Download Prelab Homework - University of Rochester

Acceleration of Gravity
Purpose
The purpose of this lab is to measure the constant acceleration g due to gravity
near the earth's surface by two different methods. The value of g at the University of
Rochester is 9.8039 m/s2. You will do this by performing two different experiments:
1. Experiment 2A - Freely Falling Body (room B267)
2. Experiment 2B - The Atwood Machine (room B266)
Introduction
The acceleration of an object due to the earth's gravitation is,
a
GM e
r
2
(2.1)
where G is Newton's gravitational constant, M e is the mass of the earth and r is the
distance to the center of the earth. Inasmuch as the radius of the earth is very large
compared to laboratory dimensions, an object near the surface of the earth, r is
approximately constant and equal to the radius of the earth, R e . Thus an object near
the surface of the earth will experience a nearly constant acceleration,
g
GM e
R e2
(2.2)
(There are instruments which are sensitive enough to measure the difference in g due
to the difference in Re of a few meteres. The experiments that you are going to do in
this laboratory, needless to say, are not that sensitive).
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Constant Acceleration
You will consider the effects of constant acceleration g . It is useful to recall the result
due to any constant acceleration a . If the body, initially at rest, is subject to a constant
acceleration a , then its subsequent displacement s, is given by,
s
1 2
at
2
(2.3)
where t is the length of time over which the acceleration has been acting.
By
measuring the displacement as a function of time elapsed after starting from rest, the
acceleration can be determined.
In Experiment 2A, Freely Falling Body, the acceleration a is g and is determined
by measurement of s and t.. In Experiment 2B, The Atwood Machine, the acceleration of
gravity is “slowed down” so that one measures a smaller acceleration a (presumably
more accurate). This acceleration is related to g by the equation given below (2.4).
Prelab Homework
The prelab homework must be done at home and handed to the lab TA before you
start the lab.
Questions
1. a) Using the known values of the mass of the earth and the radius of the earth.
Calculate the value of g.
b) What is the change in g( g ) if the measuring apparatus is raised in height by 1
meter?
2. Derive equation 2.3 for the distance that a body moves under an acceleration a. .
3. For Atwood's machine, derive the formula,
a
(M1  M2 )
g
(M1  M2 )
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(2.4)
BALL RELEASE MECHANISM
dowel
pin
press dowel pin here
thumbscrew
timer
D
target pad
Figure 2.1
where M1 and M2 are the two masses, a is the acceleration of the masses and g is the
acceleration due to gravity.
Part 2A: Freely Falling Body
While this may seem to be the most obvious method of measuring g, it was not
practical until the advent of modern electronic timers. The acceleration of gravity is
measured by allowing a steel ball to fall, after starting at rest, and then applying the
equation for accelerated motion. A free fall timer is used in this experiment.
Check List
1. The ball release mechanism should be clamped to a stand.
2. Position the target pad directly under the ball release mechanism.
3. Place the timer so that it will not be hit by a falling ball.
4. Now make a test run by clamping a ball into the release mechanism. Close the
release mechanism by pushing on the dowel pin as shown, and then tightening the
thumbscrew to lock the ball in place. Turn the timer on and press reset. Finally,
turn the thumbscrew counter-clockwise to release the ball which should then drop
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on the middle of the target pad.
The ball falls a distance
measured from the bottom of
the ball to the top of the plate.
pulley
table
top
mass 1
5. Remember to hit reset between
each measurement.
Procedure for Experiment 2A
For five different arbitrary
S
distances (measured from the
bottom of the ball in the ball
release to the top of the pad)
measure five free fall times.
Measure the distance each time
also(i.e. you make pairs of
measurements of t and h.).
mass 2
Figure 2.2
Part 2B: Atwood's Machine
Atwood's machine is shown in the diagram Figure 2.2. The arrangement of the
weights minimizes the effects of random errors so this is a fairly accurate method for
measuring g. Since this experiment can be done without the use of electronic timers, it
was one of the original ways to measure g accurately. (Had accurate timers been
available it is doubtful that this method would have found favor. So, this method is
mainly of historic interest).
Ideally, the pulley would be firmly supported and frictionless. The cord connecting
the two masses would be unstretchable and massless. To the extent that the cord
does not stretch, the motion of the system is determined by the motion of either weight
alone.(i.e. The length of the cord is constant.) The acceleration due to gravity can be
found from the formula,
g
M1  M2 
a
M1  M2 
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(2.5)
where M1 and M2 are the masses M1  M 2 , and a is the acceleration of the masses.
2
Because g is constant, we see that a must be constant. The formula, s  12 at ,
can be applied to find a, and thus we can calculate g.
The experiment works best when the sum of the masses is large when compared to
the difference between the masses. As is shown in the diagram, start off with the
heavier mass (labeled M1) above the lighter mass. Practice letting M1 fall a few times
before starting to take real measurements.
Procedure for Experiment 2B
1) For one combination of M1 and M2, use a meterstick and a stopwatch to make four
measurements of the time it takes for a weight starting at rest to traverse a distance
s( again measure s everytime you measure t ), Keep in mind that the difference in
masses should be kept small (no more than 80g) while keeping the sum of the two
masses large (400g to 500g). It is best if the same person who starts and stops the
timer also drops the mass. The other person should measure the distance. You
should have a table of t and s pairs. Also record M1 and M2 .
2) Now change M1 and M2 and
repeat the above procedure three more times,
giving you a total of four sets of measurements. Keep the sum of the two masses
constant, while changing the difference between the two masses by moving weights
from one to the other.
Data Analysis for Experiment 2A
1) Calculate the average time t and the uncertainty t (i.e. the standard deviation of t
). Also calculate s and s . Do this for each sample of five measurements.
2) Plot s including error bars, s , (i.e. the standard deviation) on the y-axis versus
the square of t including the error bars on the x-axis. (How is the standard deviation
of the square oft related to the standard deviation of t , (See Appendix, Error
Analysis). Using a ruler, find and draw the “best” straight line through the points.
The slope is g/2.
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3) Now calculate g from the data for each of the four distances and find average g ± ∆g.
Compare this average and error to the value from the slope and to the accepted
value of g.
Questions for Experiment 2A
1) By how many standard deviations do your answers differ from the accepted value?
2) Does the plot intercept the origin, and if not, what are possible explanations for this?
Data Analysis for Experiment 2B
1)
2)
For each of the four sets (i.e. different M1 , M2 ) compute s , s and t , t .
From the data compute a and a for each of the four data sets. (See Appendix on
Propagation of Errors.)
3) Plot a vs. (M1 - M2)/(M1 + M2). Show the uncertainty in a but you may assume
that there is no uncertainty in M1 or M2 . Fit a line through the points making sure
the fit line passes close to the error bars of all the points. Although the line should
pass through the origin, do not force it to. The slope of the "best" line through these
points is g .
Questions for Experiment 2B
1) How does the value of g from the slope compare to the accepted value?
2) Are there any systematic errors to be considered(Hint: friction in pulley)?
3) Does your "best" straight line go through the origin?
indicated?
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If not, what errors are
For the Lab Technician and TA
Equipment
Part 2A (room B167)
ball release mechanism(see Figure 2.1)
timer
target pad
graph paper
ruler
calculator
Part 2B(room B266)
Atwood machine(see Figure 2.2)
meter stick
stop watch
scales
For the Lab TA
Divide the class into two groups. Start one group on experiment A ; and the other group
on experiment B. ‘ Halfway’ into the session the two groups will switch experiments.
For the Person giving the Laboratory Lecture
1. Complete discussion of Appendix . Systematic errors, combination of errors
and
using the error table in Appendix
2. Describe Lab 2.
3. Error formulas needed for Lab 2.
4. Explain prelabs for Lab 2.
5. Explain about writing a lab report.
a) Refer students to model write-up in the beginning of the lab manual.
b) Stress doing calculations in either m.k.s. or C.G.S. units, preferably the
former.
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To; The student doing this experiment: Acceleration of Gravity (Lab 2; P121)
From: David Douglass
B&L 156
e-mail: [email protected]
I have been involved in the improvement of the undergraduate Physics 100-level
laboratories. This experiment is one of 15 total. Your input is important and essential.
I would appreciate any comments that you wish to make. Simply write them below,
remove this page, fold, and drop into any university mailbox. I will respond
individually to all signed comments.
DAVID DOUGLASS
B&L 156
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kj/dhd
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