Download Lab #5: The Work – Kinetic Energy Theorem

Lab #5: The Work – Kinetic Energy Theorem
Reading Assignment:
Chapter 7, Sections 7-1 through 7-6
F/A-18E/F Super Hornet
U.S. Navy photo by Photographer's Mate 3rd Class John Sullivan
http://www.chinfo.navy.mil/navpalib/images/image-cv16.html
Introduction:
Aircraft take off from the deck of ships via a catapult system. Essentially, a large force is applied
to the aircraft as it is displaced across the deck of the ship. The purpose of this is obvious. A plane needs
to reach a particular speed before it can remain airborne and the catapult provides the means to do this. In
the language of Kinematics, one would describe the motion of the plane as having acceleration. In the
language of Newton’s Laws, one would explain that the Net Force on the plane was responsible for causing
the accelerated motion. However, there is another language used by Physicists to explain situations such as
this aircraft example. This language, and approach to solving problems, is founded upon the concepts of
“work” and “energy”. The purpose of this lab is to accelerate a cart using two different mock catapult
systems and to analyze each of these systems from a “work” and “energy” perspective.
While “energy” itself is a quantity that is easy for most people to understand, it is often difficult to
describe. One of the reasons for this is the fact that there are many different forms of energy: kinetic,
gravitational potential, elastic potential, electric potential, thermal, chemical, etc. Work, on the other hand,
often seems difficult for people to understand, even though it is simple to define. The word “work” has
many connotations in the English language outside the world of Physics. These other definitions often
serve to confuse students. Keep in mind that physicists have a very specific definition of “work”. It is this
definition that will be used here.
Work is the change in energy from one form to another by means of an external force. When
work is done on an object, therefore, the object is said to have either gained or lost a certain amount of
energy of a particular type. Hence the units of work are the same as the units of energy: “joules”. Work is
considered to be positive, negative, or zero in value, depending on the direction of transfer.
Work is accomplished on an object any time a force acts over a particular displacement, d, such
that the force, or some component of it, is parallel to the displacement. If the force is constant throughout
the entire displacement of the object, then the following equation is valid:
Wsub = Fsub d cos Θ
or
Wsub = F// sub d
where Θ is defined as the angle between the force and displacement vectors,“sub” refers to the descriptive
subscript clarifying which force is doing the work, and F// sub = Fsub cosΘ. If the subscript is “friction”, for
example, then the equation describes the work done on the object by friction. It the subscript is “gravity”,
for example, then the equation describes the work done on the object by the force of gravity. If the
Lab #6 – Work-Kinetic Energy Theorem
subscript is “net”, for example, then the equation describes the Net (or Total) Work done on the object. It
is important to clarify exactly which force, and therefore, work, is being described, because most objects
are acted upon by more than one force simultaneously. The Total Work done on an object describes the
overall result of the transfer of energy caused by all of the forces combined. Work is considered to be
positive, negative, or zero in value, depending on the value of Θ. (Recall that cosΘ = 1 if Θ = 0° and cosΘ
= -1 if Θ = 180°.) In addition, it is important to keep in mind that the above equation is valid if and only if
F// sub is constant.
If the force is not constant (variable), then the work done can not be determined via the equation
above. Instead, the following integral (for the case of one-dimensional displacement along the x-axis in
which Θ = 0°) must be evaluated:
W sub = ∫
xf
xi
F
sub
( x )dx
Essentially, the work done is found by determining the area under the Force vs. Position graph. See
Section 7-5 of the text by Halliday, Resnick, and Walker for a three-dimensional analysis of determining
the work done by a non-constant force.
One example of an non-constant force is a spring force. Springs exert a force that varies in a
predictable linear fashion described by Hooke’s Law:
F
x
Fby the spring = - k x
where x is the displacement of the spring from its equilibrium (at rest) position and k represents the force
constant of the spring (N/m). Notice the negative sign and the subscript of the force, as it is important to
understand the importance of these. Section 7-6 of the text explains the significance of this in detail. The
work done by a spring force, due to it’s linear nature, is rather simple to calculate using the integral above.
Computer programs that perform integration calculations do so using various numerical
techniques. Science Workshop™ has the ability to estimate the area under a particular plot of data using
one of these methods.
The Work-Kinetic Energy Theorem describes what happens when a particular force, such as the
one supplied by the catapult, does work to cause only the kinetic energy of the object to change. It is
written as follows:
Wby a particular force = ∆K = Kf – Ki
This equation, then, would not be valid if this particular force caused another type of energy to change,
such as gravitational potential energy or thermal energy. However, the Work-Kinetic Energy Theorem can
be applied to all situations if one is very careful to define the work done as the Net (or Total) Work done on
the object. This version of the Work-Kinetic Energy Theorem is more versatile:
Wtotal = ∆K = Kf – Ki
where the Total Work is determined by the sum of the work done by each of the individual forces acting on
the object, such as:
Wtotal =Wby an applied force+Wby friction+Wby gravity+Wby a spring +Wby the normal force…(etc.)
Lab #6 – Work-Kinetic Energy Theorem
Therefore, if the work done by a particular force appears not to be equal to the change in kinetic energy of
the object, then the system should by analyzed for possible work (positive or negative) done by other
forces. (Read Section 7-3 for a thorough explanation of the Work-Kinetic Energy Theorem.)
Lab #6 – Work-Kinetic Energy Theorem
Lab #5: The Work – Kinetic Energy Theorem
Goals:
•
•
•
•
Determine the Work done by a constant & a non-constant force.
Verify the Work-Kinetic Energy Theorem.
Determine the Spring Constant, k , of a given spring and use it to calculate the work done by a spring.
Apply the Work-Kinetic Energy Theorem to the F-14A Tomcat
Equipment List:
Science Workshop
1.2 meter track with adjustable feet
Dynamics cart with force sensor attached
Ultrasonic motion detector
String
Pulley
Scale balance (for measuring the mass of the cart)
Mass hanger and mass set
Spring
Computer & Equipment Set Up:
There are many calculations to be performed in this Lab.
Therefore, it will be more efficient to take the time to completely set up Science Workshop before
starting the lab activities.
1.
Set up Science Workshop to read the data collected from the force sensor connected to the
dynamics cart and the motion detector located at the end of the 1.2 meter track. The motion
detector does not need to be calibrated but the force sensor does. Always remember to
first remove all tension and then press the TARE button to re-zero the force sensor
before data is taken for each trial.
2.
Change “Sampling Options” so that Periodic Samples = 50 Hz. Change the motion detector’s
Trigger Rate so that it is also 50 Hz.
3.
Measure the total mass of the dynamics cart and the attached force sensor. Record this mass.
4.
Open the Experiment Calculator and define the calculation for Kinetic Energy= ½ mv 2 .
5.
Create a graph of Velocity vs. Time. Once this graph is displayed, click on the input icon for the x-axis
data and change it to position so that the graph plots Velocity vs. Position. Next, click the “Add Plot
Menu” button and select “Calculation, Kinetic Energy” in order to also graph Kinetic Energy vs.
Position. Click the “Statistics” button to open the statistics area of the Kinetic Energy vs. Position
graph. Set up this area to display the maximum and minimum values of your data.
6.
Create a graph of Force vs. Time. Once this graph is displayed, click on the input icon for the x-axis
data and change it to position so that the graph plots Force vs. Position. Click the “Statistics” button
to open the Statistics area of the graph. Select “Integration” from the Statistics Menu so that the area
between the data and the x-axis will be calculated.
Lab #6 – Work-Kinetic Energy Theorem
Force Sensor
Motion Detector
Dynamics Cart
Track
7.
Set up the equipment as shown above, for Activity 1, and get ready to take data.
Activity 1: Work Done by a Constant Force
1.
Press Record and gather data as the cart moves in one direction along the track while being pulled
with a constant force by the hanging mass. Be sure that the cart is released from rest. Note its
starting position and ending position relative to the motion detector. (Use the yellow measuring tape
located on the track.)
2.
Note the region of the graphs over which this motion took place. Remember that the x-axis of each
graph is Position, not Time.
3.
Note the area calculated by the integration function on the Force vs. Position graph over the constant
force interval. Answer the following questions: (Hint: Consider the direction of the Force and the
Displacement)
1.) Why is this value negative?
2.) Is the work done on the cart by the Force (due to the hanging mass) positive or negative? Explain.
(Realize that you will need to interpret the sign of this value correctly for all further analysis.)
4.
Using the Maximum and Minimum information, determine the Change in the Kinetic Energy (∆KE)
of the cart between these values. Record the ∆KE.
5.
Highlight the region of the Force vs. Position graph over which the cart was being pulled along by the
hanging mass. (In other words, remove extraneous data from the integration calculation). Determine
and record the value of the Work done by the Force created by the hanging mass. Include the
appropriate sign and units.
6.
Copy each graph (including the statistics information) into the Word template by using “Paste
Special”. Paste each as if it were a “picture”.
7.
By what % does the Work done by the hanging mass differ from the ∆KE of the cart? (Show your
calculation.)
8.
Recall that the Work – Kinetic Energy Theorem (W = ∆KE) implies that “W” refers to the “total
work” done on the cart, not the work done by any particular individual force. Considering all of the
forces acting on the cart, why is it reasonable to assume that the work done by the hanging mass is the
“total work” done on the cart? Explain what might account for the % difference calculated above.
Activity 2: Work Done by a Non-Constant Force (ex. a spring)
1.
Unhook the hanging mass from the cart and put the string, hanger, and masses away. Carefully hook a
spring to the force probe. Do NOT, at any time during the lab, over-stretch this spring!
2.
Hold the cart at rest in front of the motion detector while carefully stretching the spring a short distance
down the track away from the motion detector. Keep the far end of the spring stationary throughout
the entire collection of data.
Lab #6 – Work-Kinetic Energy Theorem
3.
Press Record and gather data as the cart moves in one direction along the track while being pulled by
the spring. Be sure that the cart is released from rest. Make note of its starting position and ending
position relative to the motion detector.
4.
Note the region of the graphs over which the cart was being pulled by the spring. Remember that the
x-axis of each graph is Position, not Time.
5.
Using the Maximum and Minimum information, determine the ∆KE of the cart over this region.
Record the ∆KE.
6.
Highlight the region of the Force vs. Position graph over which the cart was being pulled along by the
spring. (In other words, remove extraneous data from the integration calculation). Determine and
record the value of the Work done by the spring. Include the appropriate sign and units.
7.
Copy each graph (including the statistics information) into the Word template by using “Paste
Special”. Paste each as if it were a “picture”.
8.
By what % does the work done by the spring differ from the ∆KE of the cart? (Show your calculation.)
9.
Recall that the Work – Kinetic Energy Theorem (W = ∆KE) implies that “W” refers to the “total
work” done on the cart, not the work done by any particular individual force. Why is it reasonable to
assume that the work done by the spring is the “total work” done on the cart? Explain what might
account for the % difference calculated above.
Post Lab #5: The Work – Kinetic Energy Theorem
Name: _______________
Section #: ____________
Information regarding the launch of an F-14A Tomcat off of an aircraft carrier: (Note: units vary)
FUNCTION
Carrier-based multi-role strike fighter
CONTRACTOR
Grumman
UNIT COST
$38 million
MAX. TAKEOFF WEIGHT
72,000 lb
ENGINE
Two Pratt & Whitney TF30-P-414A after-burning
turbofans with 20,900 lb of after-burning thrust (each).
LENGTH OF CATAPULT PULL
300 ft
FORCE EXERTED BY CATAPULT
AIR DENSITY (ρ ) at sea level
EFFECTIVE CROSS SECTIONAL AREA (A)
DRAG COEFFICIENT (C)
400,000 lb
1.1 kg/m3
380.0 m2
0.004344
F-14A Tomcat
Navy photo by Photographer's Mate 1st Class Craig McClure
http://www.chinfo.navy.mil/navpalib/image s/image-cv15.html
Note #1: The magnitude of the thrust force from the engines and the magnitude of the force applied by
the catapult are both constant over the entire displacement.
Note #2: The magnitude of the drag force (air resistance) is not constant. It depends upon the velocity.
The equation is given by:
F
drag
= 1 2 CρAv 2
Note #3: Analysis of a launch revealed that the velocity (measured in m/s) of the aircraft
approximately depends upon the position (measured in meters) of the airplane along the deck
according to the following equation:
v = 9.87 x
Lab #6 – Work-Kinetic Energy Theorem
1.
Draw a force diagram (free body diagram) of the F-14A Tomcat as it is being catapulted along the
deck of the ship. Label all forces.
2.
From the information above, create an equation for the Net Force as a function of Position.
Thoroughly explain your calculations. Make a graph of this function.
3.
From the information above, determine the change in Kinetic Energy of the F-14A Tomcat:
(1) … directly from the graph
(2) … by evaluating the integral:
∫ F
xf
xi
net
( x) dx
Thoroughly explain each of your calculations.
4.
Calculate the final speed of the F-14A Tomcat at the end of the deck as it begins its flight. State your
answer in units of both [m/s] and [mi/hr].
Lab #5 – Work-Kinetic Energy Theorem