Download Mechanical Energy and the Pendulum

name____________________ period _______ lab partners___________________________________
The Mechanical Energy of a Pendulum
Purpose:
To investigate the extent to which mechanical energy is conserved in a simple pendulum.
Background:
Imagine a simple pendulum--a mass m attached to one end of a string of negligible mass with the other
end suspended from some high point--being pulled over to one side and then released. As it swings
back and forth as in Figure #1, there is a continual changing of gravitational potential energy PEgrav
into kinetic energy KE, back to PEgrav, back to KE, etc. Careful inspection of the spacing of the images
in the figure reveals that the pendulum is moving fastest at the bottom of its swing--so KE is greatest-and slowest (zero for a moment) at the end of its swing--where PEgrav is greatest. Once we pick a
vertical reference level where y=0--for us it will be at the bottom of the swing--we can compute
gravitational potential energy from PEgrav = mgy. Of course kinetic energy is found from KE = ½mv2.
The sum of potential energy and kinetic energy is the mechanical energy Emech --a quantity we expect to
be conserved unless non-conservative forces act to do work on our system. That is,
PEgrav + KE = Emech = constant
equation 1
In this lab we shall investigate the extent to which equation 1 holds for two different simple pendulums
The first--the focus of Part I-- will be a real 40 gram wood sphere of radius around 4.0 cm suspended by
a string equal to one meter in length. The second--in Part II--will be a computer simulated pendulum.
Initially, the only forces acting on this pendulum generated by the equations contained in a computer
model will be the vertically downward force of gravity and the force of tension (directed radially upward
toward the fixed point of suspension). Unlike the real pendulum of Part I that moves through the airfilled lab, initially the computer screen pendulum, will move in a vacuum. After studying its behavior,
you will eventually "turn on" air resistance in an attempt to better model your Part I real lab experience.
Nonconservative forces--notably the opposing force of air resistance--rob the oscillating pendulum of
mechanical energy. Loss of kinetic energy is observed as speeds decrease with each new cycle.
Likewise gravitational potential energy declines as evidenced by decreasing vertical amplitudes of each
swing. Mechanical energy is not conserved.
Materials: two photogates, two small ring stands, spherical wooden mass, string, computer data
acquisition system, two meter sticks, tall vertical rod clamped to lab bench and holding arm a metal arm
that is used to hang the pendulum, calipers
Procedure--Part I--Investigation in the Physics Lab .
1) Start Logger Pro 3.5 and load the file Vave and Aave.
2) Look over the setup at your lab station. Note the string with one end attached to the wood sphere that
serves as the pendulum mass, has its other end secured at a high top point where it is suspended from
behind a tightened screw (part of a clamp attached to a large ring stand).
3) Without detaching it, grab the pendulum ball and use calipers to measure the diameter of this sphere.
4) Let the pendulum ball hang vertically down. Mark this undisturbed position on the lab bench top using
a pencil--this point will mark x=0. (Use meter stick as directed.) Establish the y=0 reference level at the
bottom of the pendulum mass as it hangs vertically down by measuring how far this level is above the lab
bench top.
4) Position the inverted photogates 20 cm apart at x=+10 cm and x = —10 cm as directed by your instructor.
Test them to verify that they work.
5) The pendulum ball will be released by pulling it sideways from its x=0 resting position to x=20 cm.
Check to make sure photogates are positioned so that its back and forth oscillation can be measured.
Conduct "test releases" as needed. Note you'll simply release it--not give it a push.
6) Click start to begin data gathering, release the pendulum ball and gather data for ten complete cycles
(a cycle being one swing over to the other side of x=0 and back again to approximately where it began).
Record data in Data Table I.
Procedure --Part II--Computer Simulation Aided Investigation
1) Go to http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=11 to view a simulated pendulum. Note
the display uses symbols U and K for potential and kinetic energy respectively. Click on the U= mgh
equation to enlarge the graph of potential energy vs. position U(x) and kinetic energy K(x) with position
graphs. Make a sketch of U(x), K(x), and Emech(x) on the same graph. Show this in Data Table IIa.
Click to check "Show" and velocity vector (red) and components of gravitational force will be displayed.
(Note: As you watch these, think about what forces do work on the pendulum)
2) Using Interactive Physics 5.0 (on Physics Desktop folder) open the "ex08-03" file at the computer
station you're using. This is a pendulum simulation that you start by using the mouse to click on and then
drag the big orange pendulum ball over to the desired release point. Click Run and watch the resulting
oscillatory motion. Watch ten complete cycles, then click Stop. (Note: if you want to slow down the
action and study the changing numbers in the data table, click on right arrow key at screen's lower right to
advance the time slowly forward or the left arrow key to go backwards in time.)
3) Think about the questions which follow. Put any notes related to answering them in Data Table IIb.
(Note: you'll get a chance to answer them in the Data Handling section after you have more information) Is
the mechanical energy for these computer screen pendulums of 1) and 2) above conserved? To what extent
do they behave like your pendulum in Part I? Conceivably they could configured it to do so (by adjusting
parameters such as length of pendulum string, mass of pendulum ball, its size, initial release point, etc.)
One important addition--and step toward more realism--would be to take them out of their vacuum (airless)
environment, introduce air and thus air resistance. You can do this with Interactive Physics 5.0. After
clicking Edit, and selecting Edit Mode--by clicking "World" and "Air Resistance," change the selection
here from the default "None" to "Standard." Then use the slide bar (drag with your mouse) to lower the air
resistance to a value where the computer simulated pendulum seems to behave like your Part I real
pendulum. Are you able to get a close match in this regard? Explain.
Data
record diameter of pendulum ball = ______cm= _______m.
Part I -- Data Table I
Cycle #
Time #1 L to R
Time #1 R to L
Time #3 R to L
1
2
3
4
5
6
7
8
9
10
Time #3 L to R
Part II -- Data Table IIa--Make the requested sketch / graph in the space below.
Part II -- Data Table IIb--Put requested notes (relevant to questions in Procedure Part II) below.
Data Handling / Questions
1) Using data from Data Table #1, complete the following (include units, show sample calculation below)
cycle #
Time #1, ave Time #3, ave velocity thru velocity thru KE thru
KE thru
both directions
both directions
photogate #1 photogate #2 photogate #1 photogate #2
initial cycle #1
final cycle #10
sample calculation:
2) Inspect the times in Data Table #1. As you go from cycle #1 to cycle #10, do they show a steady slowing
down of the pendulum ball, consistent with a slow but steady decrease in kinetic energy?
3) Using your results in 1) above, compute much kinetic energy is lost over the ten cycles of the pendulum
oscillation. What % of the initial kinetic energy (present in cycle #1) is lost? (Note: you can assume a
similar % of total mechanical energy is lost.) Show your work.
4) Over the ten cycles of the pendulum's oscillation, how does the work done on the pendulum by
nonconservative forces compare with the kinetic energy loss you computed in 2) above?
5) What % of total mechanical energy is lost in the unmodified computer simulated pendulums of Part II
after ten cycles? Is your graph sketched in Data Table IIa consistent with this? Explain.
6) a) What is the net work done by the gravitational force on the pendulum over a complete cycle-meaning the pendulum is released, swings to the left, then back to the right, and returns to the release point?
b) Unlike gravity, the other obvious force acting on the pendulum--tension--is a non-conservative force.
Why don't we worry about the work it does on the pendulum like we do about work done by other nonconservative forces (like air resistance)?
7) After modifying the Interactive Physics 5.0 pendulum by adding air resistance, were you able to adjust the
amount of this to a value where the computer simulated pendulum behaved like your Part I real pendulum?
Explain.
8) Answer the following question: