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Chapter 8
Rotational Equilibrium and Dynamics
Chapter 8 Objectives
Define torque
 Identify the lever arm associated with the torque of an object
 Identify the center of mass of an object
 Define second condition of equilibrium
 Recognize the moment of inertia of several objects
 Identify the six types of simple machines.
 Explain how simple machines effect work
 Calculate mechanical advantage

Definition of Torque
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Torque is a quantity that measures the ability of a force to rotate
an object around some axis.
Mr. Lent’s Definition – Torque is work in a circle.
Torque depends on a force, a lever arm, and the angle at which
the force is applied.
The lever arm is the perpendicular distance from the axis of
rotation to the line drawn along the direction of the force.
The sign on torque is based on the direction in which the force
causes the object to rotate.
Rotation in a clockwise direction is negative and
counterclockwise is positive.
Picture of Torque
Lever
Arm
Axis of rotation
is the center of
the object being
rotated.
Force
Θ
Formula for Torque
Θ
Greek “tao”
τ = Fd sin Θ
Be aware that Θ
is always less
than or equal to
90o.
This creates your lever arm so it is perpendicular
between the force and the axis of rotation.
How to Find Θ
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Θ is an angle less than or
equal to 90o
Extend the force vector in either
direction so that it will create an
acute angel with the original
lever arm.
You only need extend it so you
can draw a line from the axis of
rotation to the force vector such
that the line is perpendicular.
That perpendicular line is now
your new lever arm.
Θ
Θ
Center of Mass
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Center of mass is the point at which all the mass of a body can be
considered to be concentrated when analyzing translational motion.
The center of gravity is the point at which the force of gravity acts on
an object.
In the higher levels of physics, center of mass and center of gravity are
two different concepts and therefore can exist at two different locations
of an object.
For our purposes, we will consider them to be the same point in an
object.
For regularly shaped objects, such as a sphere, cube or solid rod, the
center of mass is located in the geometric center of the object.
Moment of Inertia
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Remember back to Newton’s 1st Law of Motion, Objects tend to stay in
motion, or at rest, unless acted upon by a net force.
Notice it says Motion, but does not specify whether the motion is linear
or rotational.
We also said that Newton’s 1st Law describes the term inertia, or the
the resistance of a change in motion.
The tendency of a body rotating about a fixed axis to resist a change in
rotational motion is called the moment of inertia of an object.
The moment of inertia of an object depends on the mass of an object
and how far away it is from the axis of rotation.
The general rule is that the further the mass is from the rotating axis,
the larger the magnitude for the moment of inertia.
Calculating the Moment of Inertia of Common Shapes
Object
Picture
Formula
Characteristics
Thin hoop about the
symmetry axis
MR2
R is the radius of the hoop and M is the
total mass. Wedding ring.
Thin hoop about the
diameter
½MR2
This is like a ring standing on its end and
spinning about the diameter.
Point mass about axis
MR2
This is spinning a weight on the end of a
string. Acts like thin hoop.
Disk or cylinder about
symmetry axis
½MR2
Flat plate or any solid cylinder, no matter
how long.
Thin rod through
center axis
1/ ML2
12
L is the total length of the rod and it
rotates like spinning a ruler on finger.
Thin rod about end of
rod
1/ ML2
3
Hold ruler at end and spin in a circle.
Solid sphere about the
diameter
2/ MR2
5
Spinning a bowling ball on your finger.
Spherical shell about
the diameter
2/ MR2
3
Spinning a basketball on your finger.
Rotational Equilibrium
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Recall the 1st Condition of
Equilibrium is that the net
force on an object is equal to
0.
That is called translational
equilibrium.
The word translation means
the refers to the translation of
an object which only
happens in a straight line, or
linear path.
Translational equilibrium
assures that the net
horizontal and the net vertical
forces sum to 0.
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We can now add the 2nd
Condition of Equilibrium.
Rotational equilibrium is
achieved when the net
torque is equal to 0.
So the sum of all clockwise
torques should be equal to
the sum of all
counterclockwise torques.
Angular Momentum
Recall that translational momentum was p=mv
 Objects still have momentum while rotating
 We account for rotating mass by using moment of
inertia
 And velocity in a circle is called angular velocity
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Angular momentum
L=Iω
Conservation of Angular
Momentum
Translational momentum is conserved, thus angular
momentum is conserved
 That is why a figure skater pulls his/her arms in order to
spin faster.
 Be sure to match the proper moment of inertia to the
rotating object(s) in the system

Li = Lf
Iiωi = Ifωf
Rotational Kinetic Energy
 Imagine a bowling ball rolling down an alley
 We know it has kinetic energy in a straight line…
 But it also has rotational kinetic energy as well
as translational kinetic energy
 The formulas look alike just use the correct
variables for the correct situation.
KEROT= 1/2Iω2
Simple Machines
A machine is an object that transmits or modifies force, usually by
changing the force applied to an object.
All machines that you may think of are actually combinations or
modifications of 6 fundamental types of machines called simple machines.
And those are broken down into 2 families.
Lever Family
1.
2.
3.
Lever – Teeter Totter,
Baseball Bat, Broom, etc.
Pulley – A rope being
pulled around a wheel.
Wheel and axle – A wheel
attached to a smaller
diameter rod.
Inclined Plane
1.
2.
3.
Inclined Plane – A slanted
surface like a wheel chair
ramp.
Wedge – Two inclined
planes back to back.
Screw – An inclined plane
wrapped around a cylinder.
Mechanical Advantage
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Because the purpose of simple
machines is to alter the direction or
magnitude of an input force, then it
must provide some type of
advantage to produce the output
force.
That advantage is called
mechanical advantage.
The mechanical advantage of a
machine is a ratio of the output
force to the input force.
It can also be a ration of the input
distance to the output distance,
which a little manipulation of the
conservation of rotational
equilibrium.
τ in = τout
Findin = Foutdout
din
dout
MA =
Fout
= F
in
Fout
Fin
din
=d
out
Lever
A lever is a type of arm that includes a pivot point, or
fulcrum.
 There are 3 classes of levers based on the orientation
of the fulcrum, input, and output forces.
 Examples of levers are teeter totters, baseball bats, pry
bars, bottle openers, golf clubs, hockey sticks,
wrenches, etc.
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Classes of Levers
1st Class Lever – The input force and output force are on opposite
sides of the fulcrum and are directed in opposite directions.
(Teeter Totter)
2nd Class Lever – The fulcrum is at one end of the lever with the
input force at the other end and the output force is in the middle. Also,
the forces are directed in the same direction. (Wheel barrow)
3rd Class Lever – The fulcrum is at
one end of the lever with the output force
at the other end and the input force is in
the middle. Again, the forces are directed
in the same direction. (Baseball bat)
Mechanical Advantage of a Lever
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The mechanical advantage of a any lever is found by
conserving rotational equilibrium.
τin = τout
Findin = Foutdout
Pulley
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A pulley is a part of the lever family because it uses a fulcrum to
redirect the force.
Fulcrum
Fixed v Movable Pulleys
The behavior of pulleys can
differ based on whether the
pulley can move or not.
A fixed pulley is one in
which the pulley itself is
attached to a wall or pole and
does not move when the load
is moved.
These types of pulleys are
often used to redirect the
applied force.
Therefore, a moveable
pulley
is one in which the pulley
moves as the load moves.
These types of pulleys are
used to spread the load out
over an additional “rope” in
the system.
This transforms the force of
the load into 2 equal tensions
that each carry half the
weight.
Mechanical Advantage of a
Pulley
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For any fixed pulley, the
mechanical advantage is 1.
It is 1 applied force (1 rope)
redirected over the pulley to
provide an advantage of pulling
versus pushing, or vice versa.
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For any moveable pulley, the
mechanical advantage is always
2.
There is now 2 applied forces
that aid in lifting the load.
So the input force is cut in half
because it only has to lift one of
the two lifting forces.
Complex Pulley Systems
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Easiest way to calculate mechanical advantage of a complex pulley
system is to identify the following:
Load bearing ropes
– These are the ropes around a moveable pulley.
Redirecting ropes
– These are the ropes around a fixed pulley.
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The mechanical advantage is the reciprocal of the tension in the load
bearing ropes.
MA = 2
MA = 2
MA = 3
MA = 3
MA = 4
Wheel and Axle
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A wheel and axle is much like two pulleys of different sizes that
spin together.
Wheel
Axle
Mechanical Advantage of a
Wheel and Axle
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The mechanical advantage is gained from the ratio of the radii of
the wheel and axle.
The best way to demonstrate the mechanical advantage is the
ratio of tire size to axle size in the rear end of a car.
τin = τout
Findin = Foutdout
R=3
r=1
Fin(2πR) = Fout(2πr)
(2πR) = Fout
(2πr) Fin
3
MA =
1
=3
Inclined Plane
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An inclined plane is simply a flat surface that is used to move
an object up or down an incline, or elevation.
The surface does not necessarily have to be flat, it must be able
to be modeled flat. (ie: a set of stairs)
Mechanical Advantage of an
Inclined Plane
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The mechanical advantage of an inclined plane is easiest to
calculate using the input distance versus the output distance.
You may need to use Pythagorean Theorem for this.
din
dout
Fout
= F
in
Output
Wedge
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A wedge is two inclined planes placed back to back.
The most common form of a wedge is an ax.
Mechanical Advantage of a Wedge
The mechanical advantage of a wedge is the ratio of inclined
surface to width of the wedge.
 It is very similar to the mechanical advantage of an inclined
plane.
 You might think that since a wedge is two inclined planes, its
mechanical advantage would be twice as large. But it is spread
over two surfaces so it divides the two back out.
 Notice that to increase the mechanical advantage of the wedge,
simply make the width smaller. That is why a sharp knife is better
than a dull knife!
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Inclined Surface
Width
Screw
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As simply put as possible, a screw is an inclined plane wrapped
around a cylinder.
It resembles a wheel and axle, except for the part that is actually
doing the work is the slope of the threads of the screw.
Mechanical Advantage of a Screw
The mechanical advantage of a screw is hard to visualize, and
therefore tricky to calculate.
 Avoiding all of the nasty calculation, it is a ratio of the
circumference of the screw divided by its pitch.
 Pitch is the distance between threads.
 If you think about it, one turn will make the screw go into the
wood the same depth that one thread of revolution. So that is
how we get the mechanical advantage.
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Circumference of Screw
MA =
Pitch