Honors Physics
Chapter 8 Notes
Mr. Fedell
(Teacher edition)
Problems
Useful diagrams and figures… This book has excellent figures and tables – most of them
have something to offer you intellectually.
Note: The concepts in this chapter are very similar to the concepts we learned in chapter 2
through 7 regarding kinematic motion in one and two dimensions, Newton’s Laws, the
conservation of mechanical energy, and the conservation of momentum.
You should try to relate the concepts to those in previous chapters. It will help you understand
this chapter.
8.1- Angular Quantities
Rigid bodies have
a)
b) purely rotational motion
Purely
motion
c) all points in the object rotate in circles
d) rotate around a fixed axis of rotation (
)
Angular Displacement
When an object moves along a straight path, we describe
how far it has moved by its
. When an
object rotates we describe how far it has rotated by its
angular displacement (
).
The mathematics of circular motion is much simpler if we
measure the angle in radians rather than degrees.
() - an angle whose arc length is equal to its radius (1 radian =
In general where r is the radius of the circle and s is the arc length
angle
∆𝜃 =
)
by the
If the angle does a complete revolution, then Δs = 2r
So… 360= 2 radians
and… 1 radian is about 57.3
Note that as the object rotates,
undergoes the
We will use the
method
… in terms of SIG FIGS-what you start with you end with!
Examples –
a. convert 80.73 degrees to radians
b. convert 2.1 radians to degrees
c. convert 1.4 revolutions to radians
d. convert 23.801 radians to revolutions
e. Find the arc length of two adjacent satellites which are in an orbit of r = 4.23 x 107m. Their
orbits are in the plane of the equator and have an angular separation of θ = 2.00.
f. The length of one day on Uranus is approximately 17 hours long. The radius of Uranus is
2.56 x 107 m. What is the Δθ in both degrees and radians that Uranus moves in 1 hour?
The angular displacement will play the same role in angular kinematics that the displacement x,
played in linear kinematics.
.
Counterclockwise rotations to have a (
and Clockwise rotations to have a (
) angular displacement
) angular displacement
Angular Velocity
Defined in relationship to the angular displacement in the same way that the linear velocity was
defined in relationship to the linear displacement.
The average angular velocity is given by the Greek letter omega (
), and is defined as the rate
of change of the angular displacement.
units:
(
/
)
𝜔=
∆𝜃
∆𝑡
Examples –
a) A rigid rotating body makes one complete revolution in 2 s. What is its average angular
velocity (in rad/s)?
b) A gymnast on a high bar swings through two revolutions in a time of 1.90s. Find the
average angular velocity (in rad/s and rev/s) of the gymnast.
c) How fast is the outer edge of a CD (at 6.0 cm) moving when it is rotating at its top speed
of 22.0 rad/s?
Angular Acceleration
Similarly, the average angular acceleration is defined as the rate of change of the angular
velocity, and is given by the Greek letter
().
units: radian per second squared (rad/s2)
Examples –
a) The angular velocity of a rotating disk increases from 2 rad/s to 5 rad/s in 5 s. What’s the
disk’s average angular acceleration?
b) Find the angular acceleration of the jets fan blades. A jet is stopped; looking at the front
of an engine, the fan blades are rotating with an angular velocity of -110 rad/s. As the
plane takes off, the angular velocity of the blades reaches -330 rad/s in a time of 14s.
Relationship between Angular and Linear Quantities
Before using these angular quantities let’s look at the relationship between them and the linear
quantities we have used. We know that velocity is defined as the change in distance divided by
the change in time. (In this case, the velocity is in the direction tangent to the circumference of
the circle, the tangential velocity (vT), so the distance is given by the arc length).
𝑣𝑇 =
∆𝑠 𝑟∆𝜃
=
= 𝑟𝜔
∆𝑡
∆𝑡
Also, the tangential component of the acceleration is
𝑎𝑇 =
∆𝑣𝑇 𝑟∆𝜔
=
= 𝑟𝛼
∆𝑡
∆𝑡
Recall the definition of centripetal acceleration.
𝑣𝑇 2
𝑎𝑟 =
= 𝜔2
𝑟
with total acceleration: a = aT + aR and since aT and aR are at right angles to each other, the
magnitude of the total acceleration is given by,
𝑎 = √𝑎 𝑇 2 + 𝑎𝑅 2
NIB – Other Useful Definitions
Frequency is defined as the number of revolutions per unit time. To change from frequency to
angular frequency, you must multiply frequency by 2 since one revolution is 2 radians. (1
rev/s)* (2 radians/rev) gives radians/sec which is the units of angular velocity.
The period is the time required to make one complete revolution, so
𝑇=
1
𝑓
NIB - Rotation and translation
All motion is a combination of rotation and translation
pure translation
pure rotation
translation and rotation
8.2 Constant Angular Acceleration
The definitions of the angular quantities are analogous to the definitions of linear quantities with
 playing the role of x,  playing the role of v, and  playing the role of a. Consequently, the
kinematic equations derived in chapter 2 are valid for rotational motion with constant angular
acceleration using the same derivations done in chapter 2. We get,
𝜔 = 𝜔𝑜 + 𝛼𝑡
𝜔2 = 𝜔𝑜 2 + 2𝛼𝜃
1
𝜃 = 𝜔𝑜 𝑡 + ( ) 𝛼𝑡 2
2
1
𝜃 = ( ) (𝜔𝑜 + 𝜔)𝑡
2
Examples –
a) A compact disk player starts from rest and accelerates to its final velocity of 3.50 rev/s in
1.50 s. What is the disk’s average angular acceleration?
b) How many revolutions does the CD from the previous problem make while coming up to
speed?
8.4 Torque
We have described angular motion, now we must talk about the dynamics of angular motion.
What causes something to rotate?
- a force caused an acceleration (a force caused an object to move)
also takes a force to start something rotating, we call this torque ( )
Important factors include:
a)
the force (torque) is applied
b) force must be applied some
from the axis of rotation
c) the lever arm is the
from the axis of rotation
to the line of action which the force acts
Torque – the force that is applied tangent to the circle and at some lever arm distance causing
rotation around a specific point (pivot)
*Note:
, but it acts like one. You could think of torque as a
“turning force”; it causes objects to rotate either clockwise or counterclockwise.*
In general we define torque as the product of
force and the length of the lever arm:
𝜏=
units:
(Nm)
= 𝑟𝐹𝑠𝑖𝑛𝜃
where  is the angle between the direction of the force
and a line drawn from the axis of rotation to the force
To produce
torque requires:
1. a force applied at some
from the pivot at
some angle other than directly through the pivot point
2.
the lever arm
3.
the force
4. maximize the torque by making the angle  =
Because torque has a distance in its definition (the
), the torque is only defined
around a certain axis of rotation (
). The torque around two different axes of rotation
may be very different.
Torques with counterclockwise rotations have a (
)
Torques with clockwise rotations have a (
)
Examples –
a) A 150kg man stands 2 m from the end of a diving board. How much torque does he apply
to the base where the board is attached assuming the board remains horizontal?
b) A student pulls down with a force of 40 N on a rope that winds a pulley of radius 5 cm.
What’s the torque of this force?
8.5 – Rotational Dynamics; Torque and Rotational Inertia
Recall from linear motion
- the property of a body that
For rotational motion
- the moment of inertia (I) resists change in
was called mass
So…I is just a measure of how
with
and
it is to start an object rotating and increases
from axis of rotation
The larger the moment of inertia the _______ an objects resistance to change its angular velocity
The smaller the moment of inertia the _________ an objects resistance to change its angular
velocity
*Remember…
produce translational motions and
produce rotational motions*
8.8 Angular Momentum and Its Conservation
Recall from linear motion
o that linear momentum is defined as p = mv
For rotational motion
o angular momentum (
) of a rigid body rotating is defined as L = rmv or L = Iω
just as linear momentum is conserved if there is no net external force,
is also conserved (it does not change) if there is no net external torque.
The Law of Conservation of angular moment states: The total angular momentum of a
rotating body remains constant if the net external torque acting on it is zero.
Some examples of this include an ice skater who is rotating and pulls her arms in or a diver who
goes into a tucked position. Why do they rotate faster?
As the
from the axis of
rotation decreases, the angular velocity
(ω) must
for L to be the same
Example –
The Johnsville Space Center’s centrifuge is the world’s largest. The arm of the centrifuge
extends 60ft from the center. The gondola, which the astronauts-in-training ride in, is at the end
of this arm. The mass of the gondola is 225 kg and reaches a top speed of 77 m/s. If run at top
speed, what is the angular momentum of the gondola?