Ch. 8: Rotational Equilibrium and Dynamics
Objectives
• Know the distinction between translational
motion and rotational motion.
• Understand the concept of torque and be able
to make related calculations.
Rolling Race
Roll various objects down a ramp: spheres, solid
cylinders, hollow cylinders, and washers.
Compare which objects are the fastest. Develop
a hypothesis regarding what factor(s) affect the
relative speeds of the different objects.
Masses and Motion
point masses (center
of mass) can have
translational motion
extended masses can have
rotational motion
Each type of motion can be analyzed separately.
Torque
• torque: the ability of a
force to rotate an object
around an axis (t)
• t = F·d·sinq
• vector quantity
• clockwise (─)
• counterclockwise (+)
• St = t1 + t2 + t3 + …
F
q
d
Net Torque Problem
Jack (244 N) and Bill (215 N) are sitting at opposite ends
of a horizontal teeter-totter. If Jack is sitting 1.75 m
from the center and Jill is sitting 1.95 m from the
center, what is the net torque? What is the net torque
if the teeter-totter is oriented upward at a 15o angle
toward Jill’s end?
Objectives
• Understand the concept of “center-of-mass.”
• Be able to find the center of mass for an
irregularly-shaped object.
• Understand the concept of “moment of
inertia.”
• Be able to compare the moment of inertia for
differently-shaped objects.
• Understand the concept of rotational
equilibrium and make related calculations.
Center of Mass
center of mass: the point around which an object
rotates if gravity is only force acting (see video)
Center of Mass
“Fosbury Flop”
An object will “topple” once its center
of mass is no longer supported by a pivot.
Finding the Center of Mass
• Follow the directions for the “Quick Lab” on
page 284.
• Predict the location of the center of mass
before you proceed.
• You don’t need to write anything—just for fun.
Moment of Inertia
• moment of inertia (I):
the tendency of an
object to resist
changes in rotational
motion
• related to mass
distribution
• This is why hoops
accelerate slowly and
spheres quickly
• torque needed to
rotate differs (try book)
Moment of Inertia
Moment of Inertia Problem
• What is the moment of inertia of a 35 gram
metal cylinder with r = 0.015 cm rolling down
an incline?
Moment of Inertia Questions
• Does a single object have a single moment of
inertia? Explain.
• What shape/axis would have the largest
moment of inertia theoretically?
• Why do bicycles have such large, yet thin
tires?
Rotational Equilibrium
• Translational equilibrium: SF = 0
(no linear acceleration)
• Rotational equilibrium: St = 0
(no rotational acceleration)
• Any axis can be used—choose for simplicity!
Rotational Equilibrium
A 5.55 N meter stick is suspended from two spring
scales (one at each end). A 9.05 N mass is hung at the
65.0-cm mark. How much force is applied by each
spring scale (scale A, scale B)?
Objectives
• Understand the concepts of angular speed
and angular acceleration.
• Be able to make angular speed and angular
acceleration calculations.
Radians
• Angles can be measured in “radians.”
• 𝜃=
𝑎𝑟𝑐 𝑙𝑒𝑛𝑔𝑡ℎ (𝑠)
𝑟𝑎𝑑𝑖𝑢𝑠 (𝑟)
• 1 radian = 57.3o
• 2p rad = 360o
s
q
r
Angular Speed
• speed = distance / time
• angular speed = angular distance / time
• 𝜔=
∆𝜃
𝑡
or 𝜔 =
2𝜋 𝑟𝑎𝑑
𝑇
• measured in rad/s
• What is the angular speed of a carousel with a period
of 8.5 seconds?
Angular to Tangential Speed
• 𝜔=
• 𝜔=
𝑠
𝜃
𝑟
=
𝑡
𝑡
𝑣𝑡
𝑟
=
𝑠
𝑡∙𝑟
=
𝑣𝑡
𝑟
• tangential speed: 𝑣𝑡 = 𝑟 ∙ 𝜔
• What is the tangential speed a child sitting 3.5 m
from the center of the carousel in the previous
problem?
Angular Acceleration
• Angular acceleration is analogous to linear
acceleration. It is a change in the rate of rotation.
• 𝛼=
∆𝜔
𝑡
=
𝜔𝑓 − 𝜔𝑖
𝑡
• Tangential acceleration: 𝑎𝑡 = 𝑟 ∙ 𝛼
• The angular speed of a camshaft increases from 145
rad/s to 528 rad/s in 0.75 s. What is a? What is
tangential acceleration of the shaft (r = 0.052 m) at
the end?
Objectives
• Understand and use Newton’s second law for
rotation.
• Understand and apply the concept of angular
momentum.
• Understand and apply the concept of
rotational KE.
Second Law for Rotation
• Translational
• Rotational
2nd
2nd
Law: Σ𝐹 = 𝑚𝑎 or 𝑎 =
Law: Σ𝜏 = 𝐼𝛼 or 𝛼 =
Σ𝐹
𝑚
𝛴𝜏
𝛪
• What is the angular acceleration of a 0.35 kg solid
sphere with radius 0.27 m if a 4.2 N net force is
applied tangential to the surface?
Angular Momentum
• Translational momentum: 𝑝 = 𝑚 ∙ 𝑣
• Rotational (angular momentum): 𝐿 = 𝐼 ∙ 𝜔
• Conservation of Angular Momentum: Σ𝐿𝑖 = Σ𝐿𝑓
• Why do skaters spin faster when they pull their arms
inward? Demo!
• Remember electron spin? Electrons really don’t spin,
but they have quantized angular momentum.
Conservation Problem
A 0.11 kg mouse rides the edge of a Lazy Susan that has
a mass of 1.3 kg and a radius of 0.25 m. If the angular
speed is initially 3.0 rad/s, what is the angular speed
after the mouse moves to a point 0.15 m from the
center?
Rotational Kinetic Energy
• Translational KE: 𝐾𝐸 =
• Rotational KE: 𝐾𝐸𝑟 =
1
𝑚𝑣 2
2
1
𝐼𝜔2
2
• Σ𝐾𝐸𝑖 + Σ𝑃𝐸𝑖 = Σ𝐾𝐸𝑓 + Σ𝑃𝐸𝑓
Rotational KE Problems
A 1.5 kg solid sphere with radius 12 cm begins rolling
down an incline. What is the translational speed of the
sphere after it has dropped a vertical distance of 2.4
meters?
Objectives
• Be able to identify simple machines.
• Be able to explain how simple machines make
doing work “easier.”
• Be able to calculate the ideal mechanical
advantage (IMA), actual mechanical (AMA)
advantage, input work (WA), output work
(WO), and efficiency (e) of a simple machine.
Simple Machines
4 kinds: lever, inclined plane, pulley, wheel and axle
Simple machines generally make doing work easier
by reducing applied force (but distance is increased).
input work:
WA = FA·dA
output work:
WO = FO·dO
If no friction: WA = WO
If friction is present: WA > WO
Simple Machines
mechanical advantage (MA): factor by which
input force is multiplied by the machine
IMA  d A
dO
“ideal”
AMA  FO
FA
“actual”
efficiency: ratio of output work to input work
(indicates amount of friction in machine)
e  WO
WA
100