Purpose: In this activity you will investigate how
hanging mass, length, the strength of gravity and starting angle affect the
period of a pendulum. Then, form a math model for the period of a pendulum.
Materials:
Pendulum apparatus Stop watch or other timing device/technique
Pendulum apparatus Stop watch or other timing device/technique
Meter
stick Various masses for pendulum bob
Balance Protractor or other means
of measuring angle
Logger Pro or Tracker Electronic force sensor Video camera or phone
Logger Pro or Tracker Electronic force sensor Video camera or phone
Background:
A
pendulum is a classic example of periodic motion, which is a motion that
is repetitive, redundant, keeps repeating itself, keeps going back and
forth…oh, I’ll stop now. A pendulum only
works when there is gravity, and of course can be used as a clock. It is also an example of circular motion,
which means there is a net centripetal force, mv^{2}/R. Because we began studying circular motion and
will be getting deeper into gravity, we will try to figure out the basic
properties of a pendulum and see how we can combine gravity and circular motion
together in a few different ways, whether it is a pendulum, an amusement park
ride, or satellite motion. Keep the
force diagram for a pendulum in mind (what does it look like?).
Research
Questions:
What effect(s), if any, do mass of the bob, length of the
pendulum, and angular amplitude of a pendulum, have on the period of the
pendulum? Our goal is to use mathematical fits to data to determine an empirical formula for the period of a pendulum.
Your group is looking to combine
your graphical fits into the form T = f(L)g(angle)h(m)s(g).
Develop Hypothesis:
Before doing any measurements, state what you think the
effect of mass, angular amplitude, length, and the acceleration of gravity have
on the period of the pendulum. Remember,
reality always tends to be a bit complex!
Procedure:
You
will be trying to determine the best fits to data using Excel. Your measurements will use four different
techniques for collection.
i.
Use a stop watch to measure the period as
a function of length of the pendulum;
ii.
Use video to determine the period as a
function of mass hanging from the pendulum. NOTE: when changing masses, you
will need to adjust the length of the string accordingly to maintain the same
total length to the center of mass;
iii.
Use an electronic force sensor to measure
the period as a function of angular amplitude;
iv.
Use the PhET simulation for pendulums to
determine the effect of gravity on the period – check out the blog post which provides the links and information
for this portion.
You will need to
get data to make four graphs:
i.
Period as function of length. Note the constant
mass and angle;
ii.
Period as function of mass. Note the constant
length and angle;
iii.
Period as function of starting angle. Note the
constant mass and length;
iv.
Period as function of g (using PhET simulation,
keep constant mass and length and angle).
Keep in mind, you should always
be thinking of estimating errors on all measurements (in any experiment you
ever do!) as well as how to minimize those errors. Think about what might be the best way of
measuring a given quantity. Also, do not
forget to include units on all measurements, along with a reasonable estimate
of the uncertainty of all measurements!
For anything measured in trials, think standard deviation.
For
your writeup: Purpose, Materials, Procedure, Data Tables, Questions
and Analysis (with appropriate graphs; these should be titled and labeled with
quantities being graphed and units!
Graphs need to be done on the computer.). Do a group report, and consider Google Docs if that works best for the
group.
Definitions:
Period
= time it takes the pendulum, when released from rest, to swing over and back
to where it started; or the time for one roundtrip.
Length
= length from the point where the string is free to swing to the center of mass
of the bob at the end of the string.
Mass: we assume massless string, so just
the mass of the hanging bob.
Angular amplitude: the starting, maximum
angle for the period, as measured from the vertical.
Questions/Analysis:
1. For
the graph of period as a function of length, figure out errors on
measurements and include them as error bars on your graph. The error bar for period measurements (some
number of trials with the stop watch) should be the standard deviation, as
always. Find the bestfit function for the graph in Excel. Include the
equation and R^{2} value.
Include the point (0.01,0.01) on your graph and in your fit, since (0,0)
won’t allow certain fit options.
2.
For the graph of period vs. mass, use five
different masses and use a video to get period measurements in Logger Pro or
Tracker. Make sure to adjust the
length of string so the overall lengths of the pendulum are always the
same. For instance, when you hang a
larger mass on the string, you will need to shorten the string a bit since the
object will be longer (think about where the center of mass is). Try to estimate errors in length and in
period from your video measurements in LoggerPro or Tracker. Find
a bestfit function for the data. Include
the equation and R^{2} value.
3. For
the graph of period vs. angle, use the electronic force sensor and measure the
period in the Logger Pro display. Plot
your data using Excel and obtain
a bestfit function to the data to get an idea of the relationship. Go from small angles around 10^{o},
and go up to around 90^{o}.
4.
Last but not least, graph period vs. g from
the PhET simulation. Find a bestfit function for the data. Include the equation and R^{2}
value.
Keep in mind you will only have 3 points, so do what you can with
this.
5. There
are only three significant forces acting on the mass at any time, those being
tension, gravity and air friction. Which
of these are constant, which are nonconstant?
If any are nonconstant, when are they the strongest, and when are they
the weakest? Explain, and include a
force diagram/freebody diagram for a pendulum.
6. Using
your graph in #1, how long should a pendulum be in order to have a period of
1.0 sec (on earth)? How long should a
pendulum be to have a period of 2 seconds? Use your fit to do so. Keep in mind the first mechanical clocks were
made with the pendulum! (way to go Galileo)
7. What
exactly does F_{centripetal} = mv^{2}/R mean? What does the value of mv^{2}/R tell
us, and what does
it
depend on specifically for a pendulum? Think
about what force or forces keep the pendulum
moving in that portion of a circle.
Consult our notes and the book for assistance.
Write a brief,
tothepoint (i.e. one paragraph) summary of your findings/conclusions about what the period of the pendulum depends
on and how it depends on specific
quantities.
*Use and
combine the results of your individual bestfit functions for your four graphs
to obtain an empirical mathematical model for the period of your pendulum as functions of
length, mass, g, and angular amplitude.*
Put it in the form: T = f(L)g(angle)h(m)s(g)
Assess the techniques. Based on your experience:

Rank the ease of the four measuring techniques.

Rank the accuracy of the four measuring
techniques.
Provide any other feedback you think is relevant to this
set of experiments!
atico export are the leading manufacturer and supplier of physics lab equipment science lab equipment in india. modern equipment
ReplyDeletenice blog !! i was looking for blogs related of Physics lab equipment . then i found this blog, this is really nice and interested to read. thanks to author for sharing this type of information.
ReplyDelete