Download Chapter 6, Week 6.

Chapter 6
Work and Energy
6.1
Lecture - The Work and Energy Theorem
In this lesson, we will study the conservation of energy, and a special case of it called the
work energy theorem. From prior work in high school physics, you are already familiar with
the concepts of kinetic and potential energy.
This week in lecture, we will conduct an analysis of a cart moving through curvilinear
path, and the trade-o↵ between work and energy that occurs during the motion. The analysis
is reminiscent of a roller coaster that you may have ridden on at an amusement park. In
Lab, we will conduct an analysis of this motion, using a digital camera as our measurement
instrument. We will continue our analysis of the data collected in Lab during the Studio
session, and we will solve work and energy related problems in recitation.
Work and energy have units of [Joules] in S.I. and [f t lbf ] in Customary U.S. Units. The
Joule is a derived unit, which means that it is defined in terms of other units. Energy is
work acting over a distance, so energy has units of force times distance:


[kg][m]
[kg][m2 ]
1[Joule] = 1[J] = 1[N ]1[m] = 1[N m] = 1
1[m]
=
1
[s2 ]
[s2 ]


[slug][f t]
[slug][f t2 ]
1[f t lbf ] = 1[lbf ]1[f t] = 1
1[f
t]
=
1
[s2 ]
[s2 ]
Now, let us learn about the strong connection between the work energy theorem and
Newton’s Laws of motion.
6.1.1
Formulate
State the Problem
Determine the speed, kinetic energy and potential energy of a vehicle as it traverses the
curvilinear path of a track. Use the di↵erence in total energy between cart positions (denoted
as locations i and j) to estimate the work done by the moving cart upon the track, to
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overcome the irreversible friction force. Estimate the requested values at a minimum of seven
positions along the path of motion of the cart. Explain any di↵erences in motion which may
be observed between the context of Newton’s Laws and the work/energy theorem and the
experimental environment employed.
Known Information
The following information is provided to describe the motion of the cart as it travels down
the curvilinear hill of your track.
m = Known
[kg]
Mass of Cart
(6.1)
Desired Information
Upon conclusion of the experiment and analysis, we shall be required to report:
KEi
P Ei
zi
si
|Vi |
Wi!j
6.1.2
=
=
=
=
=
=
?
?
?
?
?
?
[J]
[J]
[m]
[m]
[m/s]
[J]
↵
↵
↵
↵
↵
↵
Kinetic Energy of m at location i
Gravitational Potential Energy of m at location i
Vertical Position of m at location i
Curvilinear Position of m at location i
Speed of m at location i
Work Done by m between locations i and j
(6.2)
(6.3)
(6.4)
(6.5)
(6.6)
(6.7)
Assume
We will make several familiar assumptions, and one new assumption for this analysis. Most
significantly, we are going to assume that there is no heat transferred between the cart and
it’s surroundings.
Identify Assumptions
The following assumptions may be employed during the analysis.
m = Constant
!
F air
g
Qi!j
SEi
ff riction
Erotation
=0
⇡ 9.81
=0
between all locations i and j
=0
8 locations i
= Constant
=0
221
[kg]
(6.8)
[N ]
[m/s2 ]
[J]
[J]
[N ]
[J]
(6.9)
(6.10)
(6.11)
(6.12)
(6.13)
(6.14)
Justify Assumptions
We need to justify each assumption proposed for use in our analysis.
Equation 6.8 says that we assume the mass, m, of the cart to be constant. We justify that
this is a reasonable assumption as long as the cart does not break apart during it motion.
We note that the cart contains no fuel or other expendable materials which would cause its
mass to change during the time interval of interest.
Equation 6.9 says that the force of the atmosphere upon the cart is negligible. We are
neglecting drag along the direction of motion of the cart. In reality, the frictional drag of
the air upon the cart may be a significant source of error. However, we really have no way of
determining whether the friction between the cart and it’s surroundings is more prevalent as
rolling friction between the wheels and the track, or between the cart and the air. Certainly,
we can devise an experiment that will tend to increase air friction, by adding a sail to the
top of the cart, for example.
Equation 6.10 says that the local acceleration of gravity, g, near the Earth’s surface is
known at a given value. We will use the standard approximation of g as an appropriate value
for this problem.
Equation 6.11 says that there is no heat transfer between the cart and it surroundings.
As is clear from the free body diagram, we are including friction in our analysis. The friction
between the rolling wheels of the cart and the track is a force, which acts over the distance
of motion of the cart. Thus, the cart is indeed doing work upon the track. Our system
boundary was chosen carefully to include the cart but not the track. If we were to analyze
the track, we would find that the work done by the cart upon the track could have one of
two outcomes... the work could cause the track to move, or it could cause the track to heat
up slightly. That is, the track could convert the external work done upon it into thermal
energy storage as reflected by a slight rise in the temperature of the track. We do not wish
to go into that type of analysis in this course. We will defer those types of analysis until
later in our curriculum, when we study thermodynamics.
Equation 6.12 says that there is no elastic potential energy in the system. We will learn
about elastic potential energy later in the course, but for the moment, we neglect it.
Equation 6.13 says that we are treating the friction force between the wheel of the cart
and the surface as being constant over the range of motion. In reality, we know that this is not
true, since Amontons’ Law says the friction force is proportional to the normal force between
the cart and the track. Since the cart is continuously changing the angle of inclination as
it traverse the track then we know the normal force must also be changing, and hence the
friction changes. However, the ability to fully analyze this situation requires more knowledge
of calculus and di↵erential equations than we have at this point in the curriculum. Thus, we
make this simplifying assumption only to obtain a first approximation to the importance of
friction, recognizing that our analysis is incomplete.
Equation 6.14 says that we are neglecting the rotational kinetic energy stored in the
rotating mass of the wheels of the cart. While rotating wheels may contain a significant
amount of energy in real applications, we have made an e↵ort to design the wheels such that
they do not store significant rotational energy. We will develop the ability to analyze this
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form of energy storage later in the curriculum, in our dynamics class.
6.1.3
Chart
Schematic Diagrams
A schematic diagram illustrating a cart rolling down a track is shown in Figure 6.1. The
track is made from a metal ruler, so you have the ability to measure arclength along the
direction of travel. The background provides a grid against which you can manually measure
horizontal and vertical position information. You may select an origin for your coordinate
system that is convenient. The figure shows three locations, or ”states” that have been chosen
conveniently along the path of travel, and labelled sequentially. State 1 will be considered
the initial condition of the cart starting from rest at the top of the first side of the hill. State
2 will be considered to be the point when the cart is at minimum elevation, at the bottom of
the valley. State 3 will be taken to be the point at the upper most position on the second side
of the hill, when the cart is momentarily at rest before returning back downwards. While
not shown on the figure, State 4 will be the next arrival of the cart at the valley, State 5 will
be the cart’s return to the upper most position on the first side of the hill, State 6 again at
the valley, and finally State 7 will be considered the upper-most position of the cart on the
second side of the hill.
Figure 6.1: Schematic diagram of a cart rolling down a curved ramp.
Free Body Diagrams
As the cart rolls down the track, it experiences external forces due to gravity, the normal
reaction of the track, and the frictional resistance of the track to the motion of the cart as
shown in Figure 6.2. Note that the angle of inclination of the cart changes throughout the
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range of motion, since the track is not a straight line. Thus, the direction of the velocity
and the direction of the normal force reaction from the track to the cart changes as the cart
moves along the track.
Figure 6.2: Free body diagram of a cart rolling down a curved track.
Vector Diagrams
The vectors change continuously as the cart moves down the track, as shown in Figure 6.3.
This constantly changing set of vectors makes analysis of this system somewhat difficult (but
not impossible) to analyze using only Newton’s laws of motion.
Figure 6.3: Force vector diagram of a cart rolling down a curved ramp.
Data Tables
As we complete our analysis, it will be helpful to record data in the state table as shown
below in Table 6.1. Each row in the state table corresponds to one unique position and
energy status of the cart as it moves along the track, as indicated in the schematic diagram.
Each state i corresponds to a unique elevation and the energy condition of the cart when it
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is at that particular elevation. If the cart is at an elevation of 0.1[m] with a high speed at
one instant of time, but several seconds later returns to that same track elevation, with a
speed of 0[m/s], then these are two di↵erent states. Just because the cart is at the elevation
corresponding to state i does not mean that all characteristics of the cart are the same.
Thus, when we define a “state” in mechanical engineering, we use that term very precisely
as meaning “the state of a physical system fully describes the energy status of the system
at an instant of time, and is independent of the path taken to arrive at the state.” We will
estimate the track position, elevation, KE, P E, speed, and total energy of the cart at each
state. We will assume that the elastic (spring) potential energy of the cart remains zero
throughout the motion, SEi = 0. Since we are releasing the cart from rest, we declare that
the speed and KE at state i = 1 are both zero. The total energy of each state is the sum of
the components Ei = KEi + P Ei + SEi .
Table 6.1: State Table for Single Cart Motion
i
[ ]
1
2
3
4
5
6
7
si
[m]
zi
[m]
Vi
[m/s]
0.0
KEi
[J]
0.0
P Ei
[J]
SEi
[J]
0.0
0.0
0.0
0.0
0.0
0.0
0.0
Ei
[J]
As we study the work-energy theorem, we will talk about the work being done between
two states, for example, as a cart moves from one state i to another state j. The “state
table” above documents the physical condition of the cart at discrete points in time (or
space). We will now introduce another table, called the “process table” which will describe
the exchange of energy, work and heat, as we move between states. We define a “process” in
mechanical engineering as “a process of a physical system fully describes the change between
two states of the system over an interval of time, and describes the path taken between
states.” The process table for this experiment is shown in Table 6.2. We can add a row to
the process table to study the change between any two states of interest. We assume that
there is no heat transfer in our experiment, so we fill that column with zeros to remind us
of our underlying assumption. The column |si!j | is used to track the cumulative distance
travelled by the cart as it goes back and forth between hill and valley, which we denote as
two states (or, less precisely, locations) i and j.
225
Table 6.2: Process Table for Single Cart Motion
i
[ ]
1
2
3
4
5
6
6.1.4
j
[ ]
2
3
4
5
6
7
Sum
|si!j |
[m]
Ei!j
[J]
Wi!j
[J]
Qi!j
[J]
0
0
0
0
0
0
0
Execute
Recall The Governing Equations.
The first step in the execution phase is to recall the governing equations. In this case, we
have our familiar set of Newton’s laws to draw upon, and we now include the work and
energy theorem as well.
!
F N et = 0 T hen : !
a =0
!
!
!
!
p
(m V )
d(m V )
F N et = lim
= lim
=
t!0
t!0
t
t
dt
!
!
F Action = F Reaction
E2 E1 = Q1!2 W1!2
If :
Newton’s 1st Law
(6.15)
Newton’s 2nd Law
(6.16)
Newton’s 3rd Law
Conservation of Energy
(6.17)
(6.18)
Conservation of Energy and the Work / Energy Theorem.
The conservation of energy (also known as the “First Law of Thermodynamics”) was introduced at the beginning of the course. This law says that “The change in the amount of
energy stored within a system during some time interval is equal to the net amount of energy
transferred into the system by heat transfer from its surroundings during the time interval,
minus the net amount of energy transferred out of the system by work done by the system
on its surroundings during the time interval.”
Recall that we use the sign convention that “heat flowing into the system from the
surroundings is positive” and that “work done by the system on its surroundings is positive.”
Equation 6.18 tells us many useful things. First of all, if we take a system and put
thermal energy (heat) into it, then the energy contained within the system will increase.
Second, if the system does work upon its surroundings then the energy contained within the
system will decrease. Third, if the amount of heat we put into the system is equal to the
amount of work done by the system, then the energy contained within the system will be
constant. Equation 6.18 is so important to the field of mechanical engineering that we will
devote entire courses to its study! In particular, thermodynamics is an exhaustive study of
226
the law of conservation of energy. For the duration of this course we will assume that no
heat energy goes into or out of the system, other than by the action of friction.
We assume
Q1!2 = 0
(6.19)
Under this assumption, Equation 6.18 reduces to
E2
E1 =
W1!2
(6.20)
Furthermore, we will restrict ourselves in this course to only three kinds of energy storage:
kinetic energy, gravitational potential energy, and spring (or elastic) potential energy. With
these self-imposed restrictions, we can say that
E2
E1 =
KE1!2 +
P E1!2 +
SE1!2 =
W1!2
(6.21)
Equation 6.21 is called the work-energy theorem. We will use the work energy theorem 6.21
as a special case of the more general conservation of energy law, Equation 6.18 in this course.
Work is defined as a force acting over a distance. In our secondary school physics class, we
probably learned the simple definition that work equals force times distance. This definition
is sufficient if the work is constant over the entire distance of displacement. However, most
machines make use of the fact that the forces are not constant. Thus, we need to introduce
a more sophisticated and general definition of work, which says that “the net work exerted
upon an object is equal to the integral of the net force acting upon the object over the
!
distance which the force is acting.” Consider Figure 6.4, which shows some general force, F ,
acting upon mass m over some arbitrary path, !
s . From our previous work with Newton’s
Figure 6.4: Work is defined as the integral of force acting over some displacement.
first law, we know that if a body is moving along some curvlinear path !
s as shown, then
!
!
F cannot be constant. Even if the magnitude of F is constant, its direction must not be,
!
since the direction of motion of the object is changing. Force F is a vector, and so is the
displacement !
s . However, work, W , is a scalar. The work energy theorem talks about the
distance over which the force is applied – we use our vector notation and the concept of a
dot product (also called the scalar product) from mathematics to define work as:
W1!2 ⌘
Z
!
s2
!
s1
227
! !
F ·ds
(6.22)
We will study dot products in our homework problems this week. In two dimensions, we can
write this integral in terms of its vector components as:
Z !
s2
W1!2 =
(fx ı̂) · (xı̂)dx + (fz k̂) · (z k̂)dz
!
s1
Z x2
Z z2
W1!2 =
xfx dx +
zfz dz
(6.23)
x1
z1
Now that we have introduced the concept of the work energy principle and the definition of
work, let’s apply them to a familiar problem – a ball in free fall under the action of gravity.
The free body diagram for a ball in free fall is shown in Figure 6.5. From our earlier work
Figure 6.5: A ball in free fall under the action of gravity.
we know that we can write that the vertical position of a mass allowed to drop in free-fall
under the action of gravity as
1 2
z(t) = z1 + Vz1 t
gt
(6.24)
2
If we measure the position of the ball from its initial position, then
z1 = 0
when
t=0
(6.25)
when
t=0
(6.26)
and, if the ball is dropped from rest, then
V z1 = 0
and 6.24 reduces to
1 2
gt
2
At the instant t = t2 , the ball is at the vertical position z2 :
z(t) =
z2 (t2 ) =
1 2
gt
2 2
(6.27)
(6.28)
Clearly, the longer the interval of time is following release, the farther the ball will fall. Keep
in mind that we made several assumptions in our earlier free-fall analysis – that the distance
228
changes are small relative to the radius of the earth and the mass m is constant. Equation
6.28 is the theoretical distance that the ball would drop at time t2 in the absence of friction
– the idealized case when gravity is the only external force acting upon the ball, and that
the gravity force is constant, W = mg. Let’s use this assumption in the definition of work
W1!2 =
=
=
Z
z2
Zz1z2
Zz1z2
z1
W1!2
! !
F ·dz
(6.29)
⇣
mg k̂)( 1k̂)dz
Z z2
mgdz = mg
dz
= mgz|zz21
= mg (z2
(6.30)
(6.31)
z1
z1 )
(6.32)
(6.33)
with our sign convention, z2 is smaller than z1 , which says that the net work is negative. That
is, the surroundings are working on the ball. We use this observation to define a convenient
form of energy, and we call it the gravitational potential energy. The gravitational potential
energy of an object in the vicinity of the earth is defined as
P E(z) ⌘ mgz
(6.34)
where the origin of the z axis is chosen at a convenient location. Thus we can use definition
6.34 in Equation 6.33 to write
W1!2 = mgz2 mgz1
= P E2 P E1
W1!2 = P E1!2
(6.35)
(6.36)
(6.37)
which says that the work done on the ball by the surroundings is equal to the change in the
gravitational potential energy of the mass, when gravity is the only force acting upon the
ball. Defining the work done by gravity, which is a conservative force, as the gravitational
potential energy allows us to analyze many problems without having to constantly perform
the tedious integrals expressed in equation 6.29. A conservative force is defined as a force
whose cumulative action (or work) is independent of the path travelled. That is, when we
consider the work done by gravity, it does not matter what path the ball takes between the
initial state and the end state – the net work done depends only upon the two states. Let’s
develop another conveniently defined form of energy – kinetic energy. From experience in our
secondary school physics class, we know that the decrease in potential energy of the ball will
be o↵set by an increase in kinetic energy. Let’s take a closer look at this phenomena, through
study of Newton’s second law. Recall our earlier result describing the vertical position of
the ball as a function of time after its release. We refer to the release time as t1 here:
z(t) = z1
1
g(t
2
229
t1 )2
(6.38)
and the expression for the vertical component of the velocity of the ball
Vz (t) =
g(t
t1 )
(6.39)
Let’s evaluate Equations 6.38 and 6.39 at the instant of time t = t2 . First, let t ! t2 in
Equation 6.38:
1
g(t2
2
z(t2 ) = z1
z2
2(z1
s
z1 =
z2 )
g
2(z1
z2 )
g
1
g(t2
2
t1 )2
(6.40)
t1 ) 2
(6.41)
= (t2
t1 ) 2
(6.42)
= (t2
t1 )
(6.43)
t1 )
(6.44)
Now, substitute Equation 6.43 into Equation 6.39
Vz (t2 ) =
g(t2
s
V z2 =
g
2(z1
Vz22 =
2
g 2 (z1
g
Vz22
=
2
g(z1
z2 )
(6.45)
g
z2 )
(6.46)
z2 )
(6.47)
Multiply both sides of Equation 6.47 by m to get
1
mVz22 =
2
mg(z1
z2 ) = mg(z2
z1 )
(6.48)
Now, we define the kinetic energy as KE ⌘ mV 2 /2. Then
1
KE2 = mVz22
2
1
KE1 = mVz21 = 0
2
(6.49)
(6.50)
The initial kinetic energy is zero because the ball was released from a rest position. Then
we can write
KE1!2 = mg(z1
z2 )
(6.51)
But, we know already from the definition of the gravitational potential energy that
mg(z1
z2 ) =
P E1!2 = P E1
230
P E2
(6.52)
Substituting Equation 6.52 into 6.51, we see that
KE1!2 =
P E1!2
(6.53)
Let us next compare 6.53, which we developed from Newton’s laws, with our original work
energy theorem as given by Equation 6.21. The convenient definition of KE and P E allows
us to greatly simplify our analysis of problems that involve motion due to conservative forces
such as gravity. Using 6.52 in 6.21 allows us to say for the system of an object dropping in
free fall that E2 E1 = 0, or that the energy of a system is conserved when only conservative
forces are present. Thus we will include the work done by the conservative force gravity in
the KE and P E terms. The work energy theorem reduces to simply
KE1!2 +
P E1!2 +
SE1!2 =
W1!2
(6.54)
Further, if there are no springs to store elastic potential energy then in the absence of
friction we can write
KE1!2 + P E1!2 = 0
(6.55)
Equation 6.55 is a simplified version of the work energy theorem will allow us to predict motion along general curvilinear paths, without the cumbersome task of complicated integrals.
Let’s use the work energy theorem to analyze a problem that would be significantly more
difficult (but entirely solvable) using only Newton’s laws. Furthermore, the analysis of the
ball dropping in free-fall illustrates that the work-energy theorem follows directly from, and
is completely consistent with, Newton’s laws of motion!
Simplify the Governing Equations.
We know that gravity and friction are both external forces acting upon the cart. Thus, we
know from Newton’s first law that the acceleration of the cart is expected to be non-zero.
While we can use Newton’s second law to analyze the motion of the cart, it would be quite
cumbersome, since the directions of the forces are changing as the cart moves along the
track. Thus, we prefer not to employ Equation 6.16 unless we have to. We have already
used Newton’s third law, and Newton’s law of gravity to create the free body diagram for
the cart. Let’s use the conservation of energy approach to attack this problem.
The total energy of the system, Ei , at any state i is the sum of the gravitational potential
energy, kinetic energy and elastic potential energy:
SE!0
byEq.6.12
z}|{
Ei = P Ei + KEi + SEi
1
Ei = mgzi + m|Vi |2 + 0
2
[J] = [kg][m/s2 ][m] + [ ][kg][m/s]2 + [J]
[J] = [J] + [J] + [J]
[↵] = [ ] + [ ] + [ ]
231
(6.56)
(6.57)
Units Validation
Units Validation
Inventory
We indicate that the P E and KE are known, since we will measure the mass, vertical
position and speed of our object experimentally in Lab, which will allow us to compute P E
and KE respectively. After we fill in the process table for each state 1  i  7, we can
analyze the process table to study the motion of the cart between states. Now, knowing the
value of total energy at each state i, we can apply the work energy theorem 6.18 between any
state i and another state j and solve for the work done by the cart upon the surroundings:
Q!0
byEq.6.11
z }| {
Ej Ei = Qi!j Wi!j
Ej Ei = Wi!j
[J] [J] = [J]
units check
[ ] [ ]=[ ]
known values check
(6.58)
(6.59)
Inventory the Governing Equations, Known, and Desired Information.
Until this point, we have used the work and energy theorem to assemble a set of equations
with known and unknown values in them. Now, we need to use experimental observations
to solve the system of equations. In lab this week, we will use a camera to capture a series of
images of the the cart as it travels along the track. We will then be able to view this image
sequence frame-by-frame to extract meaningful data from them.
As seen in 6.1, a grid is placed in the field of view of the camera, behind the motion
of the cart. The initial position of the cart, shown as position (x1 , z1 ) in the figure, can
be determined by inspecting the video image to locate the approximate center of the cart
relative to the grid position in the background. The curved track (which is made from a
metal ruler) also has tick marks on the side of the track, corresponding to arc length position,
s, along the curved track. Using the grid and the track markings, we can know the vertical
position of the cart zi for any chosen arc position along the curved track, si . The di↵erence
in the cart’s position along the track between subsequent frames can be used to estimate
the cart speed. After we have the speed and elevation observations for each state i, we can
estimate the kinetic energy and the gravitational potential energy. Thus, from experimental
observations, we can estimate the following values, in sequential order:
zi (t) =
si (t) =
[m] Measurable from camera images
[m] Measurable from camera images
|si (t +
t) si (t)|
t
([ ] [ ])
[↵] =
[ ]
([m] [m])
[m/s] =
[s]
|Vi | ⇡
(6.60)
(6.61)
(6.62)
Inventory
Units Validation
232
Next, we use the definitions of gravitational potential energy and kinetic energy:
P Ei = mgzi
[↵] = [ ][ ][ ]
[J] = [kg][m/s2 ][m]
1
KEi = m|Vi |2
2
[↵] = [ ][ ][ ]
[J] = [ ][kg][m/s]2
(6.63)
Inventory
Units Validation
(6.64)
Inventory
Units Validation
We can apply the preceding series of equations to each state observation following the conclusion of the experiment, for all states 1  i  7. These equations are sufficient for us to
fill in every entry in the state table, presented in Table 6.1. In Studio, we will learn more
about the details of estimating the speed of the cart from the image data.
Solve
After the state table is complete, we can complete the process table, shown in Table 6.2.
The change in kinetic energy and gravitational potential energy can be computed for each
process step (the process moving from state i to state j) by:
KEi!j = KEj
[↵] = [ ]
[J] = [J]
P Ei!j = P Ej
[↵] = [ ]
[J] = [J]
KEi
[ ]
[J]
P Ei
[ ]
[J]
(6.65)
Inventory
Units Validation
(6.66)
Inventory
Units Validation
next we can substitute Equations 6.65 and 6.66 into Equation 6.54 to arrive at an expression
for the work between any two states i and j:
Ej Ei = Wi!j
(P Ej + KEj ) (P Ei + KEi ) = Wi!j
P Ei!j + KEi!j = Wi!j
[J] + [J] = [J]
[ ] + [ ] = [↵]
(6.67)
(6.68)
(6.69)
Units Validation
Inventory
Equation 6.69 allows us to fill in the last remaining column of the process table.
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6.1.5
Test
Validate
In Lab, we will conduct an experiment to measure the motion of the cart along the curvilinear
path. In Lab, you will conduct multiple trials, and each student should leave Lab with one
good series of digital images saved to your portable drive.
Verify
We have verified that the units on each result are correct. Upon completion of the experiment
and analysis, we should determine if the signs on the results are consistent with the sign
convention assumed in the analysis.
Apply Intuition
We expect kinetic energy to be at a maximum when the cart is at the lowest point in its
traverse. We expect the maximum potential energy to decrease with each cycle, since friction
requires the cart to perform work upon the track.
6.1.6
Iterate
We will validate the theoretical results obtained in our laboratory experiment.
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6.2
6.2.1
Lab - Cart in Curvilinear Motion
Scope
This week in lab we will use a digital camera to record the motion of a cart as it rolls down
a track under the influence of gravity. Each member of the lab team will conduct a separate
trial and obtain a unique series of still images for their personal analysis. After capturing the
series of still images, each student will process the images to estimate the position and speed
of the cart at several discrete positions during its traverse of the track. We will be using a
manual process to conduct the analysis, and use the analysis to gain an understanding of
how such a process works.
6.2.2
Goal
The goals of this laboratory experiment are to
1. demonstrate the applicability of the Work and Energy Theorem,
2. expand our tools for measuring physical systems,
3. acquire all data necessary for reporting the results desired, as listed in Sections 6.1.1
and 6.3.3,
4. acquire images of the cart at various instants of time using a digital camera image
capture system,
6.2.3
Units of Measurement to use
All reports shall be presented in the SI system of units. Raw data may be collected in a
variety of units.
Quantity
Length
Mass
Time
Speed
Force
Energy
Work
Basic units
[m]
[kg]
[s]
[m/s]
[kgm/s2 ]
[kgm2 /s2 ]
[kgm2 /s2 ]
Derived units
[m]
[kg]
[s]
[m/s]
[N ]
[J] or [N ][m]
[J] or [N ][m]
Table 6.3: Units of Measurement to be reported for Single Cart Motion
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6.2.4
Reference Documents
The following documents may be helpful; user’s guide for the computer, user’s guide for data
acquisition software, user’s guide for the camera.
6.2.5
Terminology
The following terms must be fully understood in order
of this laboratory experiment.
Force
Displacement
Velocity
Speed
Energy
Work
Gravitational Potential Energy Kinetic Energy
6.2.6
to achieve the educational objectives
Dot Product
Acceleration
Curvilinear
Summary of Test Method
On the myCourses site for this course you will find links to one or more videos on YouTube
for this week’s exercise. Watch all of the available videos, and complete the online lab quiz
for the week. The videos are your best reference for the specific tasks and procedures to
follow for completing the laboratory exercise.
6.2.7
Calibration and Standardization
The Lead Technologist and Assistant Technologist shall confirm that the digital image capture software is functioning properly by acquiring images of the cart at various fixed positions
along the track.
6.2.8
Apparatus
All required apparatus and equipment components are described and demonstrated in the
instructional videos for this exercise or will be familiar from common or previous use.
6.2.9
Measurement Uncertainty
The uncertainty relationships discussed here and in each Measurement Uncertainty section
of this text are summarized for your convenience in the Engineering Mechanics Reference
Table, which is posted on myCourses. These uncertainty relationships can be applied to the
energy, work, distance traveled and friction coefficient calculations and force calculations to
help quantify the range of uncertainty in your analysis. Your Studio instructor will go over
the derivation of specific uncertainty expressions needed for the analysis. Be sure to make
a note in your logbook of the instrument least count values for any instruments used in the
laboratory.
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6.2.10
Sampling, Test Specimens
Each lab group should use a lab station with unique carts and transducer for their series of
trials. Each member of the lab group should collect a unique set of photos of their cart as a
function of time. The student should check their images make sure that can be analyzed for
cart position.
6.2.11
Preparation of Apparatus
All required equipment for conducting the laboratory exercise is made available either within
one or both of the drawers attached to the lab bench or from the laboratory instructor. You
are expected to bring all other necessary materials, particularly your logbook and a flash
drive for storing electronic data as appropriate. You are to follow the general specifications
for team roles within the lab. Although there are specific, individual expectations for each
role, you are each responsible overall to ensure that the objectives and requirements of the
laboratory exercise are met, and that all rules and procedures are followed at all times,
especially any that are related to safety in the lab. When finished, all equipment is to be
returned to the proper location, in proper working order.
6.2.12
Procedure - Lab Portion
Record all observations and notes about your lab experiment in
your logbook.
Please note that you will be collecting a large set of image files, so it is important to bring a flash drive to store the results on. Additionally, pay particular
attention to the instructions in the video regarding how to verify the suitability
of your data set and how to save the image files.
The instructional videos for this exercise cover the specific procedures to follow as you set
up the apparatus to make measurements, and for actually collecting data with the various
devices and software interfaces. More generally, you should always observe the following
general procedures as you conduct any of the exercises in this laboratory.
1. Come prepared to lab, having watched the videos in detail, then completing the associated lab quiz and preparing your logbook before you arrive to class.
2. Follow the basic outline of elements to include in your logbook related to headers,
footer, and signatures.
3. As you conduct the exercise, please pay attention to the following safety concerns:
• Watch for tripping hazards, due to cables and moving elements.
• Watch for pinch points, during assembling and disassembly.
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• Be careful of shock hazards while connecting and operating electrical components
4. Every week, for every exercise, your logbook will minimally contain background notes
and information that you collect before the lab, at least one schematic of the apparatus,
various standard tables for recording the organization of your roles and equipment
used, the actual data collected and/or notes related to the data collected (if done
electronically for instance), and any other information relevant to the reporting and
analysis of the data and understanding of the exercise itself.
5. All students should create and complete a table indicating the staffing plan for the
week (that is, the roles assumed by each group member), as shown in Table 1.2.
6. All students should create and complete a table listing all equipment used for the exercise, the location (from where was it obtained: top drawer, bottom drawer, instructor?)
and all identifying information that is readily available. If the manufacturer and serial number are available, then record both (this would be an ideal scenario). If not,
record whatever you can about the component. In some, cases, there will be no specific
identifying information whatsoever either because of the simplicity of the component,
or because of its origin. In these cases, just identify the component as best you can,
perhaps as “Manufactured by RITME.” The point here is to give as much information
as possible in case someone was to try to reproduce or verify what you did. Refer to
Table 1.3.
7. For the Lab Manager only: create a key sign-out/sign-in table for obtaining the
key to the equipment drawers, as shown in Table 1.4.
8. All students should create a table or series of tables as appropriate to collect his/her
own data for the exercise, as well as any specific notes related to the data collection
activities. In those cases where data collection is done electronically, there may not be
any data tables required.
9. Many of the laboratory exercises will require the use of a specific software interface
for measurements and/or control. In all cases, these will be made available on the
myCourses site unless stated otherwise.
10. The Scribe (or a designated alternative) should take a photo of each group member
performing some aspect of the laboratory exercise for inclusion in the lab report
that will be generated during the studio session. Refer to the example lab report for
more details.
11. Record all relevant data and observations in your logbook, even those that may not
have been explicitly requested or indicated by the textbook or videos. If in doubt
about any measurements, it is better to make the measurement rather than not.
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12. When you are finished with all lab activities, make sure that all equipment has been
returned to the proper place. Log out of the computer, and straighten up everything
on the lab bench as you found it. Put the lab stools back under the bench and out of
the way.
13. Prepare for the upcoming studio session for the week by carefully read and understand
Section 4.3 of the textbook, and complete the Studio pre-work prior to your arrival at
Studio.
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6.3
Studio - Work and Energy Analysis
This week in Studio, you will complete an analysis on the Work Energy Theorem using the
curvilinear motion data for the cart you obtained in lab.
The equations for calculating energy and friction rely heavily on the Work Energy Theorem discussed in Section 6.1 of the text. Section 6.3.1 Calculation and Interpretation of
Results, provides a summary of equations that you will need to complete the Studio. Section
6.2.9 Measurement Uncertainty, describes the process for analyzing the experimental errors.
There will be uncertainty equation derivations that you need to complete prior to arriving
at studio and these are listed in the Measurement Uncertainty section.
Record all observations and notes about your studio procedures in
your logbook.
6.3.1
Calculation and Interpretation of Results
This week in Studio, you will create a State Table to show the state of the cart’s energy
(potential, kinetic and total) each time the cart reaches the top or bottom of the track for
these cycles. The process involves manually determining the vertical and arc length position
of the cart from the set images taken as the cart moved back and forth along the track over
several cycles. You will then create a Process Table to show the change in energy of the
cart for each process, where the cart moved from the top to the bottom of the track or from
the bottom to the top and so on. You will also be able to use the Work Energy Theorem
to calculate the amount of work done by the cart to overcome friction and the definition of
work to estimate the friction force between the track and the cart. The equations provided
below are summarized for your convenience from the derivations in Section 6.1 and previous
chapters.
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1
Time between frames
F rame Rate
1
[s/S] =
Units Verification
[S/s]
Qi!f Wi!f = Ei!f
Work energy theorem
[J] [J] = [J]
Units Validation
E = KE + P E + SE
Total Energy of a state
[J] = [J] + [J] + [J]
Units Validation
KEi!f = KEf KEi
Change in KE between states
[J] = [J] + [J]
Units Validation
P Ei!f = P Ef P Ei
Change in P E between states
[J] = [J] + [J]
Units Validation
Ei!f = Ef Ei
Change in Total Energy between states
[J] = [J] + [J]
Units Validation
s(f + 1) s(f )
Vi =
Cart Speed,f = frame, f + 1 = next frame
t
[m] [m]
[m/s] =
Units Validation
[s]
Z f
! !
Wi!f =
F ·ds
Work done by the system to overcome friction
t=
(6.70)
(6.71)
(6.72)
(6.73)
(6.74)
(6.75)
(6.76)
(6.77)
i
[J] = [N ][m]
6.3.2
Units Validation
Procedure - Studio Portion
Studio Pre-work
Prior to arriving at Studio, each student should have acquired the necessary data in lab,
recorded data in your notebook and stored data on a thumb drive. You should also have a
corresponding schematic that clearly identifies where each measurement was made in symbolic notation.
In addition, each you will complete several steps of the Studio exercise. This will allow
more quality time with the instructor to discuss the physical meaning of the analysis results.
You will upload your studio pre-work to your individual drop-box for the corresponding
week. You will receive a quiz grade based on the completeness of your submission.
Please complete at least steps 1-13 and upload your pre-work spreadsheet
to your individual drop-box, before coming to coming to Studio. Note that an
Excel template called ”MECE 102 Week 6 Template.xlsx” is provided this week,
which will greatly reduce the time in setting up the tables and allow you to skip
through several of the first steps in the analysis. You can work on the remaining
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portions of the exercise during Studio. All steps with the exception of the report are due
within 24 hrs after leaving Studio.
Videos
There are videos available to help with some of the excel techniques that may be new to you.
We have highlighted steps where videos might be helpful. However, you can also complete
the steps simply by following the written instructions. For those procedures that do not
have videos, you should rely on previously developed skills. You may want to review videos
from previous weeks if you feel that you need a refresher on some of the techniques.
Steps to Complete the Analysis
1. LOGBOOK: Before you begin, enter a Studio Week 6 header on a new page in your
logbook. Use Figure 1.10 as a template. After completing the analysis, you will print
out your graphs, answer questions and make observations related to your analysis. It
is important to make it clear that the work entered today is from Studio Week 6.
2. LOGIN: Login to your PC with your RIT account information. Insert your USB drive
into the USB port on your computer.
3. CREATE A NEW FOLDER: On your USB drive, create a working folder called Week 6.
Store all of your Lab and Studio files for today’s session in this folder.
4. CREATE THE HEADER, CONSTANTS AND CONVERSION FACTOR TABLES:
Refer to Figure 6.6 for the format of the tables. Notice that we have two columns
to the constants table for the instrument least count (ILC) and uncertainty in each
constant value. Fill the table in completely with constants and equations to calculate
the uncertainties. You can assume that the uncertainties for time step and gravity are
negligible compared to the other values, so enter zeros for these uncertainties. You will
need an uncertainty for the mass and can be based on the instrument least count for
the scale.
5. CREATE A STATE TABLE: Create a state table, similar to that illustrated in Table
6.7, in your spreadsheet. We will walk through how to fill in this table in the steps that
follow. Note that the video shows inches-to-m conversions for the Vertical position.
Ignore this for the analysis; we measured position in cm in our lab. Previously, the
reference grid was in inches.
VIDEO RESOURCE: Studio 06 Video - State Table
6. CHOOSE AND DOCUMENT THE STATES YOU WILL ANALYZE: Pull up the
images of your experiment and observe the cart as it progresses along the track. Choose
the frames that you will use to analyze each state of the car, 1 through 8 in the table
above. To analyze the velocity at the bottom of the ramp, you will need to choose two
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Figure 6.6: Template for tables to be created in Studio including Header, Table of Constants
and Conversion Factors.
Figure 6.7: Template for the State Table to be created in Studio.
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sequentially adjacent frames. That is what is meant by “Frame, f ” and “Next Frame,
f + 1”. There will be only one frame for each cart position at the top of the ramp.
Enter the file name into the cell for each frame you will use. Hint: If your cell value
keeps changing to a date or other number, change the data type to text and use a title
with text, for example “0.0015.jpg.” Leave all unneeded cells blank.
7. CALCULATE POTENTIAL ENERGY: By inspection of the images, determine the
vertical position for each state 1 through 8 of the system. Hint: Find an image when
your cart is at the bottom of the ramp. Locate the point on the backdrop grid where
the that aligns with the cart’s center of gravity. Locate your vertical datum, z=0 at
this point. Then your cart will have zero potential energy at the bottom of the ramp.
The vertical position of the cart should be measured from this point to the cart’s center
of gravity at that new location. Enter these data into your State Table and convert
units as needed. Enter an equation that will calculate Potential Energy (P E) for states
1 through 8 based on the data available in your spreadsheet.
8. CALCULATE KINETIC ENERGY: By inspection of the images, determine the distance traveled by the cart between the two sequentially adjacent frames for state 2, 4,
6 and 8. Hint: this is done by locating two adjacent frames (some known time apart)
and determining by visual inspection of the images the distance traveled by the cart.
You will use the ruler that is fixed along the arc of the track. In the state table there
is a column labeled s(f + 1) - s(f ). Here, s(f ) is position, s, of the car along the track
in frame f. While s(f + 1) is the position, s, of the cart in the following frame denoted
as f+1. Enter the total distance the cart traveled between frames s(f ) and s(f + 1) in
your State Table in the columns labeled s(f + 1) - s(f ). Next, enter an equation that
will calculate the instantaneous cart velocity from the distance traveled between two
adjacent frames. Finally, enter an equation that will calculate the KE at each state
1 through 8. Note, in some cases the distance traveled between two adjacent frames
is zero. In these cases, leave cells blank. For example, the distance traveled between
sequential frames when the car is at the top most recorded position is assumed zero
because the KE is zero at this point.
9. CALCULATE TOTAL ENERGY OF THE SYSTEM: Enter an equation that will
calculate the Total Energy (E) of the system at each state 1 through 8.
Congratulations! You have just completed a State Table. You will be doing more of
these tables in a follow on course called Thermodynamics.
10. PLOT SYSTEM ENERGY: Create a properly formatted and documented engineering
plot that shows the amount of KE, P E and E in the system at each state 1 through
8. Your graph might look similar to that shown in Figure 6.8. We will calculate the
error and add error bars to the plot in later steps.
Note that we are creating a line plot rather than an xy plot. An xy plot is used when
the x axis data has actual units, like meters or seconds for example. Here we are
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plotting energy on the y axis versus the state point on the x axis. The state point
has no units, therefore we choose a line plot. In a line plot, it is reasonable and often
desirable to include data markers as well as lines to help illustrate the behavior of the
system. In a line plot, it is understood that no information is known between the data
points. Trend lines are not added to line plots, because they would have no physical
meaning.
Figure 6.8: Plot showing change in cart’s energy as it moves through the states.
11. CREATE A PROCESS TABLE: Create a process table, similar to that illustrated in
Table 6.9, in your spreadsheet. We will walk through how to fill in this table in the
steps that follow.
VIDEO RESOURCE: Studio 06 Video - Process Table
Figure 6.9: Template for the Process Table to be created in Studio.
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12. CALCULATE CHANGE IN KINETIC ENERGY: In your State Table, you found the
Kinetic Energy for each state 1 through 8. Now enter an equation in your Process
Table that will calculate the change in kinetic energy KE from one state to the next.
13. CALCULATE CHANGE IN POTENTIAL ENERGY: Similarly enter an equation in
your Process Table that will calculate the change in potential energy P E during each
process.
14. CALCULATE CHANGE IN HEAT ENERGY: Recall that we are assuming in this
course that heat transferred to or from the system can be ignored. Enter zero for these
values in your Process Table.
15. CALCULATE TOTAL DISTANCE TRAVELED: Here we want to enter the total track
distance traveled by the cart during each process listed in the table. By inspection of
the images you took in lab, and the demarcations on the meter stick, estimate the
distance the traveled by the cart going from the top of the ramp to the bottom, then
from the bottom to the top, and so on.
Congratulations! You have just completed a Process Table. You will be doing more of
these tables in a follow on course called Thermodynamics. Now you will use the data
in the State and Process Tables to further analyze the system.
16. CALCULATE TOTAL ENERGY CHANGE: Using the data in your State Table, enter
an equation below the State Table to estimate the total change in energy of the system.
Recall that the change in energy is simply the value of the final state minus that of
the initial. Write the equation and results in your log book.
Test: Since the car is slowing down, the system (the cart) is losing energy. Therefore,
the change in energy should be negative. Make sure that you are subtracting the initial
energy from the final energy.
17. CALCULATE THE WORK DONE: Using the data in your Process Table, enter an
equation below the Process Table to estimate the total work done by the system to
overcome friction. Write the equation and results in your log book.
Test: Recall that the work energy principal uses the sign convention that work done
by the system is positive. Since the cart is doing work to overcome friction, the total
work done by the system (the cart) should be positive. If you are having difficulty
with this, recall that from the Work Energy Theorem we define the sum of the changes
in kinetic and potential energy between two states as the negative of the work done
between the two states or W1!2 = KE1!2 + P E1!2 .
18. ESTIMATE FRICTION FORCE: Referencing the information in Section 6.3.2 and
Section 6.3.5, enter the necessary equations in your spreadsheet to calculate the average
Friction Force on the cart. Write the equation and result in your log book.
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Congratulations! You have just complete the main calculations for this studio analysis.
Now we need to look at the error associated with each calculated value.
19. DETERMINE THE UNCERTAINTY DUE TO MEASUREMENT PRECISION: This
week we have two di↵erent uncertainties for position measurements because two different instruments were used. We used the background grid for the vertical position
and the meterstick for the distance traveled. Create the uncertainties table shown in
Figure 6.10. Enter values (in the shaded cells) and equations (in the white cells) as
needed to obtain the required information.
VIDEO RESOURCE: Studio 06 Video - Uncertainties
Figure 6.10: Template for recording uncertainties due to measurement precision.
20. DETERMINE THE UNCERTAINTIES FOR VALUES IN STATE TABLE: Create
a table of uncertainties in your spreadsheet. Figure 6.11 illustrates what your table
should include. Notice that the table has been inserted in between the State and Process Tables, because it will reference state values. Enter equations that will calculate
the uncertainties in V , KE, P E and ET OT . Note that equations for these are provided
in the header labels of the Excel template file. These uncertainties will be used to add
error bars to your plots. Note that the equations shown in Figure 6.11 assume that the
uncertainty of the time interval and mass are so small relative to that of distance, s,
or velocity. Thus, time and mass are treated as constants in the uncertainty equation
derivations.
21. ADD ERROR BARS TO THE ENERGY PLOT: Now that you have determined the
error in your energy calculations you can add error bars to the plot. Add error bars
for the KE and P E series data. Notice that adding error bars for the total energy
would be redundant. You may need to alter the marker style and size in order to see
the error bars.
You can refer to the ”Excel 13 insert error bars.pdf” document found in the ”Excel
help pdfs folder” of Content on myCourses to find help on inserting error bars. Note
for this week’s plot, you will have to select a range of cells to correspond to the positive
247
Figure 6.11: Template for recording uncertainties due to measurement precision.
and negative error bar values for the di↵erent energies. In previous week’s, we selected
just one cell. For example, when selecting the values for the KE plot, one would want
to select the range U 4 : U 11 for both the positive and negative error bar values.
Test: Notice from Figure 6.8 that the error in the KE and P E calculations should be
di↵erent. The error in KE is much larger compared to P E. Also notice that the error
bars are not the same for every state point. This is due to the nature of the uncertainty
calculations and should be represented in your equations.
22. DETERMINE UNCERTAINTY IN THE CALCULATED REULTS: Using the equations you derived before coming to Studio, program your spreadsheet to calculate the
uncertainties for change in Total Energy of the System, Ei!j , Total Work done by
the System, Wi!j , Distance Travelled, Si!j and the Average Friction Force on the
cart, F .
23. UPDATE YOUR ENGINEERING LOGBOOK: Please print out your graphs, results
table and uncertainty tables and paste them in your logbook. You may also want to
print all of the tables, but this is left for the student to decide. Sign and date your
logbook before you leave Studio.
24. SUBMIT YOUR FILES: Submit your excel spreadsheet to your individual Week 6
dropbox on myCourses before leaving the Studio. If you have not completed all the
steps, upload what you have done and within 24 hours, upload your final completed
version. Remember to save your work to you USB drive, and take it with you when
you leave the Studio.
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25. OBSERVATIONS AND ANALYSIS: Write responses to the following questions in your
logbook. Be sure to include a justification for your answer by referring to the data,
plots, and derivations that are contained within your logbook. You may want to crossreference equations from Sections 6.1, 6.3.1 and 6.2.9 in your work.
(a) Compare the values and signs for the total energy change in the system (from the
State Table) to the Work Done (from to Process Table). Describe how they are
consistent with the sign convention for applying the Work Energy Theorem.
(b) Describe and quantify the sources of error in your calculations. Which uncertainties contributed the most to the accuracy of the total work done? What would
you do if asked to minimize this error in future experiments?
(c) How do your findings show agreement or disagreement with the Work Energy
Theorem?
(d) Why could we not calculate friction force directly from the FBD of the block and
applying Newton’s 2nd Law? What additional information would have been required to estimate friction force from Newton’s 2nd Law? Describe an experiment
that would produce such results.
26. CONGRATULATIONS! You have just completed the Studio portion for week 6.
27. WRITE THE REPORT: Please refer to section 6.3.3 Report on details for the report
submission. Before leaving Studio, decide on a date and time to meet up with your
team mates to prepare the report. Reports are due Monday by 6 pm.
6.3.3
Report
Please use the same task distribution for writing the report that was outlined in Week 1.
This week we have added a conclusion section. This section is to be no more than
1/2 a page long. The scribe is responsible for compiling both the results section and the
conclusion. However, all team members should contribute ideas to drafting the conclusion.
Please use the example lab report posted in Week 6 Content on myCourses for reference on
how to write the Conclusion section.
Refine your “Team Norms” to enhance your team’s ability to work e↵ectively with one
another, particularly if your team has encountered some challenges in preparing and submitting the report. Engage in an open and frank conversation about each team member’s
expectations and whether or not they are being met.
Prepare a report to include only the following components:
• TITLE PAGE: Include the title of your experiment, “System in Curvilinear Motion”,
Team Number, date, authors, with the scribe first, the team member’s role for the
week, and a photograph of each person beginning to initiate their trial, with a label
below each photo providing team member’s name.
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• PAGE 1: The heading on this page should read Experimental Set-up. Create a
diagram of the experimental set-up. This week we will include only the diagram and
its caption. Thus, is it important that your diagram clearly communicate the set-up,
including each key component and where measurements were taken. The important
information to communicate are the variable names and datums that relate to your
measurements and results. It is a good practice to add a legend that defines any
variables or components of the schematic that are not obvious. At the bottom of the
figure include a figure caption, for example Figure 1. A brief figure caption. Refer
to the text for examples.
Note: Figure captions are required for every plot and diagram in the report, except
for the title page. Figure captions are placed below the figures, and are numbered
sequentially beginning with Figure 1 for the first figure in the report.
• PAGE 2: The heading on this page should read Results. Include the table shown in
Table 6.4 summarizing each team member’s results. At the top of the table, include a
table caption, for example Table 1. A brief figure caption. Refer to the text for
examples.
This week we include only tables and plots with no accompanying text. Thus, it is
important that your tables, graphs and captions clearly communicate to the reader
what the data represents.
Note: Table captions are required for every table in the report, except for the title page.
Unlike figure captions, table captions are placed above the tables, and are numbered
sequentially (independent of figure caption numbering) beginning with Table 1 for the
first table in the report.
Table 6.4: Results from the curvilinear motion experiment.
• PAGES 3: No heading is needed on this page, since it is a continuation of the Results
section. Include the energy plots for KE, P E and ET ot . Include one plot for each
team member, with one figure caption for each plot. Each figure caption should be
located below the graph and include the name of the test engineer responsible for that
series of trials. The first caption in the Results section should be for example Figure
2. Sponge Bob Square Pants.
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Strive for uniformity among the graphs, including axis values and plotting styles. This
will enable the reader to easily compare the results between di↵erent team members.
• PAGES 4: The heading on this page should read Conclusions. Here you will state
the major conclusions that can be drawn from this analysis. In other words, you
will qualitatively and quantitatively answer the questions posed by the experiment.
Consider the following guiding questions when preparing your conclusion. Do any of
your results violate Newton’s Laws or the Work Energy Theorem, within uncertainty
limits? In evaluating your estimates for speed, kinetic and potential energy and average
friction force, consider if there were any systematic bias present in your results. What
are the most significant contributors to uncertainty, and how would you mitigate them?
Finally, comment on whether your experimental results support the Work Energy
Theorem within reasonable uncertainty?
Your conclusion should be NO LONGER than 1/2 a page when typed in 12 pt font.
• The final report should be collated into one document with page numbers and a consistent formatting style for sections, subsections and captions. Before uploading the
file, you must convert it to a pdf. Non-pdf version files may not appear the same in
di↵erent viewers. Be sure to check the pdf file to make sure it appears as you intend.
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6.4
Recitation
Recitation this week will focus on problem solving. Please come prepared, with your attempts
at the homework problem already in your logbooks.
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6.5
Homework Problems
Complete all assigned homework problems in your logbook.
6.5.1 Given a first vector !
s1 = 3ı̂ + 3ˆ
| 2k̂ and a second vector !
s2 =
the following:
(a) a = !
s1 · !
s2
!
(b) b = s · !
s
2
1ı̂ + 7ˆ
| + 4k̂, compute
1
6.5.2 Compute the dot product of !
s1 = 1ı̂ + 1ˆ
| + 1k̂ and !
s2 = 1ı̂ + 1ˆ
| + 1k̂.
6.5.3 Compute the dot product of !
s1 = 1ı̂ + 1ˆ
| + 1k̂ and !
s2 =
1ı̂ + ( 1)ˆ
|
6.5.4 Compute the dot product of !
s1 = 4ı̂ + 3ˆ
|
6k̂ and !
s2 = 1ı̂
4ˆ
| + 8k̂.
6.5.5 Compute the dot product of !
s1 = 4ı̂ + 3ˆ
|
6k̂ and !
s2 = 0ı̂
1ˆ
| + 0k̂.
6.5.6 Compute the dot product of !
s1 = 4ı̂ + 3ˆ
|
6k̂ and !
s2 = 1ı̂ + 0ˆ
| + 0k̂.
6.5.7 Compute the dot product of !
s1 = 4ı̂ + 3ˆ
|
6k̂ and !
s2 = 0ı̂ + 1ˆ
| + 0k̂.
6.5.8 Compute the dot product of !
s1 = 4ı̂ + 3ˆ
|
6k̂ and !
s2 = 0ı̂ + 0ˆ
| + 1k̂.
6.5.9 Compute the dot product of !
s1 = 4ı̂ + 3ˆ
|
6k̂ and !
s2 =
1ı̂ + 0ˆ
| + 0k̂.
6.5.10 Compute the dot product of !
s1 = 4ı̂ + 3ˆ
|
6k̂ and !
s2 = 0ı̂
6.5.11 Compute the dot product of !
s1 = 4ı̂ + 3ˆ
|
6k̂ and !
s2 = 0ı̂ + 0ˆ
|
6.5.12 Compute the dot product of !
s1 =
5ı̂
6.5.13 Compute the dot product of !
s1 =
5ı̂ and !
s2 = 1ˆ
|.
6.5.14 Compute the dot product of !
s1 =
5ˆ
| and !
s2 = 1ı̂.
1k̂.
1ˆ
| + 0k̂.
1k̂.
2ˆ
| + 1k̂ and !
s2 = 3ˆ
|.
6.5.15 Compute the dot product of !
s1 = 436ı̂ and !
s2 = 1, 435, 789k̂.
6.5.16 Compute the dot product of !
s1 = 500, 000ı̂ + 294, 000ˆ
| + 420, 363k̂ and !
s2 = 0.7071ı̂ +
0ˆ
| + 0.7071k̂.
6.5.17 A freight elevator is used to lift a car weighing 3, 200[lbsf ] from street level at an
elevation of 900[f t] above sea level to the upper floor of a parking garage, located five
stories above street level. Assuming that each story of the parking garage is 10[f t]
apart, estimate the theoretically minimum amount of work it must take to raise the
vehicle. Express your answer in [J] and in [f tlbsf ].
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6.5.18 A heavy box containing gifts from home arrives at your dorm lobby. Assuming the
box has a mass of 35.1[kg] and the dynamic coefficient of friction between the box and
floor is µd = 0.35[ ], how much work must you expend, in [N m], to push the box
164.2[in] across the lobby?
6.5.19 A heavy box, full of parts for your pumpkin launcher, arrives at the sidewalk outside
your dorm building. There is a ramp from the sidewalk, having an angle of incline
of ✓ = 15[ ] to the horizontal, that leads from the sidewalk to the building lobby.
The change in elevation from the bottom to the top of the ramp is 62.1[in]. The box
weighs 64.9[lbs] and the dynamic coefficient of friction between the box and ramp is
µd = 0.41[ ], how much work must you expend, in [J], to push the box up the ramp?
If you are applying the force by pushing on the box, and your forearms make an angle
of = 45[ ] to the horizontal, then what force, expressed in units of [lbf ], must you
exert in order to keep the box moving at constant speed up the incline? If you weigh
140[lbs], and the static coefficient of friction between your shoes and the ramp is 0.88,
what is the heaviest box you could possible move up the ramp?
6.5.20 You want to make a pumpkin launcher that will throw a 2[lb] pumpkin 75[f t] straight
up into the air. Neglecting air friction, what velocity must you launch the pumpkin
with? Express your answer in vector form. How much energy must you provide to the
pumpkin to launch it? How much time do you have to get out of the way after you
launch the pumpkin before it will land back at your feet?
6.5.21 You are carrying a backpack full of camping gear, food, and water. It has a mass
of 12[kg]. You start out at the base of a hill, and climb to the top of the hill by a
well-worn path. When you get to the top of the hill, 150[f t] above your starting point,
how much has the potential energy of the backpack increased? How much work did
you have to do on the backpack. Now, you return to the bottom of the hill, carrying
the backpack. What is the net change in P E of the backpack? How much work did
you perform? Explain your answer.
6.5.22 A bale of hay weighs 22[lbs]. You want to lift the bale into the loft of the barn, located
25[f t] up. What force must you exert upon the bale to move it upwards as constant
speed? Over what distance must you apply this force? How much work must you
expend?
6.5.23 A piano weighs 370[lbs]. You want to lift the piano with a pulley onto the third floor
balcony of your city apartment. The balcony is 34[f t] above the sidewalk. You are
stationary, pulling a rope at an angle of ✓ = 15[ ]. The rope moves through your hands
as you pull it. What is the minimum tension in the rope to move the piano upwards
at constant speed? Over what distance must you apply this force? How much work
must you expend?
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6.5.24 A piano weighs 370[lbs]. You want to lift the piano with a simple block and tackle
onto the third floor balcony of your city apartment. The balcony is 34[f t] above the
sidewalk. You are stationary and pulling a rope at an angle of ✓ = 15[ ]. The rope
moves through your hands as you pull it. What is the minimum tension in the rope to
move the piano upwards at constant speed? Over what distance must you apply this
force? How much work must you expend?
6.5.25 A piano weighs 370[lbs]. You want to lift the piano with a simple block and tackle
onto the third floor balcony of your city apartment. The balcony is 34[f t] above the
sidewalk. You hold the rope firmly in your hands, and walk away from the building,
pulling on the rope as you move. The rope is initially at an angle of ✓ = 15[ ]. What is
the minimum tension in the rope to move the piano upwards at constant speed? Over
what distance must you apply this force? How much work must you expend? What
is the horizontal distance you have to walk in order to lift the piano to the balcony?
What will be the final value of the angle ✓?
255