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2016

Eastern Mediterranean University

Department of Physics

PHYSICS LAB MANUAL

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TABLE OF CONTENTS

Introduction 1.1 Sources of Error in Measurement ……………………………………3

1.1.1 Does Every Measurement Have an Error? ................................................3 1.1.2 Uncertainty Versus Error…………………………………………...…….3

1.2 Significant Figures, Rounding, and Uncertainty ………………...……5 1.2.1 Significant Figures ……………………………………………………….5 1.2.2 Rounding Numbers ………………………………………………………6 1.2.3 Uncertainty in Measurement ………………………………………….….6 1.2.4 Practice Questions ………………………………………………………..7 1.3 Some Basic Statistics …………………………………………………….8 1.3.1 Mean ……………………………………………………………………..8 1.3.2 Variance ………………………………………………………………….8 1.3.3 Gaussian Distribution ……………………………………………………8 1.4 Propagation of Uncertainties …………………………………………..9 1.5 The Least Squares Line ……………………………………………….12

Experiments 2.1 PHYS101 Experiments…………………………………………...………14 2.1.1 Experiment #1…………………………………………………………....15 2.1.2 Experiment #2……………………………………………………………17 2.1.3 Experiment #3……………………………………………………………21 2.1.4 Experiment #4……………………………………………………………25 2.1.5 Experiment #5……………………………………………………………28

2.2 PHYS 102 Experiments…………………………………………………..30

2.2.1 Experiment #1……………………………………………………………31 2.2.2 Experiment #2……………………………………………………………34 2.2.3 Experiment #3……………………………………………………………37 2.2.4 Experiment #4……………………………………………………………41 2.2.5 Experiment #5……………………………………………………………44

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Introduction

You are highly recommended to read this manual very carefully!

1.1 Sources of Error in Measurement

1.1.1 Does Every Measurement Have an Error? It is noteworthy to mention that not every measurement has an error. Consider the case when you wish to determine the number of students in the class. This can easily be achieved via counting, without any error. However, if we wish to determine the number of oxygen molecules in the room, it is no longer possible to count them. In this case, the number can be estimated with an uncertainty. This uncertainty determines the measurement accuracy.

1.1.2 Uncertainty versus Error

The concept uncertainty plays a vital rule during practical sessions. Thus, you must make sure that you understand it well! For this reason, let us start with a simple example. Imagine you are asked to take a measurement via using a meter ruler. Once you take your data, you must be confident about how to record it.

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Assume that you have measured the length to be 25.1 cm. Even though you may ‘somehow’ convince yourself that this is one true measurement, you cannot claim that it is 100% accurate. You must be aware of the terms uncertainty and error, alongside with their differences. They may sound like the same thing, but they actually are not! What makes this discussion even more interesting is once the word mistake is involved within the subject. Your teaching assistants have been hearing these kind of questions very frequently during the lab sessions: "I have done no mistakes today, but my percentage error has a non-zero value. I am sure I have done everything right!!! So is this logical?" The answer is no!! It indeed has no logical sense. Errors can be thought as a concept that has to exist whenever you are taking a measurement, no matter whether you wish it to be there or not, whereas the mistakes are dependent on the individual himself. You are able to control the mistakes; however, the errors are unavoidable! And what about the uncertainty? Uncertainty stands for the methodology of manifesting errors. If you can manage to minimize the errors, it will in turn have an effect on the uncertainty. As we have already pointed out, errors should not be considered as mistakes. Yet, they must exist! Reading the list below may help you to understand why it is so. There are a variety of reasons why errors arise. Some of these are listed below. Human Error Whilst taking a measurement, you may all claim you have observed something slightly different. You are all carrying out the experiments as good as you can, or scientifically speaking, up to a certain level of accuracy. The measuring techniques may vary from observer to observer. Therefore, to sum up, this type of error may occur because of being unexperienced or choosing a less accurate way of taking the concerned measurement. Instrumental Limitations on the Devices There exist calibrations of finite width. Each device has its own limitation. It is not possible to measure the length of a substance and record it as 5.16472847294728947204 cm, by using a ruler.

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Errors Due to External Influences If your system is not isolated well while you are carrying out an experiment to find specific heat capacity of a substance, some heat will be lost to the surroundings and this will make your reading less accurate. This is an example of an external influence. Parallax Error Just by examining the meaning of the word parallax, we can understand how this error arises. Parallax means alteration, hence the observations will be dependent on the angle of observation. This error can be minimized via experience.

The correct position to take the measurement in the figure above is position B. Try to take your measurements perpendicular to the observation point.

1.2 Significant Figures, Rounding, and Uncertainty

1.2.1 Significant Figures

Basic Rule: Non-zero digits are always significant. Any zeros except for at the beginning are significant.

Some Examples

26.380 5 significant figures 7.94 3 significant figures

0.00980 3 significant figures The last number can be stated in scientific notation as 9.80 x 10-3.

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1.2.2 Rounding Numbers To keep the correct number of significant figures, numbers must be rounded off. The discarded digit is called the remainder. There are three rules for rounding: Rule 1: If the remainder is less than 5, drop the last digit. Rounding to one decimal place: 5.346→5.3 Rule 2: If the remainder is greater than 5, increase the final digit by 1. Rounding to one decimal place: 5.798→5.8 Rule 3: If the remainder is exactly 5 then round the last digit to the closest even number. This is to prevent rounding bias. Remainders from 1 to 5 are rounded down half the time and remainders from 6 to 10 are rounded up the other half. Rounding to one decimal place: 3.55→3.6, also 3.65→3.6.

1.2.3 Uncertainty in Measurement

Every measurement apparatus has a limitation in precision. A Voltmeter with 4 digits cannot measure up to a precision of 5 digits. Other factors, as previously explained, may as well influence the measurement. The uncertainty of a quantity � is denoted by ∆�. Every measurement reading is written together with its uncertainty in the following form.

(� ± ∆�) ���� Absolute and Relative Uncertainty Absolute uncertainty: 3.52 ± 0.03 ��. Relative uncertainty: 3.52 �� ± 1%. Note that relative uncertainty was found via evaluating 0.03/3.52 × 100 %. Important Remark: You are expected to display uncertainties to two significant figures during the lab sessions. Having done so, you must adjust your result in such a way to have the same number of decimal places as the uncertainty.

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Let us give you an example on how to follow this procedure. Example You have taken a measurement for a quantity together with its uncertainty and wrote that

� = 3.2 �� ∆� = 0.05 ��.

How should you record this measurement on your spreadsheet? Step#1: Rewrite the uncertainty to two significant figures, i.e. ∆� = 0.050 ��. Step#2: Quote your result in a way that its decimal places match the one for the uncertainty, i.e. � = 3.200 ��. Step#3: Finalize your answer as follows.

� = (3.200 ± 0.050) �� 1.2.4 Practice Questions Rewrite the following measurements and their uncertainties in the format you are taught throughout this manual.

1) � = 5.0 cm, ∆� = 0.01 ��.

2) � = 10.42 �/��, ∆�= 1.80�

��.

3) � = 42.30 �/�, ∆�= 4.3�

�.

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1.3 Some Basic Statistics

1.3.1 Mean Example If you have repeated a measurement five times and have observed them as 5 cm, 6 cm, 5 cm, 8 cm and 6 cm respectively, the most precise way of representing your measured value is finding the average value, i.e. the mean value. This can be achieved as follows.

� =5 + 6 + 5 + 8 + 6

5= 6

Increasing the number of measurements always provides us with a higher accuracy and our final result in turn becomes more reliable.

1.3.2 Variance The variance represents a measure of how the data distributes itself about the mean or the expected value. While evaluating the variance, there is an important point to be considered.

where �� and �� represent population and sample variance, respectively.

1.3.3 Gaussian Distribution If only random errors affect a measurement, it can be shown mathematically that in the limit of an infinite number of measurements, the distribution of values follows a normal distribution (i.e. the bell curve or the Gaussian distribution at the below). This distribution has a peak at the

mean value ���� and a width given by the standard deviation �.

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Obviously, we never take an infinite number of measurements. However, for a large number of measurements, say, N~10 or more, measurements may be approximately normally distributed. In that event we use the formulae below:

Most of the time we will be using the formulae for small data sets. However, occasionally we perform experiments with enough data to compute a meaningful standard deviation. In those cases, one can take advantage of some computer packages for computing ���� and σ.

1.4 Propagation of Uncertainties1

Oftentimes we combine multiple values, each of which has an uncertainty, into a single equation. In fact, we do it when we measure something with a ruler. Take, for example,

1 † Reference for the two tables in this section: University of Pennsylvania, Department of Physics & Astronomy, Lab Manual for Undergraduates. Note that throughout the second table provided from this source, the uncertainties are not quoted to 2 significant figures. However, this will not be the case for us! These examples are only provided for you to understand how to work on error propagation.

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measuring the distance from a grasshopper's front legs to his hind legs. For rulers, we will assume that the uncertainty in all measurements is one-half of the smallest spacing.

The measured distance is where . What is the uncertainty in ��? You might think that it is the sum of the uncertainties in � and �; namely:

.

However, statistics tells us that if the uncertainties are independent of one another, the uncertainty in a sum or difference of two numbers is obtained by quadrature:

The way these uncertainties combine depends on how the measured quantity is related to each value. Rules for how uncertainties propagate are given below.

∆ ∆ ∆ 0.07cm.

∆ ∆ ∆ 0.1cm

m ∆    4.63cm 1.0cm 3.63cm. 

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1.5 The Least Squares Line

There are two possible approaches to determine the least squares line. Either we consider the square of the vertical distance between the measured points and the least square line or we consider the square perpendicular distance between the measured point and the least square line. In this manual, we mainly focus on the sum of the squares of the vertical distances between the best fit line. It is given by

Sx,y i1

n

yi mxi c2 ,

where cmxy denotes the best fit line. The problem of finding the coefficients m and c

is rather straight forward. We have to minimize the sum of the squares yx,S with respect to

the coefficients m and c , using the straightforward method we have learned in the course of

Calculus. To this end, we take the derivative of yx,S with respect to m and c , and in sequel

equate them to 0. This results in

.02

,02

1

1

cmxyc

S

xcmxym

S

ii

n

i

iii

n

i

Now, one must solve the above equations for m and c . After dividing the both equations by 2, we can rewrite the equations as following

.0

,0

11

1

2

11

cnxmy

xcxmyx

i

n

ii

n

i

i

n

ii

n

iii

n

i

The above equations can be decoupled in one equation as follows:

13

,01

1

21

1

1

n

xb

x

xa

y

yx ini

ini

ini

ini

iini

or

.1

1

1

12

1

ini

iini

ini

inii

ni

y

yx

b

a

nx

xx

The solution of the above equation can be obtained by using the Kramer’s rule:

,

1

1

1

211

1

2

12

1

ini

iini

inii

ni

ini

inii

ni

y

yx

xx

xn

xxnc

m

which finally gives the deserved results for the slope m and intercept c as follows:

.

,

2

12

1

1112

1

2

12

1

111

inii

ni

iinii

nii

nii

ni

inii

ni

inii

niii

ni

xxn

yxxyxc

xxn

yxyxnm

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PHYS101

Experiments

Experiment #1: Force Table

PURPOSE To get familiar with the vector algebra and static equilibrium of forces.

APPARATUS

A force table, pulley clamps, weight holders, weight set

THEORY When several forces act on one single point and the net force is zero, the point is said to be in

equilibrium, i.e. its acceleration is zero. Scientifically speaking,

𝐹1 + 𝐹2

+ 𝐹3 = 0

In this experiment, the forces to be considered are weight of the masses that will be placed on the

hangers. The strings passing over the pulleys are connected to a point (knot) at the center of the

force table. If the net force acting on the knot is zero, the point will remain at rest at the center of

the force table. The magnitude of each force can be calculated via 𝐹𝑖 = 𝑚𝑖𝑔 where 𝐹𝑖s are

measured in Newtons, whereas 𝑚𝑖s in kilograms. It is worthwhile to note that 𝑔 = 9.81𝑚 𝑠2⁄

and the angular position of each pulley will be read from the scale along the rim of the table.

It is possible to represent each force in unit vector notation. The x-components are given by

𝐹𝑖 cos 𝜃𝑖 and the y-components by 𝐹𝑖 sin 𝜃𝑖 in case if 𝜃𝑖 represents the angle between the

associated force and the positive x-axis. If there exists 𝑛 forces acting on a point, that point will

be in equilibrium if the following conditions are satisfied.

∑𝐹𝑋 = ∑𝐹𝑖 cos 𝜃𝑖 = 0

𝑖=𝑛

𝑖=1

∑𝐹𝑌 = ∑𝐹𝑖 sin 𝜃𝑖 = 0

𝑖=𝑛

𝑖=1

which implies that

𝐹1𝑥 + 𝐹2𝑥 + 𝐹3𝑥 = 0

𝐹1𝑦 + 𝐹2𝑦 + 𝐹3𝑦 = 0

EXPERIMENTAL PROCEDURE

1) Adjust the position of two pulley clamps to 0° and 90° (skip this part if these are already set).

2) Your assistants will provide you with the magnitude of 𝑚1 and 𝑚2. Place the masses that

match these values onto the hangers located at 0° and 90°. 3) The third pulley clamp is free to be moved between 180° and 270°. By adding appropriate

amount of mass together with making a suitable choice for the angle, the equilibrium must be

1.1

1.2

1.3

15

brought to the system. This implies that the knot should be situated at the center of the force

table.

RAW DATA

1) Record 𝑚1, 𝑚2 and 𝑚3 in the table below and find the associated forces via multiplying them

by 9.81𝑚 𝑠2⁄ .

2) Read the angle for the third pulley from the force table and record it in the table.

Mass (kg) Force (N) 𝜃(°)

Pulley 1 0

Pulley 2 90

Pulley 3

3) After subtracting 180° out of 𝜃3, you may obtain 𝜃𝑒𝑥𝑝.

𝜃𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 = 𝜃3 − 180° =............

DATA ANALYSIS

1) Evaluate x and y-components of each force and express them in unit vector notation.

𝐹1 =....................

𝐹2 =....................

𝐹3 =....................

2) Confirm that equations 1.3 are satisfied.

3) Find the quantity 𝜃𝑡ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 by†

𝜃𝑡ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 = tan−1 |𝐹2

𝐹1| =............

4) Compare 𝜃𝑡ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 and 𝜃𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 using the definition of the percentage error given

below.

%𝑒𝑟𝑟𝑜𝑟 =|𝜃𝑡ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 − 𝜃𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙|

𝜃𝑡ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙×100%

COMMENTS & DISCUSSIONS † Derive equation 1.4 and show your steps clearly.

1.4

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Experiment #2: Projectile Motion

PURPOSE The purpose of this experiment is to investigate the trajectory equation for the projectile motion. APPARATUS A steel ball, a curved track, a wooden board, a meter stick, an A4 paper, a carbon paper. THEORY In this experiment, the set up (see the figure below) shows a projectile motion in which B and D stand for the starting and ending points, respectively. This is the part of the motion that we will be focusing on.

Figure 2.1

The steel ball is projected horizontally with an initial speed o oxv v at point B. The steel ball

falls down due to gravitational acceleration, g, which is constant and directed downwards. The effect of air resistance throughout the motion is negligible. With these assumptions, the path of a projectile is a parabola.

The acceleration in y-direction is ya g and the acceleration in x-direction is 0xa , since air

resistance is neglected. Therefore, the velocity in x-direction is constant. Considering the origin of our coordinate system at point B, we can set 0o oy x . Furthermore, the projection angle ��

at this point also reads zero. The motion along x- axis is expressed as

o ox

ox

v v

x v t

where

10

7oxv gh

This is the initial velocity at point B.†

A

h

B v o

y

x y i

x i v

D

C

2.1

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Figure 2.2

The motion along the y-axis is given by

2 21 1

2 2y at gt

with

10

7ox

x xt

vgh

.

Combining the latter two equations, one can write

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20y x

h

.

The trajectory Eq. (2.4) is the equation of a parabola passing through the origin in the x-y plane.

The graph of y versus x2 is a straight line with a slope of ��

��� .

EXPERIMENTAL PROCEDURE 1) Adjust the position of the tripod in such a way that once the steel ball is released, it hits the bottom of all drawers. 2) Open the drawers and align them to one fixed vertical axis (see Figure 2.2).

2.2

2.3

2.4

18

2) Place the A4 paper on the wooden board and then put the carbon paper over it. Having done so, place them both into the first drawer such that the edge of the A4 paper remains in contact with the inner face of the drawer. 3) Release the steel ball from point A (the top point of the curved track). When the steel ball falls on the board, it will leave a mark on the A4 paper via the carbon paper showing the final position of the steel ball (point D). 4) This procedure should be repeated for the rest of the four drawers, from the second to the last one. 5. The marks on the A4 paper will be used to measure final x and y positions of the steel ball. The assistants will guide you regarding the method to be followed for measuring these values. Note that all the y values will be negative, i.e. � < 0. RAW DATA Record your data in the table provided below and fill out the third column.

y (cm) x (cm)

DATA ANALYSIS 1) Transfer the data from the first table accordingly to the second one stated below.

X(=x2) Y(=y) X2 XY ∑ X= ∑ Y= ∑ X2= ∑ XY=

The least-squares line is given by

� = �� + � in which

and

2.5

2.6

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where n stands for the number of measurements (for this experiment, n=5). 2) Calculate the experimental value of the vertical distance h from A to B by using

exp

7

20h

m

.

3) Measure h using a meter ruler and quote it as hAB. 4) Compare hexp and hAB using the definition of the percentage error given below.

5) Write the possible sources of errors in the measured value. 6) Draw the least-squares line � = �� + �, by substituting the � and � values you have already calculated. To this end, pick any two X-values from your raw data and plug into the equation of the least-squares line to obtain two corresponding Y-values. This will give you two points which can then be plotted onto the graph paper using a proper scaling. APPENDIX

A steel ball with mass � and radius � has a moment of inertia � =�

����. Since it rolls while

moving on the curved track, in addition to the linear kinetic energy, one should also consider the rotational kinetic energy. Conservation of mechanical energy requires the following relation:

�� =1

2��� +

1

2���

�� =1

2��� +

1

2×2

5��� �

�

���

�� =1

2��� +

1

5���

�� =7

10��� �� =

10

7�

Finally, we obtain the speed of the ball just before leaving the curved track as follows

� = �10

7�

2.7

2.8

2.9

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EXPERIMENT #3: Coeffi cients of Static and KineticFriction

Objective: To find the coeffi cients of static and kinetic friction between an inclined surfaceand an object.

Figure 3.1

Apparatus: Inclined plane, cart, electronic timer which enables us measure the speed,and ruler.Theory: The laws of friction are empirical. Namely, they are based on the experimental

studies. What really happens when two surfaces that are in contact slide is not fully under-stood. However, there are some commonly accepted rules about the friction that are goingto be tested in this experiment. The aforementioned rules are the following1) The forces of static and kinetic friction are proportional to the normal force.2) The force of friction does not depend on the area of the contact surfaces.3) The force of friction does not depend on the relative velocity of the two surfaces.In this experiment, a cart is firstly placed on an inclined plane with angle θ. The force on

the cart due to the gravity is the weight: mg where m is the mass of the cart. This force isdirected vertically downward as shown in Figure 3.2. Weight force can be separated into thecomponents in such a way that one of its components become parallel to the surface of theincline and the other one is normal to it. The frictional force opposes the motion, thereforeit is directed upward through the inclined plane. According to the rule 1, the proportionalityconstant between the normal force and the force of friction is known to be the coeffi cient offriction µ. In mechanics, there are two coeffi cients of friction: the coeffi cient of kinetic frictionµk (for moving objects) and the coeffi cient of static friction µs (for stationary objects).

1

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Figure 3.2After tuning the angle θ from zero to a particular angle, the cart will start to move. At thatspecific angle, the force of static friction gets its maximum value:

fs = µsN = µsmg cos θ, 3.1

and hence, one has the following equilibrium condition:

µsmg cosθ = mg sin θ, 3.2

which yields the coeffi cient of static friction as follows

µ s = tan θ. 3.3

To find the coeffi cient of kinetic friction µk our procedure is the following: First ofall, the experimenter should find the final point of cart which corresponds to the place wherethe sensor initiates the electronic timer (the associated assistant will help the experimenter inthis regard). Next, he/she should locate the cart to the various initial points (to be specifiedin Table 3.1) which will render her/him possible to find the displacements of the cart (∆x)by a ruler. Since the flag on the cart has a constant length of l = 10 cm, whenever the flagcrosses the electronic timer, the experimenter will be eligible for having the speed of the cart:

v =l

t.

3.4

After completing the data table seen in Table 3.2, the experimenter will plot v2 versus∆x graph. Using the following expressions

v2 = 2a (∆x) , 3.5

where

a = g (sin θ− µk cos θ) , 3.6

one can immediately get that the slope corresponds to 2a = 2g (sin θ−µk cos θ). Angleθ can be easily measured by a protractor. Therefore, in the obtained slope value, the only

2

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unknown is µk, which means the coeffi cient of kinetic friction between the surface and thecart.

Figure 3.3

∆x(m) v (m/s) v2 (m2/s2)

1.00

0.80

0.600.40

0.20Table 3.2

You may use the following table and formula to find the equation of the least-square line:

X (= ∆x) [m] Y (= v2) [m2/s2] X2 [m2 ] XY [m3/s2]

∑X =

∑Y =

∑X 2 =

∑XY =

Table 3.3

The least-square line is given byY = mX + c 3.7

in which

m =n (∑XY )− (

∑X) (

∑Y )

n (∑X2)− (

∑X)

2 = ............................................... 3.8

c =(∑Y ) (

∑X2)− (

∑X) (

∑XY )

n (∑X2)− (

∑X)

2 = ......................................, 3.9

where m denotes the slope of the least-square line and c is its Y -intercept.

3

23

To find the coeffi cient of static friction µs we follow this procedure: Put eachcart ( C1, C2 and C3) on the inclined plane and raise the angle until the moment when thecart will just start its motion. Record each critical angle as (θ1, θ2, and θ3) and completethe following table.

Cart No Critical angle θ µs = tan θ (see Eq. 3.3)C1 = θ1 = tan θ1 =C2 = θ2 = tan θ2 =C3 = θ3 = tan θ3 =

Table 3.3Question:a) If you add extra masses to the carts will the results for µs and µk change?b) Does friction force depend on the area in contact?c) The race cars have wide tires, what must be the reason of it?

4

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EXPERIMENT #4: Gravitational Acceleration

Objective: To find the gravitational acceleration "g" with an object moving on an airtrack.Apparatus: Air track, glider, flags, photosensor, timer, riser, meter stick.

Figure 4.1

Theory: An air track can be used to study the motion of an object on a frictionlesssurface. Metal gliders move along the air track on a layer of air. In this experiment, aphotocell gate which is connected to an electronic clock is used to measure the speed of theglider whenever it passes among them. On the top of the glider, there is a flag of lengthL. When the beams of light were abruptly cut off by the flag, the clock begins to measuretime. When the flag completely passes the photosensor and the light beam becomes againcontinuos. After that moment, the electronic timer precisely record the transition time.In this set-up, the experimenter should first level the air track in such a way that the

air track will be inclined at a small angle θ (see Figure 4.1), which satisfies the followingtrigonometric expression: Then one end- the end with a single support- is lifted by a riserof height H. The air track will be inclined at a small angle θ.

sin θ =H

L0, 4.1

where H is the height of the riser and L0 is the distance between the air track suspensionpoints. The glider with flag is released at the high end of the air track and acceleratesdownward to the other end. As it passes through the gate, the light beam is cut off by theflag within a time t. So for each measurement, the experimenter will record t, L, L0, H, andD (distance between the front edge of the flag and the photosensor; see Figure 4.1).Let us recall that the motion with one-dimension with constant acceleration is described

by the following expressions

1

25

v2 − v20 = 2a (x− x0) 4.4

Q1: What is the velocity of the glider when its flag first cuts off the light beam?A1: In this case x = D, a = g sin θ, x0 = v0 = 0. After inserting them into Eq. (4.4), we

find

v2 = 2gD sin θ. 4.5

The glider then moves through the gate in a time t.

Q2: What is the velocity of the glider as it passes the gate?A2: The clock reads t, the length of the flag is L so v = L

t. According to Eq. (4.5)

if v2 versus D graph is plotted, a straight line should have a slope of 2g sin θ. Since theangle θ is known, the experimenter will compute the experimental value of the gravitationalacceleration g. It is possible that the straight line drawn through the data points may notpass through the origin. It means that the track was not initially well-levelled.Procedure1) Carefully level the air track. Make this adjustment when the glider is on the track

with "AIR ON". Adjust the support screw until the glider comes at rest at the center of thetrack.2) The wires of the photoelectric gate should be plugged into the back of the timer in

the START position.3) Set the timer decimal point to 0.001 second. When the flag on the glider passes

through the light beam the timer must start to operate.4) Set the gate at the lower end of the track.5) Fix well the flag (length of 10cm) mounted on the glider. So that it will not slide out

from the glider when it is stopped after the gate. Hold the glider in its position. Mark theposition of the edge of the glider, using the scale on the air track. Move the glider to theposition where the counter just begins to count. Mark the position of the same edge of theglider by using the scale on the air track. The difference between those two points shouldbe the distance D.6) Measure the the distance D and the angle θ.7) Release the glider from rest and record the timer t through the gate.8) Repeat this procedure for six different values of D (with a fixed value for angle θ)

and fill the following table.

t (s) D (m) v (m/s) = 0.10t

v2 (m2/s2)0.400.600.801.001.201.401.60

2

x =1

2at2 + v0t+ x0, 4.2

v = v0 + at, 4.3

26

Analysis:1) In each case, from the average time t calculate the velocity of the glider by v = L

t.

2) Find the least square line of v2 with respect to D .You may use the following table and formula to find the equation of the least-square line:

X(= D) m Y (= v2) (m2/s2) X2 (m2) XY (m3/s2)

∑X =

∑Y =

∑X2 =

∑XY =

The least-square line is given byY = mX + c 4.6

in which

m =n (∑XY )− (

∑X) (

∑Y )

n (∑X2)− (

∑X)2

= ............................................... 4.7

c =(∑Y ) (

∑X2)− (

∑X) (

∑XY )

n (∑X2)− (

∑X)2

= ......................................, 4.8

where m is the slope of the least-square line and c is its Y -intercept.3) Find the slope of this straight line with its unit.4) Using the answer given in A2, find the experimental value of the gravitational accel-

eration: gexp.5) Since the theoretical value of the gravitational acceleration is gthr = 9.8m/s2, find the

percentage error of the experiment:

|Experimental V alue− Theoretical V alue||Theoretical V alue| × 100%

3

4.9

27

EXPERIMENT #5: Conservation of Linear Momen-tum and Kinetic Energy

Objective: To verify the conservation of linear momentum and total kinetic energy in theexperiment of an almost elastic collision.Apparatus: Air track, gliders with flags, electronic timer.

Figure 5.1

Theory : On an air track, in principle, the friction has almost an ignorable value. Forthis reason, the air track is considered as a suitable platform to explore the elastic collisions.

In this experiment, we use two gliders with variable masses on the air track and twophoto sensors to record electronically the time intervals. Each glider has a flag on it with alength of l = 10 cm. (see Figure 5.1). Prior to the collision, we denote the speeds of objects(i.e., the gliders) by v1 (for the left glider with mass M1) and v2 (for the right glider withmass M2). After the collision, the final speeds become v′1 and v

′2, respectively. Accordingly,

the conservation principle of linear momentum is given by

M1 v1 + M2v2 = M1v′1 +M2v

′2 . 5.1

Since the air track is a one-dimensional object the process takes place in one-dimension only,say the x-axis. Depending on the directions of velocities , the velocities can have ± signs.Similarly, the conservation principle for the total kinetic energy in an elastic collision canbe expressed by

1

2M1v21 +

1

2M2v22 =

1

2M1 v

′21 +

1

2M2v

′22 . 5.2

Let us recall that each kinetic energy term is a positive quantity. In order to make sure thatthere is no gravitational potential energy difference, the air track must lie in a horizontalplane surface. This must be quarantined before starting to the experiment. The masses ofeach glider will be provided by your assistant in the laboratory. The main purpose of theexperiment is to verify the provision of Eqs. (5.1) and (5.2).

The most important part of the experiment is to read the velocities, correctly. To thisend, we use photo sensors, which electronically record the amount of time required to passthe flag on each glider. If the recorded time is denoted by ti and the length of the flag of theith glider is li then the speed of the glider is found to be

1

28

In order to make the experiment simpler, one can assume that v2 = 0 throughout theexperiment. In other words, the second glider having mass M2 will assume to be initiallyat rest (before the collision). This choice leads to the following simplifications in Eqs. (5.1)and (5.2)

M1v1 =M1v′1 +M2v

′2, (5.4)

1

2M1v

21 =

1

2M1v

′21 +

1

2M2v

′22 . (5.5)

In short, the experimenter should have to read three velocities instead of four. Meanwhile,the dare mass of the each glider ismg = 0.210kg.Procedure: First, the experimenter should adjust the air track to the perfectly horizontal

position, which is possible with the zero inclination angle. Next, the various masses providedby the assistant of the laboratory will be apportioned to the gliders (the assistant will helpthe experimenter in this regard). The glider (M2), which is initially at rest, is placed betweenthe two photo sensors. In the meantime, the other glider (M1) is out from the photo sensors.Before we move the gliderM1 with initial speed v1 , we switch on the pump of the air trackso that the friction becomes almost zero. When the glider M1 passes through its nearestphoto sensors, the transition time will be recorded as t1. Then, the gliders will collide witheach other. After the collision, the direction of motion of the each glade will be reversed andin sequel they will pass the photo sensors. In this way, the associated times (t1, t ′1, and t′2)and hence the velocities (v1 , v′1, and v′2) can be easily obtained. Thus, the experimenter willfill Table 5.1.Finally, the experimenter will check whether Eqs. (5.4) and (5.5) are satisfied or not.

If Eq. (5.4) is satisfied, he/she will conclude that in the process of the collision the linearmomentum is conserved. Similarly, if Eq. (5.5) holds, he/she will remark that the totalkinetic energy is conserved during the process.It is worth noting that in such an experiment a miserable error between the left and right

hand sides of Eqs. (5.4) and (5.5) is tolerable. The students must be able to explain aboutthe possible numerical differences in those equations. For example, is the collision exactlyelastic? Is there some energy loss taking place in the collision process? How?

t1 (s) t′1 (s) t′2 (s) v1 =0.10t1

v′1 =0.10t′1

v2 = 0 v′2 =0.10t′2

m1 (kg) m2 (kg)

1234

Table 5.1Caution: If the difference between the left and right hand sides of Eqs. (5.4) and (5.5) is

big, this means the data taken is not recorded, correctly. In this case, the experimenter shouldcarefully repeat the experiment and reduce the errors as much as possible. For example, ifthe glider M1 comes to a stop after the collision, the experimenter should increase its initialspeed so that it will gain enough speed to pass through the photo sensors, after the collision.Questions:a) What are the main physical properties of the elastic, inelastic, and completely inelastic

collisions?b) Can you give an example for each collision type?

2

29

PHYS102

Experiments

30

Experiment #1: Specific Heat Capacity and Latent Heat of Fusion

PURPOSE

To evaluate the specific heat capacity of iron and brass and the latent heat of fusion of water.

APPARATUS

A beaker, a saucepan, a heater, a thermometer, an electronic balance, metal blocks, ice cubes,

and purified water.

THEORY

Once heat energy is provided to a system, this energy will lead to an increase in the average

kinetic energy per molecule, which in turn implies that the temperature of the system will rise.

At this point, it would be beneficial to introduce the quantity called specific heat capacity.

Specific heat capacity, 𝑐, is the amount of heat required to raise the temperature of unit mass

of a substance by one Celsius degree, which can be stated as

𝑄 = 𝑚𝑐∆𝑇,

where 𝑄 is the heat lost or gained in Joules, 𝑚 stands for mass in kg, and ∆𝑇 = 𝑇𝑓 − 𝑇𝑖. It is

worthwhile to note that, the unit of specific heat capacity will be expressed as 𝐽/(𝑘𝑔℃).

For the case when the heat energy is used for changing the state of a system, we require to

define a new quantity, namely the latent heat, which can be expressed as

𝑄 = 𝑚𝐿,

where 𝑄 and 𝐿 stand for the heat transferred (in 𝐽𝑜𝑢𝑙𝑒𝑠 ) and latent heat (in 𝐽𝑜𝑢𝑙𝑒𝑠/𝑘𝑔 ),

respectively. As can be noticed from Eq. (1.2), there exists no temperature change for this case,

since all the energy is used for breaking the existing chemical bonds among the molecules.

Therefore, the average internal energy per molecule remains constant.

The latent heat can be classified into two; the latent heat of fusion,𝐿𝑓 , which is the heat energy

released when 1 𝑘𝑔 of a substance in liquid form solidifies (i.e. fuses) without changing its

temperature, and the latent heat of vaporization,𝐿𝑣 , which is the heat energy needed to

vaporize 1 𝑘𝑔 of a liquid substance without changing its temperature. As a matter of fact, water

has one of the highest values of the latent heat of fusion; (𝐿𝑓)𝑤

= 333000 𝐽/𝑘𝑔. It also has a

high specific heat capacity value relative to other liquids; 𝑐𝑤 = 4186 𝐽/(𝑘𝑔℃).

Transfer of heat energy occurs when substances having different temperatures are allowed to

interact thermally. In any energy transformation for an isolated system, the total amount of

energy remains constant. This is called the law of conservation of energy. Whenever two

substances at different temperatures are allowed to be thermally exposed to each other, heat

travels from the warmer substance to the colder one. The quantity of heat given off by the

warmer substance is equal to the negative value of the quantity of heat energy gained by the

colder one, provided that heat energy does not escape to the surroundings (an isolated system).

The energy transfer will continue until both substances reach the same temperature

(equilibrium temperature). This is called the principle of heat exchange; i.e.

ℎ𝑒𝑎𝑡 𝑒𝑛𝑒𝑟𝑔𝑦 𝑙𝑜𝑠𝑡 + ℎ𝑒𝑎𝑡 𝑒𝑛𝑒𝑟𝑔𝑦 𝑔𝑎𝑖𝑛𝑒𝑑 = 0:

𝑄𝑙𝑜𝑠𝑡 + 𝑄𝑔𝑎𝑖𝑛𝑒𝑑 = 0

1.1

1.2

1.3

31

Consider the case in which a metal block at a higher temperature is mixed up with purified

water. In this case, one can write

𝑄𝑙𝑜𝑠𝑡 = 𝑄𝑏 = 𝑚𝑏𝑐𝑏(𝑇𝑓 − 𝑇𝑏)

𝑄𝑔𝑎𝑖𝑛𝑒𝑑 = 𝑄𝑤 = 𝑚𝑤𝑐𝑤(𝑇𝑓 − 𝑇𝑤)

where 𝑇𝑓, 𝑇𝑤, 𝑇𝑏, 𝑚𝑤, 𝑚𝑏, 𝑐𝑏 and 𝑐𝑤 represent the equilibrium temperature of the system, the

initial temperature of water, the initial temperature of the metal block, mass of water, mass of

the metal block, specific heat capacity of the metal block and specific heat capacity of water,

respectively. Note that the specific heat capacity of the beaker is neglected throughout the

calculations stated in Eq. (1.4). Substituting expressions in Eq. (1.4) into Eq. (1.3), one obtains

𝑚𝑏𝑐𝑏(𝑇𝑓 − 𝑇𝑏) + 𝑚𝑤𝑐𝑤(𝑇𝑓 − 𝑇𝑤) = 0

which in turn implies

𝑐𝑏 =−𝑚𝑤𝑐𝑤(𝑇𝑓 − 𝑇𝑤)

𝑚𝑏 (𝑇𝑓 − 𝑇𝑏).

For latent heat of fusion

𝑚𝑖𝑐𝑒𝐿𝑓 + 𝑚𝑖𝑐𝑒𝑐𝑤(𝑇𝑓 − 0) + 𝑚𝑤𝑐𝑤(𝑇𝑓 − 𝑇𝑤) = 0.

Rearranging for 𝐿𝑓 leads to

𝐿𝑓 =−𝑚𝑖𝑐𝑒𝑐𝑤(𝑇𝑓 − 0) − 𝑚𝑤𝑐𝑤(𝑇𝑓 − 𝑇𝑤)

𝑚𝑖𝑐𝑒.

You may take specific heat capacity of water as 𝑐𝑤 = 4186 J/(kg℃).

EXPERIMENTAL PROCEDURE

For the Metal Blocks

1) Place the beaker on the digital balance and press the calibrate button.

2) Add some purified water into the beaker and measure the mass, 𝑚𝑤 , and the initial

temperature, 𝑇𝑤, of water.

3) Press the calibrate button of the digital balance again.

4) Heat a metal block in boiling water long enough for the block to reach 100°𝐶.

5) Transfer the hot metal block into the system.

6) Record the mass of the metal block as 𝑚𝑏.

7) Measure the final temperature of the water&block system, once the heat transfer has taken

place.

For the Ice

1) Place the beaker on the digital balance and press the calibrate button.

2) Add some purified water into the beaker and measure the mass, 𝑚𝑤, and the initial

temperature, 𝑇𝑤, of the water.

3) Press the calibrate button of the digital balance again.

4) Add the ice cubes (held at the melting point) into the beaker and measure the mass of ice.

1.4

1.6

1.8

1.5

1.7

8

32

5) Measure the final temperature of the water&ice system, once the heat transfer has taken

place.

RAW DATA a) For iron

wm (kg) ironm (kg) Tw (0C) Tiron (

0C) Tf (0C)

b) For brass

wm (kg) brassm (kg) Tw (0C) Tbrass (

0C) Tf (0C)

c) For ice

wm (kg) icem (kg) Tw (0C) Tice (

0C) Tf (0C)

0

DATA ANALYSIS

1) The experimental value for specific heat of iron [Eq. (1.6)]: ……............................................

2) The experimental value of specific heat of brass [Eq. (1.7)]: .…...............................................

3) The experimental value of latent heat of fusion of water [Eq. (1.8)]: ........................................

4) Since the standard values of the associated specific heats and the latent heat are as follows:

𝑐𝑖𝑟𝑜𝑛 = 438 𝐽/(𝑘𝑔℃)

𝑐𝑏𝑟𝑎𝑠𝑠 = 380 𝐽/(𝑘𝑔℃)

(𝐿𝑓)𝑤

= 333000 𝐽/𝑘𝑔

find the percentage error for 𝑐𝑖𝑟𝑜𝑛, 𝑐𝑏𝑟𝑎𝑠𝑠, and (𝐿𝑓)𝑤

by using the equation provided below and

comment on the possible sources of errors.

%𝑒𝑟𝑟𝑜𝑟 =|𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑣𝑎𝑙𝑢𝑒 − 𝐸𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 𝑣𝑎𝑙𝑢𝑒|

𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑣𝑎𝑙𝑢𝑒× 100%

33

Experiment #2: Ideal Gas Law

PURPOSE Utilizing the ideal gas law to measure the number of moles of a confined ideal gas.

APPARATUS

The gas law apparatus

THEORY

All macroscopic pieces of matter have in common some fundamental properties. For example,

pressure ( P ), volume (V ), temperature (T ), internal energy (U ) and entropy ( S ). These

variables are not independent and for simple states of matter any of these variables can be

expressed in terms of any two of the others. These expressions are called equations of state. The

ideal gas law is the simplest equation of state:

nRTPV ,

where n is the number of moles, 𝑅 = 8.31 𝐽/(𝑚𝑜𝑙𝑒. 𝐾) is

the gas constant in SI units and T is absolute temperature, T

(𝐾) = 273 + T (℃). The first law is Boyle’s law, which is

expressed as

1KPV ( 1K is a constant)

In the case when pressure is kept fixed, Charles’ law is

2KT

V ( 2K is another constant)

and the third law (Avagadro’s law) reads

3Kn

V ( 3K is a third constant)

Throughout this experiment, we will be investigating the Boyle’s law, i.e we can express 𝑉 as a

function of 1/𝑃 as follows.

1V nRT

P ,

where in this case the slope is equivalent to 𝑛𝑅𝑇.

EXPERIMENTAL PROCEDURE

1) The apparatus is already set up in the laboratory. A fixed amount of air is closed in a glass

tube by a column of oil. The volume of the air above the oil is measured directly from the

2.1

2.2

Figure 2.1

34

35

volume scale on the apparatus, in 𝑐𝑚3. The pressure of the gas is determined directly from the

barometer in units of atmosphere. You should start the experiment by setting the gas on some

initial pressure (e.g. 2.8 𝑎𝑡𝑚 is a good starting point).

2) Decrease the pressure (in steps of 0.3 𝑎𝑡𝑚) by opening the valve on the apparatus. For each

specific pressure measurement, record the associated volume reading from the scale and write it

down in the table provided in the Raw Data section.

RAW DATA1

P (atm) V (cm3) P (Pa) V (m3)

DATA ANALYSIS

You may use the following table and formula to find the equation of the least-square line of y

versus x2.

X(=1/P) [Pa-1] Y(= 𝑉) [m3] X2 [Pa-2] XY [Pa-1m3]

∑ X= ∑ Y= ∑ X2= ∑ XY=

The least-square line is given by

𝑌 = 𝑚𝑋 + 𝑐

in which

𝑚 =𝑛 ∑ 𝑋𝑌 − ∑ 𝑋 ∑ 𝑌

𝑛 ∑ 𝑋2 − (∑ 𝑋)2

and

𝑐 =∑ 𝑌 ∑ 𝑋2 − ∑ 𝑋 ∑ 𝑋𝑌

𝑛 ∑ 𝑋2 − (∑ 𝑋)2

1 In order to complete the table in this section, you may need the following conversion factors.

1cm3=10-6m3

1atm=1.01x105Pa

2.3

2.4

2.5

36

where 𝑛 stands for the number of measurements (for this experiment, 𝑛 = 5 by default).

Draw the least-squares line 𝑌 = 𝑚𝑋 + 𝑐, by substituting the 𝑚 and 𝑐 values you have already

calculated. To this end, pick any two X-values from your raw data and plug into the equation of

the least-squares line to obtain two corresponding Y-values. This will give you two points which

can then be plotted onto the graph paper using a proper scaling.

Finally, knowing that the experiment has been made while the temperature of the gas is held at

room temperature (about 27℃ on average), determine the amount of air (in moles) enclosed in

the glass tube of your apparatus by using Eq. (2.2).

37

Experiment #3: Magnetic Force on a Wire

PURPOSE

To determine the strength of a constant magnetic field by investigating the relation between the

magnetic force and current.

APPARATUS

A cent-o-gram balance, a magnet, a low voltage AC/DC power supply, plug cord sets, a current

loop board, a tripod.

THEORY

A current carrying wire in a magnetic field experiences a force that is usually referred to as a

magnetic force. The magnitude and direction of this force depend on four variables: The magnitude

and direction of the current (𝐼); the strength of the magnetic field (�� ); the length of the wire (�� ); and the angle between the field and the wire (𝜃). This magnetic force can be described

mathematically by the vector cross product:

𝐹 𝑚 = 𝐼�� × ��

Or in scalar form,

𝐹𝑚 = 𝐼𝐿𝐵 sin 𝜃

In the special case of 𝜃 = 90°, when the magnetic field and the current carrying wire are normal

to each other, Eq. (3.2) turns into

𝐹𝑚 = 𝐼𝐿𝐵,

which will be the case for the incoming experiment.

3.1

3.2

Figure 3.1. Front View Figure 3. 2. Side View

3.3

38

As in Figure 3.1, the lower part of a rectangular current loop is lying in a magnetic field. The

magnetic field produced by a permanent magnet is constant, uniform, and directed outwards. Once

the current flows through the wire, the magnet starts to exert a force upon the wire via its magnetic

field. Direction of this force is upwards and in case if you need assistance regarding how to prove

this, you may ask for your assistants’ help. Figure 3.2 illustrates the same picture but from another

point a view. In this rotated picture, the magnet itself together with the forces acting on it are

shown. One of these forces is the so-called gravitational field 𝐹 𝑔, while the second one, 𝐹 𝑚′ , is

nothing but the reaction force exerted on the magnet by the wire. Based on Newton’s third law,

the magnitude of this force is equal to the magnitude of the original force 𝐹 𝑚, whereas its direction

is opposite to it. Therefore, the magnitude of the net force exerted on the magnet can be expressed

as

𝐹𝑛𝑒𝑡 = 𝐹𝑚′ + 𝐹𝑔 = 𝐼𝐿𝐵 + 𝑚𝑔.

This implies that the changes of 𝐹𝑛𝑒𝑡 with respect to 𝐼 is linear, with its slope and y-intercept being

equal to 𝐿𝐵 and 𝑚𝑔, respectively.

EXPERIMENTAL PROCEDURE

1) Set up the apparatus as shown in Figure 3.3 and make sure that the magnet and the current loop

board are not in contact.

Figure 3. 1

2)The magnet should be situated on the scale pan in such a way that the white side (south pole) of

the magnet faces the tripod.

3) Determine the mass of the magnet with no current flowing. The procedure to be followed for

this step is explained in Appendix.

4) Convert this value into kg and record it into the table provided in the Raw Data section. This

number represents the true mass of our magnet.

5) Turn on the power supply and set first the current to 1A. Once the current starts flowing, the

measurement to be taken will be equal to the apparent mass of the magnet. Your aim now is to

readjust the position of the center windows in a way to make the two indicators match again.

6) Having completed step 5, continue taking the measurements and record each value of apparent

masses in your table.

3.4

39

RAW DATA

Current (A) Mass (kg)

DATA ANALYSIS

X(=I) [A] Y (F=Mg) [N] X2 [A2] XY [AN]

∑ X= ∑ Y= ∑ X2= ∑ XY=

The least-squares line is given by

𝑌 = 𝑚𝑋 + 𝑐

in which

𝑚 =𝑛 ∑𝑋𝑌 − ∑𝑋 ∑𝑌

𝑛 ∑𝑋2 − (∑𝑋)2

and

𝑐 =∑𝑌 ∑𝑋2 − ∑𝑋 ∑𝑋𝑌

𝑛 ∑𝑋2 − (∑𝑋)2

where 𝑛 stands for the number of measurements (for this experiment, 𝑛 = 9 by default).

3.5

3.6

3.7

40

Draw the least-squares line 𝑌 = 𝑚𝑋 + 𝑐, by substituting the 𝑚 and 𝑐 values you have already

calculated. To this end, pick any two X-values from your raw data and plug into the equation of

the least-squares line to obtain two corresponding Y-values. This will give you two points which

can then be plotted onto the graph paper using a proper scaling.

APPENDIX

As can be observed from the apparatus, you will be able to determine the mass values via using

the scaling provided on the cent-o-gram balance. The poise show weights through the center

windows and the beams are stepped up from front to rear. Each poise will enable you to determine

which number you shall quote for each digit. Adjust the position of the center window on the rear

beam to ‘100’. Then, by trial and error, figure out which numbers shall be chosen for the second,

third and front beams so that the indicator aligns with the reference point marked at the right side

of the cent-o-gram balance. Once this is achieved, your system is in equlilibrium.

Figure 3. 2

41

Experiment #4: Ampere’s Law

PURPOSE

To determine an experimental value for the constant ��

�� via Ampere’s law.

APPARATUS A low voltage AC/DC power supply, a Teslameter with probe, and a square wooden frame with ten windings of wire.

THEORY An electric current passing through a conductor produces a magnetic field or flux density in the nearby region. The distribution and magnitude of the magnetic flux density (Ψ) is given by Ampere's law:

Ψ = � �� . �� = ���

where �� is the permeability of free space of which standard value reads, 4� × 10�� ���

�� i.e.

���

4��

��������= 1 × 10�� �

��

��.

An infinitely long straight wire carrying a current � will produce circular lines of magnetic field centered at the position of wire. The strength of this magnetic field �, at a radial distance � away from the wire can be deduced from Ampere's law to be

� =���

2��

From Eq. (4.2), one may state that the behavior of � with respect to � is linear with a slope

equal to ��

���. Exploiting this fact, we will try to acquire an experimental value for

��

��.

EXPERIMENTAL PROCEDURE 1) The apparatus is already set up in the lab (figure 4.1). Before switching on the power supply, turn on the Teslameter and calibrate it to zero. The slight likely fluctuation of the value displayed by the Teslameter is due to its sensitivity to the unrelated peripheral magnetic fields (e.g. the magnetic field of the Earth or your mobile phones) and is rather ignorable.

4.1

4.2

Figure 4. 1

42

2) Bring the probe next to the bandaged part of wired wooden frame. 3) Switch the power supply on and adjust the value of current such that the current reading shows 1.5 �. Since the wire is winded ten times around the wooden frame, this starting value has to be multiplied by 10 and recorded as 15 � in the table provided in Raw Data section. You must repeat the same procedure for each reading, before quoting in your table. 4) Register the corresponding value for � into the table. Note that this reading is in millitesla and will need be converted into tesla during data analysis. You may start taking data by setting the current on some initial value (e.g. 15 � is a good starting point), and then you may increase the current (in steps of 5�). 5) Record the strength of magnetic field corresponding to each current value. This can be observed from the display of the Teslameter and written down in the following table. RAW DATA

� [�] 15 20 25 30 35 40 45 50

� [��] DATA ANALYSIS 1) Transfer the data from the first table accordingly to the second one stated below and fill in the remaining columns by computing the required calculations.

�(= �) [�] �(= �) [�] �� [��] �� [��]

∑� = ∑� = ∑�� = ∑�� =

The least-squares line is given by

� = �� + � in which

� =� ∑ �� ∑ � ∑ �

� ∑ �� (∑ �)�

and

� =∑ � ∑ �� ∑ � ∑ ��

� ∑ �� (∑ �)�.

where � stands for the number of measurements (for this experiment, � = 8 by default).

4.3

4.4

4.5

43

2) Draw the least-squares line � = �� + �, by substituting � and � values you have already calculated. To this end, pick any two �-values from the first column of the table in this section and plug into the equation of the least-squares line to obtain two corresponding �-values. This will give you two points which can then be plotted onto the graph paper using a proper scaling. Then, show your raw data points on the graph paper you are provided with. Make sure that you specify the scaling you have used and also do not forget to label your axes (together with the concerned units).

3) Determine the experimental value of ��

��. Call this �

��

���

������������. Show your steps clearly.

4) Compare ���

���

������������and �

��

���

�������� using the definition of the percentage error

given below.

%����� =

����4��

���������

��4��

�������������

���

4����������

× 100%

5) List at least two sources of errors in your experiment.

4.6

44

Experiment #5: Magnetic Induction

PURPOSE

To verify Faraday’s law of induction and determine an experimental value for ��

��.

APPARATUS A function generator, two solenoids, an ammeter and a digital voltmeter. THEORY In 1820, the Danish physicist Oersted noticed that a current carrying wire produces a magnetic field. Not so long after, Michael Faraday asked the converse question to himself: Can current be produced by a magnetic field in the vicinity of any power supply? He figured out that the answer is yes, but only if the magnetic flux through the current loop changes by time. Faraday realized that once a closed conducting loop, (figure 5.1), is situated in a region containing a static magnetic field (i.e. a magnetic field independent of time), the voltmeter will show zero volts. In other words, not only the current in the loop, but also the potential difference between the ends of the loop will be zero.

Figure 5.1

On the other hand, if the magnetic field is not static (i.e. if it is time-dependent), the voltmeter will show a non-zero voltage. The non-zero voltage value indicates that there exists a current flowing through the loop. Namely, there is a potential difference between the ends of the loop, which is called the induced electromotive force and is abbreviated as ��� or �. The definition of this quantity also represents Faraday’s law which can be written as

� =�Φ�

��

where Φ� stands for the magnetic flux through the loop. The negative sign in Faraday’s law tells us that the flux produced by the induced current opposes the change in the external flux. This result is called Lenz’s law, and often provides an easy way to find the direction of the induced current or voltage. It is worth mentioning that the magnetic flux can be evaluated via

Ψ = � �� . �� = ���.

5.1

5.2

45

To gain some intuition about the flux, consider a loop placed in a uniform magnetic field �� . In this case, the flux through the loop will be equivalent to

� = �� cos �,

where � and � represent the area of the loop and the angle between �� and the normal to the plane of the loop, respectively. Thus, one may define flux as the amount of magnetic field which passes through the loop perpendicularly. The instrument that we will use in this experiment is shown in figure (5.2). Inside a long solenoid, the magnetic field can be obtained by

� = ������ sin(2���),

where �� is the amplitude of the current flowing through the windings, �� shows the number of windings per unit length of the outer solenoid and � represents the frequency of the function generator. Therefore, when an induction coil (inner solenoid) of �� windings, whose area is represented by the � is placed inside the solenoid, the induced ��� in the induction coil can be calculated via

��� = ���������(2��) cos(2���).

Hence, the relation between the induced ��� and the current output [� = ��cos(2���)] can be written as

��� = �������(2��)�.

EXPERIMENTAL PROCEDURE 1) Prior to starting the experiment, it is worthwhile to mention that AC current, which is a function of time, will be applied to the outer solenoid. Using AC current creates a magnetic flux through a loop. In a sense, the magnetic field goes straight through the outer solenoid. Hence, when you insert the inner solenoid (induction coil) into the outer solenoid, a current is induced on the inner solenoid. Consequently, you can observe that the digital voltmeter will display a non-zero value. 2) Your lab Assistant will show you how to use the function generator to fix the frequency of the AC current to 10.7 × 10����. Having done so, you must alter the amplitude of the current.

Figure 5.2

5.3

5.6

5.43

5.5

46

During this step, you must make yourself familiar with reading the ammeter and the digital voltmeter. 3) Read the windings �� and �� written on the outer and inner solenoids, respectively. On the inner solenoid, there exists a � symbol. Here, � indicates the diameter, in millimeters, of the inner solenoid. In order to evaluate the cross-sectional area, � (in ��), of the induction coil, you should use the following formula.

� = � ��

2× 10���

�

.

4) Now, you may start taking your data by setting the current on some initial value (e.g. 20 �� is a good starting point) 5) Increase the current in steps of 10 �. For each current value, you are expected to read the corresponding ��� value from the display of the digital voltmeter. RAW DATA

� [��] 20 30 40 50 60 70 80 90

��� [�] DATA ANALYSIS 1) Transfer the data from the first table accordingly to the second one stated below and fill in the remaining columns by computing the required calculations.

�(= �) [�] �(= ���) [�] �� [��] �� [��]

∑� = ∑� = ∑�� = ∑�� =

The least-squares line is given by

� = �� + � in which

� =� ∑ �� ∑ � ∑ �

� ∑ �� (∑ �)�

and

� =∑ � ∑ �� ∑ � ∑ ��

� ∑ �� (∑ �)�.

where � stands for the number of measurements (for this experiment, � = 8 by default).

5.7

5.8

5.9

5.10

47

2) Draw the least-squares line � = �� + �, by substituting � and � values you have already calculated. To this end, pick any two �-values from the first column of the table in this section and plug into the equation of the least-squares line to obtain two corresponding �-values. This will give you two points which can then be plotted onto the graph paper using a proper scaling. Then, show your raw data points on the graph paper you are provided with. Make sure that you specify the scaling you have used and also do not forget to label your axes (together with the concerned units).

3) Determine the experimental value of ��

��. Call this �

��

���

������������. Show your steps clearly.

4) Compare ���

���

������������and �

��

���

�������� using the definition of the percentage error given

below.

%����� =

����

4����������

���

4��������������

�

���

4����������

× 100%

5) List at least two sources of errors in your experiment.

5.11