042 Laboratory Four: Marble mass, velocity, and momentum
Introduction
This laboratory explores the concepts of momentum
Questions
Do marbles in equal marbles out?
Does velocity in seem related to velocity out?
Is momentum in related to momentum out?
Existing theory asserts that momentum is conserved. At first you will qualitatively explore the conservation of momentum. Then you will make some quantitative measurements of momentum for marbles inbound and outbound from a collision.
In physics:
Momentum is the mass (grams) multiplied by the velocity (cm/s). The letter P is used for momemtum, m is used for mass, and v is used for velocity (speed).
The arrows above the P (momentum) and the v (velocity, also known in this class as speed) mean that these are quantities with an amount and a direction. If the amount of velocity changes, then the momentum changes. If the direction changes, then the momentum also changes. Both affect the momentum. In this experiment the intent to keep the marbles moving in the same direction, so the arrows will be dropped from the formulas.
Conservation means "stays the same." Usually this means, "the momentum after an event is the same as the momentum before an event." For this lab the "event" is a collision between marbles.
Equipment
marbles
rulers
meter sticks
stopwatches, two per group
banana leaf guide tracks
I. Qualitative exploration
Five marbles sit touching each other on the flat portion of a marble track. The marble track is made using rulers. One or more marbles are released from an elevated end of the track and allowed to collide with the five stationary marbles.
Release one marble into the group of five. How many marbles are ejected?
Release two marbles into the group of five. How many marbles are ejected from the group?
Repeat for three, four, five... marbles.
How is the number of marbles in related to the number of marbles out?
If zero marbles are sent in, how many marbles come out?
Marbles in
Marbles out
As you work on the above questions, experiment. Play with the marbles. How to the marbles know what to do? How does a marble know whether to go or to stay? How do the marbles count? Just how smart is a marble? Play gently - marbles can and do break - but do play.
II. Quantitative explorations of a single marble inbound
Said "mathematically," the momentum before is equal to the sum of the momentums after is written:
Pbefore = Pafter minbound × vbefore
=
moutbound × vafter
where m is the mass of the marble, v is the speed of the marble.
Find the masses of both the inbound and outbound marbles.
Following guidance in class, measure the distance and time for the inbound and outbound marbles. Use the table to calculate the inbound and outbound momentum. Repeat for different inbound speeds.
After obtaining five pairs of momentum in and momentum out, prepare a graph of the data. Consider whether zero cm/s in is also zero cm/s out and whether this is a possible data point.
Data tables
mass inbound marble
×
velocity inbound
=
momentum in [x]
mass outbound marble
×
velocity out
=
momentum out [y]
mass m (g)
×
distance (cm)
÷
time (s)
=
momentum (g cm/s)
mass (g)
×
distance (cm)
÷
time (s)
=
momentum (g cm/s)
×
÷
=
×
÷
=
×
÷
=
×
÷
=
×
÷
=
×
÷
=
×
÷
=
×
÷
=
×
÷
=
×
÷
=
Transfer the momentum results to the following table. The following is the data to be graphed.
momentum in p (g cm/s) [x]
momentum out p (g cm/s) [y]
0
0
Graph
Make an xy scattergraph with the momentum in on the x-axis, momentum out on the y-axis
Analysis [a]
Was momentum conserved?
What was the slope?
Based on your slope, was momentum lost or gained?
[Notes from the
field for instructors: In this laboratory we explore conservation of
linear momentum. Another momentum that is conserved is angular
momentum. Angular momentum is the momentum of spinning. Spinning
objects tend to continue to spin. Objects that are not spinning tend
to remain at rest– to not spin. Think of a child's toy top.
In the experiments above we considered only linear momentum, but the
marbles are spinning as they move on the track. In part two a
spinning m1 duck hits a non-spinning m2 duck. The m1 duck loses
speed and thus spin, the m2 duck goes from not spinning (sitting
still on the track) to spinning very quickly. These changes in spin
momentum are related to why linear momentum is consistently "lost"
in these collisions.
Where linear
momentum is p = mv, the angular momentum L = Iω where I =
0.4mr² and ω = v/r. Thus the angular momentum of a marble
is L = 0.4mrv. One cannot just add all the momentums and hope for
the best: the units are different. Ultimately one has to retreat to
an energy position noting that the potential energy must appear as
both linear and rotational kinetic energy in both of the marbles
post-collision, along with losses to friction, sound, and any heat
produced in the collision.
The thought
occurred as to what to try to reduce the impact of external torque
exerted by the track. One idea was to lubricate the ruler track with some form of
greaseless lubricant such as WD-40®.
WD40 was tried. The first complication is the tape no longer holds the tracks in place.
This problem proved rather insurmountable. In addition, WD40 wound up everywhere -
on hands, table tops, soaked into paper that slid into the WD40. Would need a
greaseless lubricant. Even, the loss of taping ability would remain fatal.
Why not simply use pucks on an air table? Two key reasons. The puck and air table
are unfamiliar to students - this raises the probability that the students will,
in their own minds, see the whole thing as magic. Another mysterious thing in the
modern world. Secondly, the lab should be as reproducible as possible by any teacher
in the nation. Part one requires nothing more than what an instructor on an atoll might
be able to get their hands on. Part two adds only one unlikely element - a stop watch.