LABORATORY EXPERIENCE of PHYSICS 1 COURSE

Description of the Experiments

Alessio Verna, Ph.D

email: alessio.verna@polito.it

1
reference

http://polilabhome.polito.it/en/

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lab rules

Refer ONLY to this presentation for detailed instructions

Lab experience must be done in groups!

Look at LAB PARTECIPANT LIST FILE to see group division and proposed calendar

With your group co-members, choose an experence between pendulum and inclined plane (...or perform
both)

Follow deliverables at the end of each experience

Prepare a ~30 min discussion (supported by a report in .pdf) of the experiments for each group (it is not an
exam! - see schedule on LAB PARTECIPANT LIST FILE ).

If you wish to bring the experience to the exam, re-send your report (after discussion and revisions), by email,
to your co-tutor (look LAB PARTECIPANT LIST FILE to see who is your co-tutor) and we will decide for the
extra points.
(In case, me and the co-tutors will be available for help by email or skype)
(If a part of the group decide to prepare a report, while the other group members are not interested, we will accept
reports prepared and signed by those students who effectively write the report.)
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experiment 1: determination of g with a pendulum

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activity overview
1. Measurement of the pendulum period and the acceleration of gravity with Tracker.
[Single student]
I. Construction of the pendulum, creation of a video of the oscillations,
II. analysis of the video with Tracker,
III. determination of the period through the sinusoidal fit of the diagram x(t).
2. Repeated measures for determining the period of a pendulum and the
acceleration of gravity. [Single student]
I. Pendulum construction, repeated measurement of the period with a stopwatch,
II. determination of the average value and uncertainty on each individual
measurement,
III. calculation of the uncertainty on the average value,
IV.evaluation of the acceleration of gravity and its absolute uncertainty.
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3. Study of the dependence of a pendulum’s period on its length and
determination of the acceleration of gravity. [Collaboration between at least 5
students]

I. Pendulum construction, each component of the collaboration determines the period of

a pendulum with different length,
II.linear fit with the least squares method of the relationship between the squared period
and length,
III.determination of the acceleration of gravity and its absolute uncertainty starting from
the parameters of the fit.

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list of needed material

Thin sewing thread.

1 light mechanical nut or a metal washer ( mass between 5 and 15 grams).

Adhesive tape (preferably paper tape).

(For activity n ° 1) Smartphone or Tablet with video camera.

(For activity n ° 1) Free "Tracker" software (physlets.org/tracker/)

PC with spreadsheet software (Excel, Libre Office, ...).

extensible meter (precision 1mm).

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pendulum construction

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remarks

Build the pendulum so that it is as long as possible (from 40 cm upwards).

When conducting an experiment it is essential to pay attention to the minimization of
relative errors. Values of the quantities we will measure large with respect to the errors
we will make.

Limit to oscillation amplitudes that are less than 1/10 of the pendulum length L.

In the Physics 1 course the simple pendulum is modeled in the approximation of small
oscillations (sinθ ∼ 0), to consider the motion as harmonic and one-dimensional.

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
Place in the background, in the immediate vicinity of
the pendulum (e.g. on one table leg), two notches with
adhesive tape (see figures).
 Measure and note the distance dref between the two
notches. This will be used to inform the software about
distances in the image.
 To minimize the relative error it is good that the dref
occupies a high percentage of the height of the frame
of the movie.

place yourself in a position where the shooting direction
lies on the plane formed by the 2 wires of the
pendulum, in a position from which the two wires
appear superimposed on each other.

Finally, make sure that the amplitude of the swing
occupies a good percentage of the frame.

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
Import video into Tracker (File / Import)

A position reference system (x, y) is defined. In the
image it is represented by the magenta axes.
 The reference length dref is inserted into the
program.

The position of the body is tracked in each frame of
the movie of interest.

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data analysis

Tracker offers both the x(t) graph of the
horizontal component of the position and the
numeric data.

By opening the "Data Tool (Analyze)" window
(View / Data Tool (Analyze)) you have the
possibility to fit the experimental points with a
sinusoid of the type
x(t) = A sin(Bt + C) .
Where A represents the amplitude, B the angular
frequency ω, C the phase of the harmonic
motion.

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
Fit the curve manually by changing the parameters A, B, C by hand

At the end of this operation we determine the value of ω, the parameter B, which
guarantees the best fit.

From the value of ω we find the frequency f, the period T and the estimate of the
absolute value of the acceleration of gravity g:
ω 1 4 π2 L
f= , T = , g= 2
2π f T

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evaluation of uncertainties

To evaluate the absolute uncertainty on g we need the usual calculation techniques of
the error propagation on the formula.
4 π2 L
g= 2
T

The temporal cadence is given by the acquisition frequency of our video camera
(frame-rate or frame per second) which can be seen from the time data column. If for
example, the frame-rate is 30 Hz, the instrumental uncertainty will be equal to δt = 1/30
s = 0.03 s

The uncertainty on L, measured with an extensible meter, can be considered equal to
0.5 cm. This choice guarantees us not to underestimate the errors due to a possible
misalignment of the meter.

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repeated measures for determining the period of a
pendulum and the acceleration of gravity

Measure and note the pendulum length L.

Establish and write down the uncertainty on L (we can assign
the value δL to the uncertainty by 5 mm)

Determine the number of oscillations whose duration will be
measured (we recommend working with 5 complete
oscillations).

Position yourself beside the trajectory at a motion inversion
point. The pendulum there will have zero speed so the
measurement of the duration of the 5 oscillations will be more
accurate.

Better to use a stopwatch or a smartphone app (es:
Timestudy Stopwatch) to record and export data

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
Repeat the measurement of the duration of 5 oscillations 100 times and write down the
100 values T5(i) measured (with i ranging from 1 to N = 100).
 Working on a spreadsheet (OpenOffice, Excel ....) evaluate the average T5.

Evaluate the uncertainty associated with each individual measurement, or the standard
deviation of the 100 measurements

Calculate the uncertainty on the mean
 Create an histogram of T5(i) (that should look like this:)

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
Evaluate the average of the pendulum period T, that is the duration of the single
oscillation (divide by 5)
 Propagate uncertainty from T5 to T

The outcome of the experiment is: T = T ± δT .

Use this formula to calculate g starting from the period T and the length L derived from the
experiment. 4 π2 L
g=
T2

Calculate the uncertainty on the value of g thus obtained with error propagation
techniques.

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 dependence of a pendulum’s period on its length and
determination of the acceleration of gravity

The activity involves the collaboration of several students

By collecting the data of each individual student, the group will have a data set
consisting of at least 5 pairs of values of the pendulum length and period (Li, Ti)
where Li is the pendulum length of the i-th student and Ti the corresponding
period.
 The purpose of this activity is to represent on a graph with the length Li of the
pendulums in the x-axis and the square of the periods, Ti2 in the y-axis.
Then, with the least squares method (using EXCEL), find the parameters a and b
and R2 (goodness of fit) of the line that fits the experimental data
y = ax + b
where y = T2 and x = L.

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
Everything will be done knowing that the 2 4 π2
relationship between T2 and L is a straight line T = L
g
through the origin with slope inversely
proportional to g

check b to have a value close to zero, that is the
interval [b − δb, b + δb] contains zero inside it

We can then estimate g using the slope a with
this relationship:
4 π2
a=
g


Evaluate the error on a

Propagate the error on g

The activity ends by expressing the numerical
value of g obtained with this procedure:
4 π2
g= ±δ g 19
a
deliverables
You report (a .pdf or .ppt... ) should include:

1. g obtained with Tracker, with error analysis

2. g obtained with “stopwatch”, with statistics, error analysis and histogram

3. g obtained by least-squares using method 1. or 2. or both (in group)

4. Pictures, (videos), comments and ideas on how to do better

5. Row data available on request (But don't incude them in the final report!)

only 1 report for group...no problem if all group members don't present both 1 and 2, but just
one of them

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experiment 2: friction determination with a inclined plane

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activity overview
1. Acceleration determination with Tracker. [Single student]
I. You have to realize a video of an object that slides on an inclined plane and analyse
the motion with the free software "Tracker".
II. With Tracker, make a parabolic fit of the data x(t) and obtain the acceleration of
the body, the friction and their absolute uncertainties.
2. Acceleration and dynamic friction coefficient with the least squares. [Single
student]
I. Using data collected with Tracker, fit the experimental data with least-squares
method.
II. Obtain the acceleration of the body and the dynamic friction coefficient from the fit
parameters. Evaluate the uncertainty of the estimated quantities.

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list of needed material

An inclined plane (a bench to be tilted, a formica or wooden board, with uniform
surface)

A body to slide over it

Smartphone with "bubble level" application to measure the angle of inclination

Free "Tracker" software (physlets.org/tracker/)

PC with a spreadsheet (Excel, Libre Office, ...).

extensible meter (sensitivity 1mm).

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construction

Create an inclined plane (by tilting a bench, a
shelf, an axis, ...).

Keep in mind that:
➢ The body mass will play an important role
in determining the friction, light bodies will
have less friction (with the same inclination)
➢ The tilt angle will affect both the
component of gravity acceleration along the
plane and the friction.

DO NOT USE ROTATING OBJECTS!

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remarks


Excessively fast motion will make less precise the tracking of the position in the
various frames of the video.

One possible way to determine the inclination is as follows: increase the angle until
slightly exceeding the inclination which allows the component of the weight force along
the plane to exceed the maximum static friction force. In this way, when the body is
released with zero initial speed, it will begin to slide without the need for external
intervention.
 Measure the length of the inclined plane (dref ) that will allow the software to convert
lengths from pixels to meters.

If possible, choose a place that allows you to have a uniform background, with a color
in contrast with those of the body that will slip.

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
Place the camera at about the midpoint of the body path.

Care must be taken in order to fit the entire inclined plane in the frame, since its length
will be a reference for Tracker.

Before let the body slip along the surface, measure its angle of inclination. For
example with a "bubble level" app. A mobile phone can be placed along the inclined
plane with the two opposite orientations and the average of the measured angles is
evaluated.

Alternatively, the tilt angle can be measured using trigonometric formulas measuring the
length of the plane and the horizontal projection.

Place the body at the top of the inclined plane and leave it free to slide
downwards from a standing position. The descent of the body is filmed. You may need
to get help from a second person.

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
Import video into Tracker (File / Import)

Define a position reference system (x, y). The x-
axis is directed along the inclined plane and the
origin is placed in the initial position of the body.
 The reference length dref is inserted into the
program.

The position of the body is tracked in each frame
of the movie.

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good and bad setups

Bad setup: the camera is not in front of

the diagonal plane

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data analysis

As body position is tracked, Tracker offers both x(t)
graph and numeric data.

You can determine the acceleration of the body
motion by trying to fit the experimental data with a
parabolic function of the type
x(t) = At2 + Bt + C

Open the analysis window offered by Tracker (View /
Data Tool (Analyze)) and the fit manually the
experimental points.

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
During the fit operation it will be necessary to keep in
mind that the sliding body starts from a position x = 0
and still. Therefore parameters B and C can be set
directly to zero. The parabola will be of the type:
x = At2

we can derive the acceleration of the body from it:
a = 2A
and we can estimate the component of the
acceleration of gravity along the inclined plane and we
can derive the acceleration due to friction (oriented in
the opposite direction):
afr =g·sinθ−a

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evaluation of uncertainties

Export data from Tracker: under the graph x(t), select the data columns with time and
position. Clicking with the right button select "Copy Selected Cells" and "Full Precision".

Open a spreadsheet (Excel, OpenOffice, ...) and paste the copied data columns.

We are assuming that the equation of motion is
1 2
x= a t
2
therefore, evaluating for each data pair (t, x) the quantity 2x/t 2 we will have many
estimates of the acceleration of the body during the motion.

Create in your spreadsheet, next to the column of the x, a new column where you will
calculate the value 2x/t2 expressed in m/s2.

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
Create an histogram of data

Good histogram Bad histogram


By neglecting first data which are affected by a very large relative uncertainty,
evaluate the average value of a which should be close to the fitted value with Tracker.

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
The uncertainty of a will be the standard deviation of the column of values 2x/t2.

The acceleration will be expressed by the value obtained with the Tracker fit and by the
uncertainty just calculated.

To complete the experience, we can calculate the uncertainty of friction acceleration
a fr = g⋅sin (θ)−a
and the uncertainty on the afr will be due to the propagation of the error of θ and a

To determine the uncertainty on θ we can follow two ways depending on how the angle
has been measured.
1. If the angle is the average of two measurements with the smartphone, their semi-
dispersion can be considered as uncertainty:
∣θ −θ ∣
δ θ= 1 2
2
2. If the angle has been established starting from the measurement of the sides of a
right triangle, the error of the measurements of the sides on the angle must be
propagated (see reference for detail). 33

At the end of these calculations the friction acceleration will be expressed as

a fr =( g⋅sin (θ)−a)±δ a fr

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Acceleration with the least- squares method

It is necessary to previously perform the body tracking operation with Tracker (see
above). We need time and position data.

With the least squares method we will determine the parameters of the equation of the
straight line which expresses the relationship between the position x and time squared
t2. 1
x= a t 2
2

We make this substitution
1
x→ y, a → c , t2 → x
2

and so out fit will be in the form y=cx+ b

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
Determine parameters c and b with their
uncertainties (using EXCEL)

Represent your data on a Cartesian plane (t2, x).
(The line should adapt to our experimental points. If it
passes away from them or shows a slope in striking
disagreement with experimental data probably some
calculations are wrong)
The theoretical model tells us that the linear
relationship is passing through the origin. We
therefore expect a small value for b.

Propagate the uncertainty of c on the uncertainty of
the acceleration. Remember that a = 2c.

Verify if the range of the parameter b, that is the
interval [b - δb, b + δb] includes zero.

Evaluate R2 (goodness of fit) always with EXCEL
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examples

Bad fit:
Good fit The object is not sliding
correctly!

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Friction acceleration and the friction coefficient

We can exploit the results obtained with the least squares method to calculate the
friction acceleration and the dynamic friction coefficient.

Acceleration due to friction is given by expression
a fr = g⋅sin (θ)−a
that we can evaluate using the value of a obtained with least squares, the value of θ
measured and assigning g a measured value, (for example, with the pendulum) and the
uncertainty about afr will be estimated by propagating the errors of g, θ and a
 As for the dynamic friction coefficient μD, remember that
a fr = g sin (θ)−μ D g cos(θ)
so we can estimate its value. By propagating the uncertainties of g, θ, afr, we get the
uncertainty δμD.
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deliverables
Your report (a .pdf or .ppt... ) should include:

1. Acceleration due to friction obtained obtained with Tracker, with error analysis, statistics
and histogram

1. Acceleration due to friction obtained with the least-squares method

1. friction acceleration and friction coefficient (with error analysis)

1. Pictures, (videos), comments, ideas on how to do better

1. Row data available on request

Activity should be individual, but the report will be by groups, each group must wirite a
common discussion on the esperiment, and present all members results together.
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Something about g...
In 1901 the third General Conference on Weights and
Measures defined a standard gravitational acceleration for
the surface of the Earth:
g = 9.80665 m/s2.
It was based on measurements done at the Pavillon de Breteuil
near Paris in 1888.

This definition is not a value of any particular place or carefully

worked out average, but an agreement for a value to use if a
better actual local value is not known. Deviations from the theoretical
Gravity on the Earth's surface varies by around 0.7%, from
gravity
9.7639 m/s2 on the Nevado Huascarán mountain in Peru to
9.8337 m/s2 at the surface of the Arctic Ocean.

For more info see: https://en.wikipedia.org/wiki/Gravity_of_Earth 40

To calculate local gravity, you can use a calculator. There are several free calculators available online. Here is a
list of the best local gravity calculators online:

NOAA Surface Gravity Prediction Tool

BGI Prediction of Gravity Value Calculator
SensorOne Local Gravity Calculator
FlowSolv Local Gravity Calculator
Walter Bislin’s Earth Gravity Calculator

And depending on the model used, they sligly differ...

For more info see:

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