Translation and Rotation


10
Torque
Torque is the tendency of a force to rotate an
object about an axis. Torque, t, is a vector
quantity.
Consider an object pivoting about the point P by
the force F being exerted at a distance r.
The line that extends from the tail of the force
vector is called the line of action.
The perpendicular distance from the pivoting
point P to the line of action is called Moment
arm.
Magnitude of torque is defined as the product of
the force exerted on the object to rotate it and
the moment arm.
When there are more than one force being exerted
on certain points of the object, one can sum up
the torque generated by each force vectorially.
The convention for sign of the torque is positive
if rotation is in counter-clockwise and negative
if clockwise.
11
Moment of Inertia
Measure of resistance of an object to changes in
its rotational motion. Equivalent to mass in
linear motion.
Rotational Inertia
For a group of particles
For a rigid body
What are the dimension and unit of Moment of
Inertia?
Determining Moment of Inertia is extremely
important for computing equilibrium of a rigid
body, such as a building.
12
Torque Angular Acceleration
Lets consider a point object with mass m
rotating on a circle.
What forces do you see in this motion?
The tangential force Ft and radial force Fr
The tangential force Ft is
The torque due to tangential force Ft is
What do you see from the above relationship?
What does this mean?
Torque acting on a particle is proportional to
the angular acceleration.
What law do you see from this relationship?
Analogs to Newtons 2nd law of motion in rotation.
How about a rigid object?
The external tangential force dFt is
The torque due to tangential force Ft is
The total torque is
What is the contribution due to radial force and
why?
Contribution from radial force is 0, because its
line of action passes through the pivoting point,
making the moment arm 0.
13
Example for Torque and Angular Acceleration
A uniform rod of length L and mass M is attached
at one end to a frictionless pivot and is free to
rotate about the pivot in the vertical plane.
The rod is released from rest in the horizontal
position. What are the initial angular
acceleration of the rod and the initial linear
acceleration of its right end?
The only force generating torque is the
gravitational force Mg
Since the moment of inertia of the rod when it
rotates about one end
Using the relationship between tangential and
angular acceleration
We obtain
What does this mean?
The tip of the rod falls faster than an object
undergoing a free fall.
14
Rotational Kinetic Energy
What do you think the kinetic energy of a rigid
object that is undergoing a circular motion is?
Kinetic energy of a masslet, mi, moving at a
tangential speed, vi, is
Since a rigid body is a collection of masslets,
the total kinetic energy of the rigid object is
Since moment of Inertia, I, is defined as
The above expression is simplified as
15
Example for Rigid Body Moment of Inertia
Calculate the moment of inertia of a uniform
rigid rod of length L and mass M about an axis
perpendicular to the rod and passing through its
center of mass.
The line density of the rod is
so the masslet is
The moment of inertia is
What is the moment of inertia when the rotational
axis is at one end of the rod.
Will this be the same as the above. Why or why
not?
Since the moment of inertia is resistance to
motion, it makes perfect sense for it to be
harder to move when it is rotating about the axis
at one end.
16
Example for Moment of Inertia
A system consists of four small spheres as shown
in the figure assuming the radii are negligible
and the rods connecting the particles are
massless, compute the moment of inertia and the
rotational kinetic energy when the system rotates
about the y-axis at w.
Since the rotation is about y axis, the moment of
inertia about y axis, Iy, is
This is because the rotation is done about y
axis, and the radii of the spheres are negligible.
Why are some 0s?
Thus, the rotational kinetic energy is
Find the moment of inertia and rotational kinetic
energy when the system rotates on the x-y plane
about the z-axis that goes through the origin O.
Does it make sense that KE is bigger?
17
Calculation of Moments of Inertia
Moments of inertia for large objects can be
computed, if we assume the object consists of
small volume elements with mass, Dmi.
The moment of inertia for the large rigid object
is
It is sometimes easier to compute moments of
inertia in terms of volume of the elements rather
than their mass
How can we do this?
Using the volume density, r, replace dm in the
above equation with dV.
The moments of inertia becomes
Example Find the moment of inertia of a uniform
hoop of mass M and radius R about an axis
perpendicular to the plane of the hoop and
passing through its center.
The moment of inertia is
The moment of inertia for this object is the same
as that of a point of mass M at the distance R.
What do you notice from this result?
18
Parallel Axis Theorem
Moments of inertia for highly symmetric object is
easy to compute if the rotational axis is the
same as the axis of symmetry. However if the
axis of rotation does not coincide with axis of
symmetry, the calculation can still be done in
simple manner using parallel-axis theorem. where
D is the distance from the new axis to the
center-of-mass.
Moment of inertia of any object about any
arbitrary axis are the same as the sum of moment
of inertia for a rotation about the CM and that
of the CM about the rotation axis.
What does this theorem tell you?
19
Parallel Axis Theorem
Moments of inertia for highly symmetric object is
easy to compute if the rotational axis is the
same as the axis of symmetry. However if the
axis of rotation does not coincide with axis of
symmetry, the calculation can still be done in
simple manner using parallel-axis theorem.
Moment of inertia is defined
Since x and y are
One can substitute x and y in Eq. 1 to obtain
D
Since the x and y are the distance from CM, by
definition
Therefore, the parallel-axis theorem
What does this theorem tell you?
Moment of inertia of any object about any
arbitrary axis are the same as the sum of moment
of inertia for a rotation about the CM and that
of the CM about the rotation axis.
20
Example for Parallel Axis Theorem
Calculate the moment of inertia of a uniform
rigid rod of length L and mass M about an axis
that goes through one end of the rod, using
parallel-axis theorem.
The line density of the rod is
so the masslet is
The moment of inertia about the CM
Using the parallel axis theorem
The result is the same as using the definition of
moment of inertia. Parallel-axis theorem is
useful to compute moment of inertia of a rotation
of a rigid object with complicated shape about an
arbitrary axis
21
Torque and Vector Product
Lets consider a disk fixed onto the origin O and
the force F is exerted on the point p. What
happens?
The disk will start rotating counter clockwise
about the Z axis
The magnitude of torque given to the disk by the
force F is
But torque is a vector quantity, what is the
direction? How is torque expressed
mathematically?
What is the direction?
The direction of the torque follows the
right-hand rule!!
The above quantity is called Vector product or
Cross product
What is the result of a vector product?
What is another vector operation weve learned?
Another vector
Scalar product
Result? A scalar
22
Properties of Vector Product
Vector Product is Non-commutative
What does this mean?
If the order of operation changes the result
changes
Following the right-hand rule, the direction
changes
Vector Product of two parallel vectors is 0.
Thus,
If two vectors are perpendicular to each other
Vector product follows distribution law
The derivative of a Vector product with respect
to a scalar variable is
23
More Properties of Vector Product
Vector product of two vectors can be expressed in
the following determinant form
24
Similarity Between Linear and Rotational Motions
All physical quantities in linear and rotational
motions show striking similarity.
Quantities Linear Rotational
Mass Mass Moment of Inertia
Length of motion Distance Angle (Radian)
Speed
Acceleration
Force Force Torque
Work Work Work
Power
Momentum
Kinetic Energy Kinetic Rotational