We then apply Newtons second to each object neglecting the mass of the string and the pulley
and choosing the downward direction as the positive direction.
We omitted for simplicity the vertical projection for the cart. Because the two objects are
Position on Yaxis (cm)
3. Representing the Error bars:
We use the Y axis scale which has been defined so that (231.2 161.3)
corresponds to 23 cm. We
have then (we have already found the error b
Experimental General Physics for Engineers I
Laboratory Report PHYS 192 Fall 2011
Evaluation (For Instructor use only)
Measurement and Errors Analysis II
1 Aim of the experiment
The aim of this session is to introduce you to error and error propagation.
The material needed for this lab has been explained in detail in the introductory chap
To fix the sign in Eq. 8 we reason as follow: When the pendulum moves from
the angle is decreasing so that
being positive, we have to chose the
The motion from
takes one quarter of the period , thus:
The free body diagram in this figure shows that three forces act on the ball:
1. The gravity force
exerted by the earth on the ball. Near the earth surface this is
radius of the sphere, respectively, and
the density and
a. Error bars: Since
is plotted on the Y axis, the error bars represent the error on
been already calculated above and found to be
b. Excel graph:
. This has
y = 466.55x + 106.49
R = 0.9633
and thus, using Eq.4, we get
Similarly one can show that if
9. One can continue in a similar way with other kind of functions.
7 Significant Figures
Significant figures in a given number are the figures that we know reliably, i.e. we are not
Note that this formula is similar to that of case 2 (see Eq. 6).
Example: A cube of volume of (27.00 0.05)
and mass (250.0 0.1)g. What is the error on its density
are constants. The derivative with respect to
y = 9.4857x - 158.52
R = 0.9535
Figure 10: Data, best fit and residuals are shown. Errors bars are equal to 5 (i.e.
At the minimum the partial derivatives should be equal to zero:
These two equations ca
Figure 5: accuracy versus precision
9 Comparing Two Measurements15
If two quantities and
equal if their difference
are exactly known (i.e. no errors is attached to them) then they are
is equal to zero:
But if it happens that we know the two quantities
are the torques due to the tension and frictions, respectively. Here
as was the case for speeds, the linear and angular acceleration are related:
Please note that here refers to the radius of the pulley and not that of the platter (see
Figure 6: Joule's experiment setup
James Prescott Joule46 did, in 1843, an experiment where he measured the mechanical equivalent of
heat constant . In his setup, see Fig. 1, a falling known mass generates heat, by friction, in a known
mass of water. The
Fit linearly your data and get the slope and the intercept.
Using the Excel function linest, calculate the errors on the slope and intercept.
Does your data represent the theory given by Eq. 4? Look at the fit quality variable .
7. Calculate the experimental values of , . This is the first method for measuring .
8. Calculate, and this is the second method, the theoretical values of ,
using Eq. 6
where is the radius of the platter.
9. How do the two values of compare?
10. What can
4 Experimental procedure
1. Measure the mass of calorimeter (i.e. water recipient) when it is empty.
2. Put some water in the calorimeter (the water should fully cover the resistor) and
measure the new mass. Deduce the mass of the water.
3. Connect the ci
We would like to calculate the slope , the intercept and the error on them using these
measurements and the formulae developed above. First, as suggested, let us calculate the
needed sums (as we see on the table, the number of measurement
is equal to 6).
is the proper angular frequency34, and
are constants determined by the
initial conditions of the object (i.e. its position and velocity at
). A such sinusoidal motion
is called a harmonic. The object thus will oscillate with a frequency:
Once we have measured, in this way, the specific heat capacity of a metal, we can use it to
make the calorimeter. We can then measure of any other substance (solid or liquid especially)
by using the following equation (deduced from step 5 above):
5 How to estimate the error
We can estimate the error committed when using a Vernier caliper in the following way. We
notice first that the error comes in because we are unable to locate exactly the first graduation
on the Vernier scale that coincides wit
2. A series of measurements of a physical quantity gave us the following numbers (in some
17.47468 17.47392 17.47616 17.47499 17.47257
a. Calculate the best estimate of the physical quantity (i.e. calculate the mean value of
the series of measureme
The last equation can be re-written as
which is a linear relation between
and . The slope is therefore
and the intercept is
3 Equipment needed :
Force sensor, Computer, Pasco interface, Stand, thread and bar, ruler, masses and mass ho
the derivative of the curve at
. This is given by the slope of the tangent to the curve
at that point (
. To calculate the acceleration from this curve we can find the speed
at two distinct moments and , by finding the slope of the tangents at those point
1 Aim of the experiment
You study in this experiment the harmonic motion of simple pendulum.
A simple pendulum consists of a mass (bob) and an inelastic massless string (see Fig.1). when
the bob is given a sma
Appendix C: Least-square fit
We will use the method of Least Square to fit our data to a straight line. The aim is to find the
line that represents best our measurements. For this we have to minimize the following
are our measured
Phys-192 is a laboratory experimental course designed to accompany the theoretical lectures
given in the Phy-191 course. It therefore presents the student with unique opportunities to
have a firm grasp of the abstract concepts and laws he/she
2. Zeros on the right, usually, are not significant for a non-decimal number but they are significant
for a decimal number:
Examples: 1200 has 2 significant figure (by the first part of this rule). 0.10 has also two significant
numbers (the second part of