The Block and Pendulum∗
Objectives
This lab has three purposes. (1) To give you experience in estimating the experimental
uncertainty on measured values. (2) To see how an error in a measurement can influence
a final result. (3) To investigate the interpretation of a discrepancy—a difference that cannot be explained by experimental inaccuracies—between a measured value and a theoretical
prediction.
Instructions
• Read
the
online
notes
on
error
analysis:
Short-Craft.pdf
found
at
http://www.physics.queensu.ca/∼phys106/labs.html
• You should record theory that might be useful while performing the lab. For some
labs this will involve a summary of the Theory section of the lab instructions, while
for others you will have to read specific sections of your text book. The important
thing is to have the relevant formulae recorded and explained for easy reference while
performing the lab.
• Once you have left the laboratory, complete the Worksheet and hand it in one week
after the lab.
Introduction
One of the foundations of physics is that nature responds according to a set of underlying
rules; the goal of physics is to discover those rules. Unfortunately, the connection between
∗
An appropriate name for a pub perhaps?
2
the assumed form of the underlying laws and the physical phenomena that result is often
complicated and involves a number of further assumptions. We will call this combination of
fundamental laws and any approximations or simplifications necessary to apply them to a
particular situation The Theory.
The goal of experimental physics is to make measurements on systems that challenge the
validity of particular theories and thus point us in the direction of improved theories. It has
been said that “the further a measurement is from theory, the closer it is to a Nobel Prize.”
We call the difference between a measured result, including any experimental uncertainty,
and a theoretical prediction a discrepancy.
Experiments are always performed with the goal of reducing experimental uncertainty, often
with the intention of observing a discrepancy. However, before challenging the validity of a
theory, an experimentalist must be confident that any apparent discrepancy is indeed real
and not the result of experimental inaccuracies.
Once satisfied that a real discrepancy has been observed, there are essentially three possible
explanations for the failure of the theory. Either:
1. Another physical effect played a role, but was not accounted for in the theory.
2. Mathematical approximations made in applying the theory are not valid.
3. The physical law is incorrect.
Obviously the careful physicist considers reasons 1 and 2 thoroughly before speculating that
a violation of a well tested and universally accepted law has been observed.
Theory
In part A (The Block ) you will need the formula for mass density, ρ:
ρ=
M
V
(1)
3
where M is the mass and V is the volume. You should also record the densities of a few
common materials for comparison, such as wood, air, and lead.
In part B (The Pendulum) you will be measuring the period of oscillation for a simple
pendulum and comparing it with the standard formula,
s
T = 2π
L
,
g
(2)
where T is the time in seconds for the pendulum to complete one complete cycle (called the
period), L is the length of the string in metres (measured to the centre of the weight), and
g is the acceleration due to gravity in m/s2 . If you want to know the precise value of g for
Stirling Hall you can find it on the plaque describing the Foucault pendulum, which is in
the middle of the building.
Although you are not responsible for deriving this result (interested readers are referred
to the text) it is important to appreciate that this represents an approximate solution.
Firstly, it does not take into account the effect of air resistance, an effect that becomes
more important as the speed of an object through the air increases. Secondly, to derive this
simple result from the basic equations governing the motion of the pendulum we must make
the mathematical simplification that sin θ ≈ θ where θ is measured in radians (recall that
2π rad = 360◦ ).
simple
pendulum
θ
L
4
Procedure
PART A: THE BLOCK
This part of the lab will be in the form of a competition. Each group has a different
wooden block in which a random amount of lead shot has been added so that each block
has a different average density. Your task is to make the appropriate measurements on your
block to determine its density with an estimate of the experimental uncertainty. You will
be using intentionally crude instruments for this so you can expect to have rather large
uncertainties. The “true” values are known (these have been measured with much more
accurate instruments and, for our purposes, will be considered to be perfectly accurate) and
the winner of the competition will be the group whose value has the smallest experimental
uncertainty and is in agreement with the true value. It is important therefore that each
measurement be made as carefully as possible and that uncertainty estimates be as small as
is justified. But, beware of being overly optimistic because this could lead to the true result
lying outside the range of your measured value.
Follow the instructions below, and fill in the appropriate the values on the attached worksheet.
(a) Use the centimetre ruler to measure the lengths of the three sides of your block. State
your results in metres, the S.I. base unit for length. For each measurement assign an associated estimate of the uncertainty. Compute V and ∆V .
(b) Use a balance to measure the mass of your block (in kilograms), again record the result
on the worksheet.
(c) Calculate the density, ρ, and ∆ρ.
(d) When you are satisfied that you have the best possible estimate of the density, ask your
T.A. to record your answer. Once all groups are finished, a winner will be announced. How
did you do?
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PART B: THE PENDULUM
(a) Suspend the metal ball from about 1 m of string and fasten the other end securely to
the clamp. Measure the distance from the centre of the ball to the point of attachment of
the string (do not forget to estimate the uncertainty). Compute the expected period T from
equation 1.
(b) Pull the ball back about 10◦ from the vertical and measure the time for 10 complete
oscillations. Record the measured period for this pendulum.
(c) Repeat step (b) for initial angles of about 45◦ and 75◦ .
(d) Repeat steps (b) and (c) using next the wooden ball and the cork ball.
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Name:
Partners Name:
Student ID:
Date:
WORKSHEET: To be handed in:
Block
VALUE UNCERTAINTY
L1
L2
L3
V
M
ρ
1. How does your result compare to the quoted densities for common materials?
2. Do a sample calculation for ρ and ∆ρ.
Pendulum
3. To investigate the validity of the approximation sin θ = θ, create a table in your lab
book giving θ (degrees), θ (radians), and sin θ for values of θ of 1◦ , 5◦ , 10◦ , 25◦ , 50◦ ,
and 75◦ . What does this tell you about the validity of our approximation?
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Metal Ball
VALUE UNCERTAINTY
L
Expected T
Measured T10◦
Measured T45◦
Measured T75◦
4. How does your measured value for the pendulum compare to your predicted value for
T10◦ , T45◦ , and T75◦ ?
5. Do you notice a trend as you increase the angle? Is this what you expected?
6. Answer questions 4 and 5 for the Cork Ball.
Cork Ball
VALUE UNCERTAINTY
L
Expected T
Measured T10◦
Measured T45◦
Measured T75◦
7. Answer questions 4 and 5 for the Wooden Ball.
Wood Ball
VALUE UNCERTAINTY
L
Expected T
Measured T10◦
Measured T45◦
Measured T75◦
8
8. Do you notice any qualitative differences in the motion using the different balls? Can
you account for these?
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Error Worksheet
Let A=(10.8 ± 0.2) cm
B=(3.75 ± 0.08) cm
C=8.3 cm ± 5%
θ = (67± 2)◦
1) What are the absolute uncertainties on A, B and C?
2) What are the relative uncertainties on A, B, and C?
3) Calculate
(a) A + C
(b) B − A
(c) A × B
(d) B ÷ C
(e) C2
√
(f) 4π C
(g) ln(B)
(h) Asin θ
(express all results with uncertainties both in absolute and relative (percentage) form)