Phase, Pendulums, & Damping An investigation into swingers .docx

Phase,
Pendulums, & Damping
An investigation into swingers
4/11/2108
1
Last Time…
Our model:
2
frictionless surface
(spring equilibrium position)

The Phase Constant
If the block is pulled to A and released from rest, the phase
constant is zero.
3
A
phase
phase constant
The Phase Constant
What if the block isn’t released from rest at A?

There is still an oscillation between ±A,
but now we need a phase constant
The phase constant exists to provide that shift away from A
when t=0
4
Note: This thing moves when you run the power point, the x
location at t = 0 is no longer A
Example Problem (15-4)
The position vs. time graph for a SHO is shown below. What is
the phase constant ?
5

What does your
calculator say?
Which one is it? We
need a way to figure
this out.
T
T = 4 s
Finding Φ
To determine the sign of the phase constant, we’ll turn to
uniform circular motion
Now what if the object doesn’t start on the x-axis?
6

The x-component of the object’s position vector is:
Now, the x-component of the object’s position vector is:
Great, but what do I need to memorize here?
Nothing… but…
We know initial conditions x(0) and v(0)

So we can determine the correct quadrant for Φ
Finding Φ
7
Consider that you can draw a circle and determine these things
without memorizing them. The motion starts at the x axis, and
the x velocity starts out negative and becomes positive once you
start to come back around the circle. Furthermore, you can tell
when the x velocity is increasing or decreasing in magnitude.
7
Example Problem (15-4)
The position vs. time graph for a SHO is shown below. What is
the phase constant ?
8
T

For this problem:
slope < 0
Whiteboard Problem 15-5
Below is a velocity vs. time graph of a particle in simple
harmonic motion.
What is the amplitude of the oscillation?
What is the phase constant? (LC)

What is the position at t = 0?
9
Note this is a plot of velocity, it’s a little different than the one
that we just did.
Tick… Tock…
Swing on over to Mastering Physics and start the PhET Lab:
Pendulums
Work as a group, but submit individually.
Sometime before you finish the PhET, calculate the length of
the pendulum in the lobby of Kreger (LC) using the equation
below:
You’ll have about 20 minutes to finish the PhET now, and
whatever isn’t finished will be due at 6 today
10

A simple pendulum is a point mass on a massless string that can
swing around a pivot point
The motion of the pendulum can be
described with torque
The Simple Pendulum
11
+
This is not SHM, but if we consider only small angles:

What is the period of a 1.0m long pendulum on:
Earth?
Venus? (LC)
12
Pivot
point
Physical Pendula
A physical pendulum is a real object that can rotate about some
pivot point:
13
Center of
mass
The text uses the same steps that we used for a simple pendulum
to show that for small angles, this is also SHM where:
A uniform rod of mass M and length L swings as a pendulum on
a pivot at a distance of L/4 from one end of the rod. Find an
expression for the frequency of the oscillation for small angles.
(LC)

Your expression should contain only g, L, and numbers
14
Chaos
15
Double pendula exhibit chaotic behavior; the progression of the
motion is very sensitive to initial conditions.
Other neat things:
https://www.youtube.com/watch?v=B6vr1x6KDaY
15
Damped Oscillations
Our model so far has excluded all non-conservative forces
So what’s the rub?
Here’s one model of a damped oscillator:
16
FBD:

b = damping constant
This is not an easy differential equation to guess a solution
for… we’ll just jump to the solution.
The Answer
The solution for this model of a damped oscillator is a decaying
exponential:
17
42
oscillating part
decaying part
The angular frequency
of the undamped
oscillator.
A 250g air track glider is attached to a spring with spring
constant 4.0N/m. The damping constant due to air resistance, b,
is 0.015kg/s. The glider is pulled out 20cm from equilibrium
and released.
How many oscillations will it make during the time in which the

amplitude decays to e-1 of its initial value? (LC)
18
Driven Oscillations & Resonance
Nonconservative forces can take energy out of an oscillator, but
they can also put energy in
The driving force should be applied with the same frequency as
the oscillator; the oscillator’s natural frequency
When this condition is met, it is called resonance
19
Examples include swinging, shattering glass, and the Tacoma
narrows bridge
19
Lavf54.63.104
Lavf54.63.104
Lavf54.63.104
Oscillatory Motion
There and back again.
4/9/2108
1
Motion
Things can move in lines

They can also move around a fixed point
Sometimes, really far away things can change motion
We’re going to take a look at the lines again, but now over and
over
And over again
2
A Spring Refresher
Hop on to Mastering Physics and start the PhET Lab: Masses &
Springs
Work on this with your group, but make sure each person
completes the assignment
You’ll have about 15 minutes to finish the PhET now, and
whatever isn’t finished will be do at 6 today
3

Simple Harmonic Motion
Hereafter commonly referred to as SHM
SHO will refer to a simple harmonic oscillator
Simple? � Motion? � Harmonic? …
References harmonic functions, like sines and cosines
Variety of different descriptors, but first; dynamics
4
Dynamics
Our first model of SHM will be a spring:
If we pull the mass, the spring will pull back…
This is a linear restoring force: linear as it varies with x,
restoring since it points toward an equilibrium position
Anytime you have a linear restoring force, you can have SHM
5

So the spring always pushes or pulls the mass back to x = 0, the
equilibrium position of the spring.
The Equation
Let us see what the forces acting on the mass are:
So we have an equation of motion, we just need to solve…
6
Free Body Diagram for m:
(Nothing real interesting here.)
(What kind of animal is this?)

Second order linear differential equation
6
What we have is a differential equation: some relationship
between a function and some of the function’s derivatives
What we need to do is find a function that satisfies the
relationship presented
We will do this by using the most sophisticated mathematical
tool available: guessing
Solving The Equation
What we have is a differential equation: some relationship
between a function and some of the function’s derivatives
What we need to do is find a function that satisfies the
relationship presented
We will do this by using the most sophisticated mathematical
tool available:
If you’ve guessed correctly, you’ve found the only solution
7
Uniqueness theorem states that linear differential equations
have only one unique solution.
7
The Old College Try
We need a function whose second derivative is the negative of
the function, with some other stuff
From calculus you should know that trigonometric functions
have this property:

So, as a guess, why not a trig function?
Note that we aren’t necessarily dealing with angles, we’re using
trig functions for the oscillating nature
8
Show by direct substitution that our guess:
Is a solution of the differential equation:
For x(t) to be a solution, what must ω be equal to? (LC)
9
Picking Things Apart
Consider our model:
10

The solution:
Note: be careful using these equations in your calculator. The
argument of the sine and cosine is in radians, and your
calculator has to be set to radians.
SHM Graphed
If the mass is pulled to x = +A
and released from rest at t = 0,
then A = the amplitude and Φ0 = 0
11

We’ll handle the phase shift later, but it will still be useful to
know where it is
11
An air track glider is attached to a spring and oscillates between
the 10cm mark and the 60cm mark on the track. The glider
completes 10 oscillations in 33s.
Determine:
Period
Frequency
Angular Frequency
Amplitude
Maximum speed (LC)
12
12
Energy
Like gravity, springs provide conservative forces
With no friction, all forces are conservative, so energy is
conserved

Kinetic and potential energies change with time, but their sum
is constant
Max potential?
Max kinetic?
13
Energy Plots
14
A 300g oscillator has a speed of 95.4cm/s when its displacement
is 3.0cm, and 71.4cm/s when its displacement is 6.0cm.
What is the oscillator’s amplitude and maximum speed? (LC)
15
You’re welcome to work as a table for this one

Wave Motion
And a bit on damped oscillations
4/13/2108
1
Phase Constant
2
Cosine reflection
Sine reflection
Tangent reflection
Determining Quadrant:
X varies with cos, and velocity with negative sin, so take a look
at those graphs to see which quadrant your initial conditions are
in.
2
Damped Oscillations
Our model so far has excluded all non-conservative forces
So what’s the rub?
Here’s one model of a damped oscillator:

3
FBD:
b = damping constant
This is not an easy differential equation to guess a solution
for… we’ll just jump to the solution.
The Answer
The solution for this model of a damped oscillator is a decaying
exponential:
4
42
oscillating part
decaying part

The angular frequency
of the undamped
oscillator.
A 250g air track glider is attached to a spring with spring
constant 4.0N/m. The damping constant due to air resistance, b,
is 0.015kg/s. The glider is pulled out 20cm from equilibrium
and released.
How many oscillations will it make during the time in which the
amplitude decays to e-1 of its initial value? (LC)
5
Driven Oscillations & Resonance
Nonconservative forces can take energy out of an oscillator, but
they can also put energy in
The driving force should be applied with the same frequency as
the oscillator; the oscillator’s natural frequency
When this condition is met, it is called resonance
6
Examples include swinging, shattering glass, and the Tacoma
narrows bridge
6
Wave Motion
What if the oscillator wasn’t fixed in place?
The oscillation could travel through space

A wave is an organized disturbance that travels at a well
defined speed.
A wave is not the same thing as the medium
Like particles, waves can carry energy and momentum
7
‘Dex Entries
There are 3 main categories of waves, and 2 types
Mechanical Waves:
Require a medium to propagate
Wave speed is determined by the medium
Oscillations can be transverse or longitudinal
Electromagnetic Waves:
Medium not required
Wave speed fixed ~ 3x108 m/s
Matter Waves:
Light is both a particle and a wave? That means particles can be
waves too
8
Wave Speed
Consider a single traveling down a string
The wave moves through the medium, displacing
the string from it’s equilibrium position
The text goes through a derivation of the speed of a wave on a
string:

9
Where:
The wave speed on a string is 150m/s when the tension is 75N.
What tension will give a speed of 180m/s? (LC)
10
The Sine Wave
A sinusoidal (harmonic) disturbance creates a sinusoidal
travelling wave
At a given point in space, a single particle undergoes simple
harmonic motion (a); for a snapshot in time, the whole wave is a
sine wave (b)
11
Where D(x,t) is the general disturbance from the equilibrium
state. Note: it is a function of two variables.
Modeling Sine
With a snapshot in time, we can describe a few things about a
wave:

Many of these variables are fundamentally related:
With that relationship, we can describe the motion of a traveling
wave…
12
Travelling in +x direction
Wave number and spring constant are different!
12
One more thing…
The differential equation for the harmonic oscillator gave us a
clue as to what ω was:
Waves have a similar differential equation:
13

Expression for a Travelling Wave
Wiggle on over to Mastering Physics and finish the assignment
that’s the same as the title
The goal is to become more familiar with the pieces and parts of
the wave equation
When you’re done, consider starting your homework, or playing
with the wave on a string PhET
14
15
Write the displacement equation for a sinusoidal wave that is
traveling in the negative y-direction with a wavelength of 50cm,
speed of 4.0m/s, and an amplitude of 5.0cm. Assume the phase
constant is zero.
16
Rotational Kinematics
& Center of Mass
3/26/2108
1

Get to Know Your Group Members
Introduce yourselves within your group
Suggested topics:
Name
Year in school
Why are you taking physics?
Major
Hometown
Goal for this semester
Favorite Pokemon
What do you hope to gain from your physics experience?
Would you rather fight 100 duck sized horses or 1 horse sized
duck?
2
Rotation of a Rigid Body
So far, motion has been treated rather thoroughly in this class
But that was only one type of motion: translational
Motion has many other forms, namely rotational, vibrational,
and wave-like
The next few classes will be covering rotational motion
They recap everything we’ve done so far, but for spinning
3
Good news, you’ve used nearly all of these concepts before
Bad news, all the letters are going to change, and they’ll mix
together
3
Nonuniform Circular Motion
For circular motion up to this point, we’ve considered

accelerations only in the radial direction
These produce uniform circular motion
In nonuniform circular motion there is an acceleration in the
tangential direction as well
The accelerations are then…
For constant at :
(maintains the circular motion)
(changes the speed)
4
Consider Chapters 4.7 and 8.5 for more information
Divide regular kinematics by r to get circular

4
Rigid Body Rotation
A rigid body is an extended object whose components don’t
move relative to one another.
Can be a small group of particles at fixed positions, or a
continuous body.
Pivot Point
Or Rotation axis
If the body is pivoted at some point, it is free to rotate about an
axis through that point.
All points in the body have the same w, but different v.
5
Rotational Kinematics
Rotational kinematics are essentially the same as circular
motion, it’s just harder to see the circle.

The rotational kinematic equations are the same as the linear
ones, just use the rotational counterparts:
Pivot
Point, P
Sign Convention:
CounterClockWise (CCW) is positive
6
An ice skater holds their arms outstretched as they spin at 180
rpm.
What is the speed of their hands if they are 140cm apart? (LC)
7
A high speed drill starts from rest and steadily increases its
speed until it reaches 2000 rpm in a time of 0.50s.
What is the drill’s acceleration?
How many revolutions does the drill make in this time? (LC)
8
Center of Mass

The center of mass is the point about which an unconstrained
rigid body will rotate.
For a discrete collection of point masses…
The center of mass coordinates are:
N masses
9
Since the distance from the origin is a vector, it is most useful
to find the components of center of mass and then find the
resulting distance.
9
The masses shown below are connected by massless, rigid rods.
Mass A is centered on the origin.
What is are the coordinates of the center of mass of this system?
(LC)
10

Good News about Continuous Bodies
To find the center of mass of a system, you consider the mass
and location of each object.
What if you have a continuous object?
The center of mass coordinates are:
This type of problem can make for unpleasant integrals, but we
won’t be doing (m?)any.
11
Objects that are symmetric with uniform densities have the CoM
at the geometric center.
11
Vector Cross Product
larger

angle
Where:
What does the cross product mean in words?
12
The Right Hand Rule
13
The Right Hand Rule
Use your right hand.
No really.
With your four fingers extended and together, point them in the
direction of the first vector of the cross product.
Rotate your hand until your palm is facing the second vector in
the cross product.
Curls your fingers into a fist, moving through the angle θ to get
to the second vector.
Your thumb will be the direction of the resultant vector.
14
(for winners)

Evaluate the following cross products, giving direction as in to
or out of the page.
15
LC (mag only)
LC (mag only)
Where is 12-4?
15
Vector Cross Product using a Determinant
Sometimes, you will be given the components of two vectors
and be asked to find their cross product. This is done by taking
a determinant; you may learn more about these in linear algebra
but here’s the gist…
a, b, c, and d are just numbers
A 3x3 determinant can be expanded in to 3 2x2 determinants:
16
Note, the negative sign; when you expand a determinant, the

signs on the terms always alternate.
Consider the vectors:
This method directly gives you the components of the cross
product vector. No angle, no RHR.
Vector Cross Product using a Determinant
17

17
Whiteboard Problem 12-5 Cont.
Find the magnitude of the cross product between A and B. (LC)
18
19
Torque & Moment of Inertia
One thumb up and one thumb down, unless you have two left
thumbs.
3/28/2108
1
Moment of Inertia
So far we have angular equivalents for standard kinematic
variables

Angular position, velocity, and acceleration
What about dynamics? F = ma?
Moment of Inertia is the equivalent of mass
It can be calculated somewhat like
center of mass:
2
Axis of Rotation
(perpendicular to x-y plane)
2
Find the moment of inertia of the four masses below about an
axis that passes through mass A and is perpendicular to the
page. (LC)

3
Rotation Axis:
Continuous Bodies
Just like the center of mass, the moment of inertia of a
continuous body is found through an integral.
And, like center of mass, we won’t
do any of these integrals.
Problems will either:
Tell you I
Tell you a known shape
Have you change the axis
Have a discrete number of masses
4
Axis of Rotation
5

Parallel Axis Theorem
If you know an object’s moment of inertia about an axis through
its center of mass, you can find I for any parallel axis.
6
Axis through the center of mass,
know the moment of inertia about
this axis,
Axis parallel to axis through CM
Torque
If I is the equivalent of m, what is the equivalent of F?
Torque (tau), also called moment in engineering.
An applied force on an object can cause an acceleration
Applied forces on rigid bodies can cause angular accelerations
7
Pivot
Point, P

Pivot
Point, P
Or,
The forces have equal magnitudes,
which force causes the most rotation? (LC)
The forces have equal magnitudes,
which force causes the most rotation? (LC)
The greater the distance from the pivot point to the force, the
larger the torque
The more perpendicular to the line from the force to the pivot,
the larger the torque
7
Torque can be conceptualized many ways: all produce the same
result
Three Paths
8
Pivot
Point, P
Vector from P to the point
where force is applied
Torque of F about P:

All of these produce the same expression for torque, which may
or may not look familiar.
8
The 20cm diameter disk shown below can rotate about an axle
through its center. What is the net torque about the axle? (LC)
9
Axle
In General
Having calculated torque using all of the equivalent methods,
we can now utilize the tool we learned in the last class.
The torque about some pivot:

Thus, the angular acceleration of an object is:
10
Pivot
Point, P
Vector from P to the point
where force is applied
11
LC (magnitude only)
A 1.0kg ball and a 2.0kg ball are connected by a 1.0m long
rigid, massless rod to form a dumbbell. The dumbbell is rotating
CW about its center of mass at 20rpm.
Find the location of the center of mass.
Find the moment of inertia about the center of mass.

Find the constant torque that will bring the balls to rest in a
time of 5.0s (LC)
12
Orbital Motion
♪♫ Like a record player ♫♪
4/6/2108
1
With the acceleration caused by gravity changing, Ug = mgh is
not true for every situation
Your text describes a new form of potential energy which works
on larger scales:
Some notes:
Zero potential energy is located at infinity
The negative sign is important, and expresses that gravity is
attractive
The potential energy varies as 1/r, not 1/r2
Gravitational Potential Energy
2

Circular Orbits
In the PhET lab, you may have calculated the speed of an object
in a circular orbit…
From this, we can calculate circular orbital speed:
3
Free Body Diagram of m:
For uniform circular motion:

Orbital Period
The text describes a law which describes how long an orbit
takes:
The square of the orbital period is proportional to the cube of
the semimajor axis of the orbit.
We can do better
than proportional
So Newton’s form of Kepler’s third law:
4
The above equations work when m << M. If that is not the case,
both masses orbit about their common center of mass.
Consider two stars with masses M1 and M2 separated by a
distance d and orbiting about their center of mass in circular
orbits.
Find an expression for the orbital speed of M1 in terms of M1,
M2, r1, and r2. (LC)
What is the orbital speed of M2?
5

Elliptical Orbits
The speed of an object in an elliptical orbit
changes in response to the changing force
The force varies with the distance from M
Since there are no nonconservative forces acting, the total
mechanical energy of the system is conserved:
This information can connect any two points in an orbit:
6
Orbital Angular Momentum
Gravity between M and m always acts

toward the center, so there is no torque
So angular momentum is conserved
Calculating this at any point is cumbersome, but there are two
points where the angle β is 90°
7
perihelion
aphelion

The dwarf planet Pluto moves in a fairly elliptical orbit. At is
closest approach to the sun of 4.45 x 109 km, Pluto’s speed is
6.12km/s.
What is Pluto’s speed at aphelion, which is 7.30 x 109km? (LC)
8
Pluto as seen by New Horizons
8
Clouds of Mass
What if instead of a star, there was just a cloud of mass?
Like a dust cloud, or Dark Matter
For spherically symmetric clouds, you can calculate the
effective mass by finding the mass “inside” of the orbit.
All of the mass inside the orbit is treated as a point mass in the
middle.
9

r > R
m
m
Mass, M
R
r < R
The Dark Matter Problem
We’ve seen that for circular orbits, the orbital speed is:
If we consider our Solar System, we see this rotation curve:
If we consider the whole Milky Way, we would
anticipate something similar…
Even though the composition is similar, the
curve is different, why?
10

The Dark Matter Problem
As we saw in the video after the last exam, the standard
explanation is that there is some “dark” matter which changes
the mass of the system.
The issue is, there is a lot of the stuff, and we haven’t been able
to detect it.
An alternate explanation is that Newton’s law of gravitation
changes at sufficiently small accelerations,
There’s an extra credit assignment on Canvas exploring this
problem, you can work with up to 3 other people on it, and it is
due on Monday.
11
Exploring Orbits
To better prepare for calculating orbits, there is a PhET
assigned where you can play with various parameters of a solar
system.
Use a computer with Flash (Version 8 or newer, but really just
update your flash player) to run the simulation. You can find a
link to the simulation in the google doc under the assignment on
Canvas.
Finish the assignment as a group, submitting 1 copy per group.
If you finish early, you’re free to go. If you don’t be sure to
submit the finished document by today at six.
12

Rolling Motion &
Angular Momentum
4/2/2108
1
Table Challenge 5
A pencil has mass M and length L. It is standing straight up on
a table on its eraser end. A slight push causes the pencil to fall
over. Friction between the eraser and the table provides a pivot
point.
Find an expression for the speed of the tip of the pencil as it
hits the table.
2
Mass M
Length L
Pivot
Point
Rolling motion without slipping is difficult to describe with
kinematics
So instead, use energy
The text describes rolling as:
Rotational kinetic energy can be added in with translations to
find the total kinetic energy of an object:

Rolling Motion
3
Gotta Go Fast
4
We’ll be taking a look at the rolling motion of 4 different
objects:
Solid Sphere, Hollow Sphere, Solid Cylinder, Hollow Cylinder
Using the materials provided, set up an incline plane and rank
the speeds of the objects from fastest (shortest time) to slowest
(longest time)
When finished, record your ranking on LC
4

An Analysis of Motion
There are several ways you can find how long it takes an object
to slide down an incline without friction…
What about for rolling motion?
5
0
1
L
y = 0
From 1D Kinematics for x-axis parallel to incline:
Conserve energy 0 to 1:
Find an expression for the time it takes for a solid sphere to go

down an incline with no rolling friction. Assume that the object
is rotating about its center of mass, so that the moment of
inertia can be found in a table. Your expression should look
like:
Enter your number in (LC)
6
A marble starts from rest and rolls down the track shown around
the loop-the-loop of radius R. The marble has mass m and
radius r.
Find an expression for the minimum height h that the track must
have for the marble to make it around the loop without falling
off. (LC)
7
Angular Momentum
When an object rotates, it has rotational kinetic energy
It will also have angular momentum.
8
For a particle:

Uniform Circular Motion:
Rigid Body rotating about a fixed symmetry axis:
The vectors in the rigid body momentum just indicate direction
as ccw or cw
8
Conservation of Angular Momentum
Recall that Newton’s second law could be written in terms of
momentum
So then for a rotating system:
Which if there are no net torques…
9

In Summary
Kinematics:
Dynamics:
Energy:
Momentum:
10
Constant Acceleration

(particle)
(rigid body)
10
A 2.0kg, 20cm diameter turntable rotates at 100 rpm on
frictionless bearings. Two 500g blocks fall from above, hit the
turntable simultaneously at the opposite end of a diameter, and
stick.
What is the turntable’s angular velocity, in rpm, after this
event? (LC)
11
Gravity
Warping the very fabric of reality.
4/4/2108
1
Consider the following…

Gravity is an inverse square force
Gravitational forces act along the line connecting the centers of
mass
Gravity is always attractive
Newton’s Theory of Gravity
2
Only works with spherical symmetry
Inverse square give elliptical orbits
2
What is the ratio of the Sun’s gravitational force on the Moon to
the Earth’s gravitational force on the Moon? (LC)
3
Earth’s Pull

The acceleration caused by gravity has a value of 9.8m/s2
This is merely an approximation, the effect changes as you
move around Earth
We now have the tools to calculate the effect of gravity some
distance away from the Earth:
4
So, the acceleration of gravity is:
Earth
(from the
Earth’s center)
1 gal == 1cm/s2, or .01m/s2, so a miligal is a difference of
.00001m/s2, or .00001% of our daily gravity
4
Planet Zed is 10,000km in diameter. The free-fall acceleration
on the surface of Zed is 8.0m/s2.
What is the mass of Zed?
What is the free fall acceleration 10,000km above Zed’s North
pole? (LC)
5

With the acceleration caused by gravity changing, Ug = mgh is
not true for every situation
Your text describes a new form of potential energy which works
on larger scales:
Some notes:
Zero potential energy is located at infinity
The negative sign is important, and expresses that gravity is
attractive
The potential energy varies as 1/r, not 1/r2
Gravitational Potential Energy
6
The escape speed is the minimum speed needed to escape from
the surface of a planet and leave its gravitational influence.
Use conservation of energy to find an expression for the
minimum initial speed the rocket needs to escape to infinity.
Calculate the escape speed from the surface of the Earth. (LC)

(ME = 5.98 X 1024 kg; RE = 6.37 X 106 m)
7
Exploring Orbits
To better prepare for calculating orbits, there is a PhET
assigned where you can play with various parameters of a solar
system.
Use a computer with Flash (Version 8 or newer, but really just
update your flash player) to run the simulation. You can find a
link to the simulation in the google doc under the assignment on
Canvas.
Finish the assignment as a group, submitting 1 copy per group.
If you finish early, you’re free to go. If you don’t be sure to
submit the finished document by Friday at noon.
8
Rotational Energy
& Static Equilibrium
3/30/2108

1
A Summary So Far
Kinematics:
Dynamics:
Energy:
Momentum:
2
Constant Acceleration
Rotational Kinetic Energy

Consider our old pal the uneven dumbbell…
What kinetic energy does the system have?
So, rotational kinetic energy must then be:
3
Energy of a Rigid Body
The energy for a cluster of masses can be generalized to a
continuous object:
4

Axis of Rotation
A 300g ball and a 600g ball are connected by a 40cm long
massless, rigid rod to form a dumbbell. The dumbbell rotates
around its center of mass at 100rpm.
What is the rotational kinetic energy of the dumbbell? (LC)
5
Static Equilibrium
For objects that aren’t points, equilibrium is a bit different.
This is a future you problem.
15
FBD:
x
y

Previously in PHY191…
6_2, slide 15
6
Static Equilibrium
A body is in static equilibrium if:
Torque about any point must be zero.
Side note: forces acting at a pivot point produce 0 torque.
7
A note on Gravity
Gravity can exert a force, but what about a torque?
Gravity acts over the entire body.
Whenever you’re solving a problem, know that gravity will
effectively act at the center of gravity of an object.

This is the same location as the center of mass for all of the
objects we will consider in this class.
8
Solving Static Equilibrium Problems
Picture
Reference Frame
Including rotation direction (CCW +)
Draw a FBD
Sum the forces in all directions
They sum to zero
Sum the torques about a point (usually a pivot)
They sum to zero
Solve
9
A beam of mass M and length L is resting on a pivot as shown
below:
What must the force F be in order to keep the beam still?
Example
10

The two blocks of citrine shown below have uniform density
and are balanced on the pivot.
Draw Free Body Diagrams for both blocks. What is the force of
the upper block on the lower block?
Use the FBD of the lower block
to find the distance d. (LC)
11
That’s about 15,000$ of citrine.
11
In the figure below, an 80kg construction worker sits down
2.0m from the end of a 1450kg steel beam to eat his lunch. The
cable supporting the beam can withstand a maximum tension of
15,000N.
Draw a FBD of the beam.
Determine the tension in the cable (LC) – does the cable break?
12
Live Action Science
What is the mass of a meter stick?
On Canvas, find the assignment labeled Meter stick mass, make
a copy of the google document and share it with your group.
Using the materials on the wire rack on the East side of the
room, determine a method using what we’ve learned so far this
week to determine the mass of a meter stick. Record your
process on the google doc.
13

Phase, Pendulums, & Damping An investigation into swingers .docx

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