The Evergreen Gravito-Adobe Super Collider
Introduction: In
fundamental physics, one pictures the world as composed of elementary particles
of matter (quarks, electrons, neutrinos and the like) acting on each other
through four kinds of influence (gravity, electromagnetism, weak interaction,
and strong interaction)[1].
One things these interactions do is transfer momentum from one particle of
matter to another. The Moore physics text discusses this concept at length, and
from it you have heard that momentum is a vector associated with each isolated
particle or group of particles, and specifically: (a) the vector points in the
direction of the particle’s motion, and (b) the magnitude of the vector is
equal to the product of the particle’s speed with its mass. A standard
shorthand is
p = m v.
This lab investigates
the transfer of momentum. It has two portions, involving you in two different
kinds of experiment:
(1)
investigating the momentum acquired by a pendulum as a function of its mass and
the height from which it is released to swing; this is an inductive experiment, i.e. one which looks for a pattern in
nature’s behavior;
and
(2) investigating the transfer of momentum when two pendulum bobs travelling in
different directions collide and stick together; this is a deductive experiment—it reasons from a theoretical claim (that the
momentum of isolated systems stays constant) to make predictions about the
direction of the combined motion after the collision—and the lab is to compare
these predictions with observations.
Preparation: Read
Moore C2 and C3, as well as all of this lab description. Make sure to read the
footnotes, too.
Part I of the lab uses
a computer-linked photogate to measure speed. There are only enough set-ups for
about half the program at a time, so half will need to start with Part II.
Either Part can be done first.
The instructions in
this lab leave you to work out details of the procedure. It is a good idea for
you and your partner to do a full “dry run” of each Part before taking data,
and looking for improvements in procedure that make things more convenient or
reliable.
Part I: Transfer of Momentum Into a Pendulum
The basic idea is
simple: a ball of clay, swinging on the end of a string, gains momentum from
gravity when it is released from a position away from its rest position at the
bottom. You can find out the magnitude of the ball’s momentum by measuring its
mass and speed. Your task is to look for simple patterns (if any) in the way
momentum at the bottom of the swing (where the ball is moving fastest) is a
function of the ball’s mass and of the height at which it is released.
Equipment: § ceiling
hook
§
nylon string or equivalent
§
modeling clay or equivalent
§
screw eyes, plain screws, or equivalent, to serve as anchors in the bobs for
attaching string
§
balance
§
photogate, data interface, and computer
§
ring stand and clamp
§
meter stick or equivalent
§
calipers
§
protractor or equivalent
§
straws
§
scissors
Theory: The pendulum
will swing so that a small segment of straw, protruding from the bottom of the
bob, will briefly interrupt a beam of light in the photogate. The photogate
measures the time the beam is interrupted; from the measured width of the straw
and the definition of speed, one can calculate what speed of the bob
corresponds to the measured time.
Procedure: (A) The
photogate, interface, and computer should already be hooked up; log onto the
computer, start the Vernier software (Logger Pro, inside the Data Acquisition
entry of Programs in the Start menu), and follow the instructions (Open
File>Probes & Sensors>Photogates) to open the Vernier experiment
template called One Gate Timer. This should open with two graph windows and a
data table showing. Click the Collect button on the computer screen and check
that the photogate is working by interrupting the light beam briefly with a
finger. Figures should appear in the data table. (Note that the Velocity column
does not give true results until you have entered the right width for your
straw, which you do by measuring the straw then select the Velocity column and under
the ‘Column Definitions” tab manually enter your value for length in the column
equation (originally 0.05/”gate time”; change the 0.05 to your value of straw
length.)
Before proceeding,
make sure you understand the Collect/Stop button (not hard, but essential) and
also that you can clear data from the table.
Use modeling clay and
an anchor to make a roughly spherical pendulum bob about 300 g in mass. (It is
a good idea from this point on to have different people handling clay and using
the computer: the computer isn’t made to handle clay between the keys.) Measure
the bob’s mass. Suspend it from the ceiling hook and arrange the photogate in
such a way that a small segment of plastic straw stuck in the bottom will pass
through the photogate at the bottom of the pendulum’s swing. Swing the bob back
and up to a known height above the lowest point [[10 centimeters or so]], and
release.[2]
Use the time recorded by the photogate, and the basic definition of speed, to
calculate the speed of the pendulum bob at the bottom of its swing. Be sure to
do this by hand with at least the first several data points, rather than
trusting the computer. If the numbers agree, use the computer’s thereafter, if
you like. Record this data by hand in your notebook.[3]
Repeat for several release heights, well spaced between 5 and 25 cm, and for
several different masses between 100 and 500 g.
Analysis for Part I:
The goal is to find simple functions, if possible, that give the way pendulum
speed depends on pendulum mass for a given height, and the way pendulum speed
depends on release height, for any given mass. First, “eyeball” the data to see
if any patterns are quickly apparent; then use your calculator’s “power law”
fitting routine (called PwrReg on the TI-83) to find the parameters of a
function in the form axb that approximates your data.
Calculate the momentum
of the bob at the bottom of its swing for each of your combinations of mass and
release height.[4]
Part II: Momentum Transfer in Collisions
Additional Equipment:
§
Velcro tape, both kinds
§
large plain paper sheets
Theory: Two pendulums
will collide at the bottom of their swing and stick together. During the time
of the collision, they are essentially isolated from outside influences (in the
sense that the collision interaction is much stronger than anything else, like
small air currents that may be acting on them at that time). Therefore, their
total momentum vector (the vector sum of their two individual momenta) does not
change in the collision, and the resulting combined object has the same
momentum, in magnitude and in direction. This experiment measures the direction
of the momentum after the collision, by finding the line of swing of the
combined mass, and compares it with the direction predicted by conservation of
momentum. You make the prediction by finding the vector sum of the two
individual momenta, as indicated in the diagram:
To define the two
vectors, you need a coordinate system in which each direction can be specified.
You also need the mass and speed of each bob. You can measure the mass
directly, and you can determine the speed from your results in Part I. (As you
read through the procedure, you will see that you can record all the data you
need for this part before you do Part I; you just can’t do all the calculations
until Part I is complete.)
(B) You do not need
the photogate/interface/computer rig for this part of the lab. Make a second
bob, to be suspended from the same support as the first. Measure its mass. The two
pendulums are to be released from different locations so as to collide at the
bottom, stick together, and move off at some new direction. You will measure
the direction for a series of different mass combinations, and compare with the
predictions of momentum conservation. Wrap the Velcro tape around each of the
two bobs, so they will reliably stick together when they collide. Devise a way
of recording the directions of the two swings before the collision and also of
the direction the combined mass moves afterward. (Suggestion: the analysis is
simplest when the two bobs come in along perpendicular lines.)
After practicing so
that you can reliably get the two bobs to collide, record the initial
conditions (release heights and directions) and the results of a series of
collisions of bobs of different masses.
Analysis for Part II:
Notice that your results from part (A) will allow you to know the magnitude of
the momentum of either of your bobs, provided you measure their release heights
and masses. Since the two bobs are moving in different directions here, their
momentum vectors are different even
if their masses and release heights are the same. Conservation of momentum
predicts that the total momentum of the two bobs, i.e. the vector sum of their
momenta, does not change in the collision. And since there is only one body
after the collision, the stuck-together lump of clay, it must move in the
direction of that total momentum vector, if conservation of momentum is right.
Choose a coordinate
system; calculate the momentum vector for each bob; find their vector sum,
specifically its x and y components, as outlined in the Theory section. Compare with your observations.
Optional Dessert Course
The Logger Pro
software allows a number of graphing options and has a built-in curve-fitting
routine. If you like, explore these and other features of the interface.
Clean-up: Return clay to the supply table and make sure your desktop is
clean.
Write-up: Follow the format of previous labs. Write out a typical example
of each calculation, but there is no need to show additional calculations in
detail when they repeat a typical example with different numbers.
[1] “Influence” is a purposely general word (not to
say vague), chosen to give one a way of referring to all the different effects
that the four basic interactions can have.
[2] Since
the bob is a blob, not a mathematical point, you have to decide what feature of
it to measure the height of. Note also that it is the height above the bottom
of the swing, and not the height above the floor, desktop, or sea level, that
counts.
[3] You
will also be able to save this data in a file, but it is important that the raw
data appear in your notebook, and we strongly advise direct recording until you
become thoroughly familiar with data saving in this lab environment. Lost
electronic data is nearly impossible to recover.
[4] Note
that you can use your data directly for this step; you don’t need to use your
fitted power function.