Physics Laboratory Manual

UMM AL-QURA UNIVERSITY - College of Applied Science- Department of Physics

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Physics

Laboratory Manual Preparedby

CommitteeofLaboratories

General Physics Lab. (102)


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Introduction (1): Safety and importance of laboratory work

Objective: By the end of this lesson, the student should know the followings:

I- Safety and Security in the Laboratory.

1. The Name of the supervisor of the Lab.

2. The telephone No.s of the emergency within the faculty and the University and Kingdom of Saudi Arabia.

3. The place of the first aid bag and how to use it

4. The place of Fire-extinguisher and how to use it

5. How to exit the Lab and the building in an emergency

6. How to give the first aid to the others

7. How to protect your eyes

8. How to protect your skin

9. The general instruction of the safety and security

II- Introduction to the Laboratory.

1. Aim of the Experiments

2. The importance of the experimental work

3. General Instructions for Performing Experiments

4. How to Record an Experiment in the Practical File

Aim of the experiments

In Physics, an experiment is an empirical procedure that arbitrates between competing

models or hypotheses. Researchers also use experimentation to test existing theories or

new hypotheses to support or disprove them. Experiments form the foundation of the

growth and development of science. The chief aim of experimentation in science is:


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1- Discovering the law which governs a certain phenomenon,

2- Verifying a given law which has been derived from a theory

3- Determination of physical constants

4- Determination of the physical properties

A general scheme of scientific investigation known as the Scientific Method involves

the following steps:

1- Observations: Qualitative information about a phenomenon collected by unaided senses.

2- Experimentation: Quantitative measurements (with the help of instruments) of certain physical quantities which have some bearing on the phenomenon.

3- Formulation of hypothesis: Analysis of the data to determine how various measured quantities affect the phenomenon and to establish a relationship between them, graphically or otherwise.

4- Verification: The hypothesis is verified by applying it to other allied phenomena.

5- Predictions of new phenomena.

6- New experiments to test the predictions.

7- Modification of the law if necessary

The above discussion shows that experimentation is vital to the development of any

kind of science and more so to that of Physics.

Importance of Laboratory work

Physics is an experimental science and the history of science reveals the fact that most

of the notable discoveries in science have been made in the laboratory. Seeing

experiments being performed, i.e., demonstration experiments are important for

understanding the principles of science. However, performing experiments by one’s

own hands is far more important because it involves learning by doing. It is needless to

emphasize that for a systematic and scientific training of a young mind; a genuine

laboratory practice is a must.

General Instructions for performing experiments


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1. Before performing an experiment, the student should first thoroughly understand the

theory of the experiment. The objectives of the experiment, the type of apparatus

needed and the procedure to be followed should be clear before actually performing

the experiment. The difficulties and doubts if any, should be discussed with the

supervisor.

2. The student should check up whether the right type of apparatus for the experiment

to be performed is given to him or not.

3. All the apparatus should be arranged on the table in proper order. Every apparatus

should be handled carefully and cautiously to avoid any damage. Any damage or

breaking done to the apparatus accidentally, should be immediately brought to the

notice.

4. Precautions meant for the experiment should not only be read and written in the

practical file, but they are to be actually observed while doing the experiment.

5. All observations should be taken systematically, intelligently and should be honestly

recorded on the fair record book. In no case an attempt should be made to cook or

change the observations in order to get good results.

6. Repeat every observation, number of times even though their values each time may

be exactly the same. The student must bear in mind the proper plan for recording the

observations.

7. Calculations should be neatly shown using log tables. The degree of accuracy of the

measurement of each quantity should always be kept in mind so that the final result

does not show any fictitious accuracy. So the result obtained should be suitably

rounded off.

8. Wherever possible, the observations should be represented with the help of graph.

9. Always mention the proper unit with the result.


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Introduction (2): General Instruction

Objective: This lesson is an introduction to the experimental work. By the end of this

lesson, the student should know the followings :

1. Error and Observations

2. Accuracy of Observations

3. Accuracy of the Result

4. Permissible Error in the Result

5. How to Estimate the Permissible Error in the Result

6. Estimating Maximum Permissible Error

7. Percentage Error

8. Significant Figures (Precision of Measurement (

9. Logarithms

10. Graph

11. Calculations of Slope of a Straight Line

12. Rules for Measurements in the Laboratory

13. How to write a Lab report.

1- Errors and Observations

We come across following errors during the course of an experiment:

Personal or chance error: Two observers using the same experimental set up,

do not obtain exactly the same result. Even the observations of a single

experimenter differ when it is repeated several times by him or her. Such errors

always occur inspite of the best and honest efforts on the part of the

experimenter and are known as personal errors. These errors are also called

chance errors as they depend upon chance. The effect of the chance error on the

result can be considerably reduced by taking a large number of observations

and then taking their mean.


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Error due to external causes: These are the errors which arise due to reasons

beyond the control of the experimenter, e.g., change in room temperature,

atmospheric pressure etc. A suitable correction can however, be applied for

these errors if the factors affecting the result are also recorded.

Instrumental errors: Every instrument, however cautiously designed or

manufactured, possesses imperfection to some extent. As a result of this

imperfection, the measurements with the instrument cannot be free from errors.

Errors, however small, do occur owing to the inherent manufacturing defects in

the measuring instruments. Such errors which arise owing to inherent

manufacturing defects in the measuring instruments are called instrumental errors. These errors are of constant magnitude and suitable corrections can be

applied for these errors.

2- Accuracy of Observations

The manner, in which an observation is recorded, indicates how accurately the physical

quantity has been measured. For example, if a measured quantity is recorded as 50 cm,

it implies that it has been measured correct ‘to the nearest cm’. It means that the

measuring instrument employed for the purpose has the least count (L.C.) = 1 cm .

Since the error in the measured quantity is half of the L.C. of the measuring

instrument, therefore, here the error is 0.5 cm in 50 cm. In other words we can say, the

error is 1 part in 100.

If the observation is recorded in another way, i.e., 50.0 cm, it implies that it is correct

‘to the nearest mm. As explained above, the error now becomes 0.5 mm or 0.05 cm in

50 cm. So the error is 1 part in 1000.

If the same observation is recorded as 50.00 cm. It implies that reading has been taken

with an instrument whose L.C. is 0.01 cm. Hence the error here becomes 0.005 cm in

50 cm, i.e., 1 part in 10,000.

It may be noted that with the decrease in the L.C. of the measuring instrument, the

error in measurement decreases, in other word accuracy of measurement increases.

When we say the error in the measurement of a quantity is 1 part in 1000, we can also

say that the accuracy of the measurement is 1 part in 1000. Both the statements mean

the same thing. Thus, from the above discussion it follows that:


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a) The accuracy of measurement increases with the decrease in the least count of the

measuring instrument; and

b) The manner of recording an observation indicates the accuracy of its measurement

3- Accuracy of the Result

The accuracy of the final result is always governed by the accuracy of the least

accurate observation involved in the experiment. So after making a calculation, the

result should be expressed in such a manner that it does not show any superfluous

accuracy. Actually the result should be expressed up to that decimal place (after

rounding off) which indicates the same accuracy of measurement as that of the least

accurate observation made.

4- Permissible error in the Result Even under ideal conditions in which personal errors, instrumental errors and errors

due to external causes are some how absent, there is another type of error which creeps

into the Introductory observations because of the limitation put on the accuracy of the

measuring instruments by their least counts. This error is known as the permissible

error.

5- How to estimate the permissible error in the result

Case I: When the formula for the quantity to be determined involves the product of

only first power of the measured quantities: Suppose in an experiment, there are only

two measured quantities say p and q and the resultant quantity s is obtained as the

product of p and q, such that q.ps (1)

Let p and q be the permissible errors in the measurement of p and q respectively.

Let s be the maximum permissible error in the resultant quantity s. Then

q.pp.qq.pq.p)qq(.)pp()ss(

(2)

q.pq.pp.qq.pq.ps)ss(


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q.pp.qq.ps (3)

The product of two very small quantities, i.e. (the product p.q) is negligibly small as

compared to other quantities, so the equation (3) can be written as

p.qq.ps (4)

For getting maximum permissible error in the result, the positive signs with the

individual errors should be retained so that the errors get added up to give the

maximum effect. Thus, equation (4) becomes

p.qq.ps (5)

Dividing L.H.S. by s and R.H.S. by the product p. q, we get

q.pp.q

q.pq.p

ss

pp

qq

ss

.max

(6)

Expressing the maximum permissible error in terms of percentage, we get

%100pp

qq

ss

.max

(7)

The result expressed by equation (7) can also be obtained by logarithmic

differentiation of relation (1). This is done as follows:

On taking log of both the sides of the equation (1), one gets

qlogplogslog (8)

On differentiating (8), one gets


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pp

qq

ss

(9)

The result (6) and (9) are essentially the same.

Case II: When the formula for the physical quantity to be determined contains higher

powers of various measured quantities. Let

cba r.q.ps (10)

Then taking log of both sides of (10), we have

rlogcqlogbplogaslog (11)

On differentiating equation (11), we get

rrc

qqb

ppa

ss

(12)

Thus

%100rrc

qqb

ppa

ss

.max

(13)

Since maximum permissible error can be conveniently estimated by logarithmic

differentiation of the formula for the required quantity, so, the maximum permissible

error is also called as the Maximum Log Error.

In actual practice the maximum permissible error is computed by logarithmic

differentiation method.

6- Estimating the maximum permissible error

For determination of the resistivity of a material, the formula used is

ldR 2


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The maximum permissible error is computed as follows. The resistivity is a function of

three variables R, d and l. Taking logs of both sides, we get

lloglogdlog2Rloglog

Differentiating with respect to the variable itself, we get

)constnatisce(sinll0

dd2

RR

Changing the negative sign into positive sign for determining maximum error, we have

ll

dd2

RR

The error is maximized due to error in physical quantity occurring with the highest

power in the working formula.

In an experiment, the various measurements were as follows:

R = 1.05 , R = 0.01

d = 0.60 mm, d = 0.01 mm = least count of the screw gauge

l = 75.3 cm, l = 0.1 cm = least count of the meter scale

0.044 = 0.0442 = 0.0013 + 0.0334 + 0.0095 =3.751.0

6.001.2

05.101.0

The permissible error, in in the above case is

4.4%1000.044 =

Out of which 3.3% is due to error in d.

7- Absolute error


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The absolute error (X) is the difference between the mean value of the experimentally

determined quantity (Xm) and the standard (correct) value of the quantity (XS). It is

done as follows:

value standard - valuemeasured=Error Absolute

Sm X - X=XErrorAbsolute

8- Relative error

The relative error is the ratio between the absolute error (X) to the standard (correct)

value of the quantity (XS), i.e.,

valuestandard

Error Percentage=Error Relative

9- Percentage error

The mean value of the experimentally determined quantity is compared with the

standard (correct) value of the quantity. It is done as follows:

100 valuestandard

valuestandard - valuemeasured=Error Percentage

100ErrorRelative=Error Percentage

10- Significant figures (precision of measurement)

No measurement of any physical quantity is absolutely correct. The numerical value

obtained after measurement is just an approximation. As such, it becomes quite

important to indicate the degree of accuracy (or precision) in the measurement done in

the experiment. Scientists have developed a kind of shorthand to communicate the

precision of a measurement made in an experiment. The concept of significant figures

helps in achieving this objective. To appreciate and understand the meaning of

significant figures let us consider that in an experiment, the measured length of an

object is recorded as 14.8 cm. The recording of length as 14.8 cm means (by

convention) that the length has been measured by an instrument accurate to one-tenth

of a centimeter. It means that the measured length lies between 14.75 cm and 14.85


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cm. It also indicates that in this way recording of lengths as 14.8 cm, the figures 1 and

4 are absolutely correct whereas the figures ‘8’ is reasonably correct. So in this way of

recording a reading of a measurement, there are three significant figures. Let us now

consider another way of recording a reading. Let the measured length be written as

14.83 cm. This way of writing the value of a length, means that the measurement is

done with an instrument which is accurate up to one-hundredth of a centimeter. It

means that the length lies between 14.835 cm and 14.825 cm, which shows that in

14.83 cm, the figures 1, 4 and 8 are absolutely correct and the fourth figure ‘3’ is only

reasonably correct. Thus, this way of recording length as 14.83 cm contains four

significant figures. Thus, this significant figure is a measured quantity to indicate the

number of digits in which we have confidence. In the above measurement of length,

the first measurement (14.8 cm) is good to three significant figures, whereas the second

one (14.83 cm) is good to four significant figures. From the above discussion, we

should clearly understand that the two ways of recording an observation such as 15.8

cm and 15.80 cm represent two different degrees of precision of measurements.

ROUNDING OFF

When the quantities with different degrees of precision are to be added or subtracted,

then the quantities should be rounded off in such a way that all of them are accurate up

to the same place of decimal. For rounding off the numerical values of various

quantities, the following points are noted.

1. When the digit to be dropped is more than 5, then the next digit to be retained should

be increased by 1.

2. When the digit to be dropped is less than 5, then the next digit should be retained as

it is, without changing it.

3. When the digit to be dropped happens to be the digit 5 itself, then (a) the next digit

to be retained is increased by 1, if the digit is an odd number. (b) the next digit is

retained as it is, if the digit is an even number.

After carrying out the operations of multiplication and division, the final result should

be rounded off in such a manner that its accuracy is the same as that of the least

accurate quantity involved in the operation.


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11- Graph

A graph is a pictorial way to show how two physical quantities are related. It is a

numerical device dealing not in cm, ohm, time, temperature, etc. but with the

numerical magnitude of these quantities. Two varying quantities called, the variables

are the essential features of a graph.

Purpose: To show how one quantity varies with the change in the other. The quantity

which is made to change at will, is known as the independent variable and the other

quantity which varies as a result of this change is known as the dependent variable.

The essential features of the experimental observations can be easily seen at a glance if

they are represented by a suitable graph. The graph may be a straight line or a curved

line.

The advantage of Graph: The most important advantage of a graph is that, the average

value of a physical quantity under investigation can be obtained very conveniently

from it without resorting to lengthy numerical computations. Another important

advantage of the graph is that some salient features of a given experimental data can be

seen visually. For example, the points of maxima or minima or inflecion can be easily

known by simply having a careful look at the graph representing the experimental data.

These points cannot so easily be concluded by merely looking at the data. Whenever

possible, the results of an experiment should be presented in a graphical form. As far as

possible, a straight line graph should be used because a straight line is more

conveniently drawn and the deduction from such a graph are more reliable than from a

curved line graph.

Each point on a graph is an actual observation. So it should either be encircled or be

made as an intersection (i.e. cross) of two small lines. The departure of the point from

the graph is a measure of the experimental error in that observation.

How to Plot a Graph? The following points will be found useful for drawing a proper

graph:

1. Examine carefully the experimental data and note the range of variations of the two

variables to be plotted. Also examine the number of divisions available on the two axes

drawn on the graph paper. After doing so, make a suitable choice of scales for the two


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axes keeping in mind that the resulting graph should practically cover almost the entire

portion of the graph paper.

2. Write properly chosen scales for the two axes on the top of the graph paper or at

some suitable place. Draw an arrow head along each axis and write the symbol used

for the corresponding variable along with its unit as headings of observations in the

table, namely, d/mm, R/W, l/cm, T/S, I/A etc. Also write the values of the respective

variables on the divisions marked by dark lines along the axes.

3. After plotting the points encircle them. When the points plotted happen to lie almost

on a straight line, the straight line should be drawn using a sharp pencil and a straight

edged ruler and care should be taken to ensure that the straight line passes through the

maximum number of points and the remaining points are almost evenly distributed on

both sides of the line.

4. If the plotted points do not lie on a straight line, draw freehand smooth curve passing

through the maximum number of points. Owing to errors occurring in the observations,

some of the points may not fall exactly on the free hand curve. So while drawing a

smooth curve, care should be exercised to see that such points are more or less evenly

distributed on both sides of the curve.

5. When the plotted points do not appear to lie on a straight line, a smooth curve is

drawn with the help of a device known as French curve. If French curve is not

available, a thin flexible spoke of a broom can also be used for drawing a smooth

curve. To make the spoke uniformly thin throughout its length, it is peeled off suitably

with a knife. This flexible spoke is then held between the two fingers of left hand and

placed on the graph paper bending it suitably with the pressure of the fingers is such a

manner that the spoke in the curved position passes through the maximum number of

points. The remaining points should be more or less evenly distributed on both sides of

the curved spoke. In this bent position of the spoke, a smooth line along the length of

the spoke is drawn using a sharp pencil.

6. A proper title should be given to the graph thus plotted.

7. Preferably a millimeter graph paper should be used to obtain greater accuracy in the

result.


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12- Calculations of slope of a straight line

In order to compute the value of slope m of the straight line graph, two points P (x1, y1)

and Q (x2, y2) widely separated on the straight line are chosen. PR and QS are drawn

to x-axis and PN QS.

The slope of the line

12

12

xxyy

PNQN= m

This method of calculating the slope emphasizes the important fact that: QN and PN

which are (y2 – y1) and (x2 – x1) must be measured according to the particular scales

chosen along y- and x-axis respectively.

13- Rules for measurements in the laboratory

Measurements of most of the physical quantities in the laboratory should be done in

the most convenient units, e.g., the mass of a body in gram, measurements using a

micrometer screw in mm, small currents in electronic tubes, diode and triode in mA…

Calculations: All the measured quantities must be converted into S.I. units before

substituting in the formula for the calculation of the result.

14- How to write a Lab report

y

xO R

N

S

P

Q

y1

y2

x1 x2


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A neat and systematic recording of the experiment in the practical file is very

important in achieving the success of the experimental investigations. The students

may write the experiment under the following heads in their fair practical notebooks.

Date: ............. , Experiment Name: ..............

Experiment No.: ……………….. , Page No.: ..............

1. Object/Aim: The object of the experiment to be performed should be clearly and

precisely stated.

2. Apparatus used: The main apparatus needed for the experiment are to be given

under this head. If any special assembly of apparatus is needed for the experiment, its

description should also be given in brief and its diagram should be drawn.

Diagram: A circuit diagram for electricity experiments and a ray diagram for light

experiment is a must. Simply principle diagram wherever needed, should be drawn

neatly.

3. Theory: The principle underlying the experiment should be mentioned.

4. The Formula used: The formula used should also be written explaining clearly the

symbols involved. Derivation of the formula may not be required.

5. Procedure/Method: The various steps to be followed in setting the apparatus and

taking the measurements should be written in the right order as per the requirement of

the experiment.

6. Observation: The observations and their recording, is the heart of the experiment.

As far as possible, the observations should be recorded in the tabular form neatly and

without any over-writing. In case of wrong entry, the wrong reading should be scored

by drawing a horizontal line over it and the correct reading should be written by its

side. On the top of the observation table the least counts and ranges of various

measuring instruments used should be clearly given. If the result of the experiment

depends upon certain environmental conditions like temperature; pressure, place, etc.,

then the values of these factors should also be mentioned.


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7. Calculations: The observed values of various quantities should be substituted in the

formula and the computations should be done systematically and neatly. Wherever

possible, graphical method for obtaining results should be employed.

8. The Results: The conclusion drawn from the experimental observations has to be

stated under this heading. If the result is in the form of a numerical value of a physical

quantity, it should be expressed in its proper unit. Also mention the physical conditions

like temperature, pressure, etc., if the result happens to depend upon them.

9. Error: Percentage error may also be calculated if the standard value of the result is

known.

10. Sources of error and precaution: The possible errors which are beyond the control

of the experimenter and which affect the result should be mentioned here. The

precautions which are actually observed during the course of the experiment should be

mentioned under this heading.

Exercise

1- Draw the following data in graphs

2- Calculate the slope of each line.

3- Determine the constants b and a in the following equation using the graph. Use

the data of columns C and D.

bxay

A B C D

x y x y x y x y

1 2 7 21 2 1 25 31.4

2 4 8 24 4 2 30 34.39697661

3 6 9 27 6 3 35 37.15298104

4 8 10 30 8 4 40 39.71820741

5 10 11 33 10 5 45 42.1275207

6 12 12 36 12 6 50 44.40630586


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Experiment: THE SIMPLE PENDULUM

Objective: The study of the physical properties of the basic simple pendulum

Apparatus: String, pendulum bob, meter stick and Stop watch

Theory: A simple pendulum consists of a

small bob suspended by a wire of length L

(massless) fixed at its upper end. When pulled

back and released, the mass swings through

its equilibrium (center) point to a point equal

in height to the release point, and back to the

original release point over the same path. The

force that keeps the pendulum bob constantly

moving toward its equilibrium position is the

force of gravity acting on the bob. The period,

T, of an object in simple harmonic motion is

defined as the time for one complete cycle.

For small angles (θ < ~15°), it can be shown that the period of a simple pendulum is

given by:

Where g is the acceleration due to gravity, 9.8 m/s2. Equation 1 indicates that the

period and length of the pendulum are directly proportional; that is, as the length, L, of

a pendulum is increased, so will its period, T, increase. However, it is not a linear

relationship. The period increases as the square root of the length. Thus, if the length of

a pendulum is increased by a factor of 4, the period is only doubled. This is a

logarithmic relationship. A more general form of equation 1 is:


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Whereg

2k and the exponent n is ½. Rearranging this expression for k yields:

Taking the log10 of both sides of equation 2 yields:

Comparing this to y = mx + b (the equation of a straight line), we can see that if the

period vs. the length of the pendulum were plotted on a graph with logarithmic axes,

then the slope of the line would equal n and the y-intercept would be equal to the value

of log10 k. Also from equation 2, it can be seen that for a pendulum whose length were

1 m (L = 1), then Ln = 1. Therefore,

Procedure:

1. part I : Dependence of Period on Mass

A mass is attached to the other end of the string and pulled back and let go, so that it

executes (approximately) simple harmonic motion. The time required to complete 50

cycles (t) is measured with a stopwatch and recorded. To improve accuracy, three trials

are completed for each measurement. The average of the recorded values of t for the

three trials is then divided by 50 to obtain the period (T) of the motion. In the first part

of the experiment, the length L of the string is kept constant at 30 cm and different masses are attached to its end to determine the dependence of the period on mass. All

times are measured in seconds. The mass is measured in grams, and then converted to

kilograms.

a) Record the data in the following format:


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m(g) m(Kg) t1(s) t2(s) t3(s) tavg (s) T= tavg/50

Table 1: Period T vs. Mass m for Simple Pendulum. ( L= 30 cm)

b) Plot T vs. m. (fig.1)

c) From the graph (question b), the period of a simple pendulum is dependent of

mass?

2. part II : Dependence of Period on Length

The mass (m) is held constant at 100 g and the length (L) is varied from 30 cm to 100 cm, in 10 cm increments. Times are measured and period calculated as in part I.

As before, all times are in seconds. Lengths are measured in centimeters, then

converted to m.


L(cm) L(m) t1(s) t2(s) t3(s) tavg (s) T(s) 30 40 50 60 70 80 90 100

Table 2: Period vs. Length for Simple Pendulum. (m = 100g)

b) Plot T vs. L. (fig.2).

c) From the graph (question b), the relationship between T and L is linear? Can

you deduce the value of n and k? (see eq. 2)

d) To determine the values of n and k, we utilize eq.4. Complete the table below:


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L(m) T (s) log (L) log(T)

Table 3: log (L) vs. log(T) for Simple Pendulum. (m = 100g)

e) Plot log(T) vs. log(L). fig (3).

f) From fig.3, find the value of k. Explain whether this agrees with the value of k

reported from estimating the period at a length of 1 m?

g) From fig.3, find the value of n.

Find the sources of Error in this experiment.

Make a conclusion.


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Experiment: TORSION PENDULUM

Objective: To determine the torsion constant, the rigidity modulus of the material of

the rod and the moment of inertia of the metallic disc.

Apparatus: Torsion pendulum, annular ring, meter stick and Stop watch.

Theory: A torsional pendulum, or torsional

oscillator, consists of a disk suspended from a

thin rod or wire. When the mass is twisted

about the axis of the wire, the wire exerts a

torque on the mass, tending to rotate it back to

its original position. If twisted and released,

the mass will oscillate back and forth,

executing simple harmonic motion. This is the

angular version of the bouncing mass hanging

from a spring. This gives us an idea of

moment of inertia. We try to calculate the

moment of inertia of a ring given the moment

of a disc. We can also compare the

experimental values with theoretically

calculated values.

The working is based on the torsional simple harmonic oscillation with the analogue of

displacement replaced by Angular displacement θ, Force by Torque τ and the torsion constant κ. For a given small twist θ (sufficiently small), the experienced

reaction is given by

The minus sign indicates that the direction of the torque vector is opposite to the angle

vector, so the torque tends to undo the twist. This is just like Hooke’s Law for springs.


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If a mass with moment of inertia I is attached to the rod, the torque will give the mass

an angular acceleration α according to

Combining (1) and (2) yields the equation of motion for the torsional pendulum,

Hence on solving this second order differential equation we get

The period T of oscillation is given by:

The torsion constant κ can be determined from measurements of T if I is known:

The torsion constant κ can be determined also from measurements of the radius r of

the rod, the length l of the rod and the constant n (shear modulus or modulus of rigidity):

l2rnk)7(

4

Procedure:

3. Part I : Determination of the torsion constant κ and the shear modulus n.

Hang the disc alone and give a small angular displacement to the system and leave it to

oscillate. The time required to complete 50 cycles (t) is measured with a stopwatch and

recorded. To improve accuracy, three trials are completed for each measurement. The


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average of the recorded values of t for the three trials is then divided by 50 to obtain

the period (T) of the motion.

d) Record the data in the following format:

t1(s) t2(s) t3(s) tavg (s) T= tavg/50

Table 1: Period of oscillation for Torsion Pendulum (disk only).

e) Measure the theoretical values of the moment of inertia of the disk. Record

below.

Mdisk = …………………….Kg

Rdisk = …………………….m

2diskdiskdisk RM

21.)theo(I = ………………………….

f) Calculate the torsion constant κ (From eq. 6)

The torsion constant κ = …………………………….

g) Calculate the modulus of rigidity n :

rrod = …………………….m

lrod = …………………….m

The modulus of rigidity n (From eq.7) = …………………………….

4. Part II : Determination of the moment of inertia of the ring.

Now hang the ring along with disc and follow the same

procedure as before to find the Time Periods Tdisk + ring. .

We can find the theoretical values of the moment of

inertia of ring by knowing the mass and the radius by the

following equations:

)RR(M21.)theo(I 2

ring22ring1ringring

h) Record the data in the following format:


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t1(s) t2(s) t3(s) tavg (s) Tdisk + ring (s)

Table 2: Period of oscillation for Torsion Pendulum (disk+ring).

i) Use equation 5 (k

I2T ring +disk

ring +disk ) to measure the experimental values

of the moment of inertia of the ring. Record below.

.)(expIring = ………………………….

j) Calculate the theoretical values of the moment of inertia of the ring. Record

below.

Mring = …………………….Kg

R1ring = …………………….m

R2ring = …………………….m

.)theo(Iring = ………………………….

The percentage error of the moment of inertia of the ring is ………………


Make a conclusion.


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Experiment: MOMENT OF INERTIA

Objective: To measure the moments of inertia of several objects

Apparatus: Torsion pendulum, several objects, meter stick and Stop watch.

Theory: For a point mass, like a very

small rock tied to a string, the moment

of inertia is

Where m is the mass of the object and r

is the distance from the mass to the

center of rotation. If several masses

rotate about an axis, like a cheerleader's

baton, the net moment of inertia is

simply the sum of the individual

moments such that

Figure 1

Moments of inertia can be added and subtracted as long as they are about the same axis

of rotation. For an extended body, the summation becomes an integral and the moment

of inertia becomes

Where r is the distance from each mass element, dm, to the axis of rotation. For an

extended object, r is variable and dm must be expressed as a function of r so that the

expression can be integrated.


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A torque will be applied to the base by putting weight on a string that winds around the

axe. Applying Newton’s Second Law to the masses allows us to show that the total

tension in the string is related to the acceleration of the masses by: T = mg – ma. The

moment of inertia of each system can be experimentally found using the formula:

)1ag(RmI 2

Procedure:

The setup for the experiment is depicted in Figure 1. A string is tied around one hub of

the pulley, and the other end is tied to a weight hanger, where we suspend (including

the hanger) masses of roughly 10 g, 25 g and 50 g. The mass is allowed to fall a

distance h.

We have 2 bodies:

A thin rod with constant cross-section

The moment of inertia 2barbarbar lm

121I

A parallelepiped The moment of inertia

)bl(m121I 2

.par2

.par.par.par

5. Part I : Determine the moment of inertia of the bar

k) Put the bar above the disk.

l) Complete the following table.

m(Kg) y(m) t1(s) t2(s) t3(s) tavg (s)

m) Calculate the acceleration a. a=…………………………m/s2

l

l


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n) From equation: )1ag(RmI 2

diskbardisk , bardiskI =…………………Kg m2

o) 2disk

2diskbardiskbar mKg.................Rm

21II

p) Calculate the absolute and the percentage error.

q) Remove the bar from the rotating assembly. Replace it with the parallelepiped.

r) Repeat the previous steps to determine the moment of inertia of the parallelepiped.


Make a conclusion


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Experiment: PROJECTILES MOTION

Objective: Study two dimensional projectile motion of an object in free fall

Apparatus: Projectile launcher, plastic ball, carbon paper and meter stick

Theory: In this lab we will study two dimensional projectile motion of an object in

free fall - that is, an object that is launched into the air and then moves under the

influence of gravity alone. Examples of projectiles include rockets, baseballs,

fireworks, and the steel balls that will be used in this lab. To describe projectile

motion, such as the trajectory (path), we will use a coordinate system where the y-axis

is vertically upward and the x-axis is horizontal and in the direction of the initial

launch (or initial velocity). To simplify projectile motion, we assume that the

gravitational acceleration (g = 9.8 m/s2) is constant, such that ax = 0 and ay = −g, and

we will ignore any air resistance. The equations of motion in the x and y directions for

a projectile launched with a velocity v0 at an θ are given as:


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Procedure:

a) Measure the time ts required for the ball travels a distance ab = 10 cm (see fig1)

b) From eq.2, measure the initial velocity vx0.

c) The carbon paper will be used to mark where the projectile lands, giving you an

measurement of the total horizontal distance traveled (xexp). To improve

accuracy, three trials are completed for measurement. Measure the xexp.

d) From eq. 1 and 4, calculate x theoretical (xtheo.).

e) Repeat steps for different values of h.

f) Record the data in the following format:

h(m) ts(s) vx0 (m/s) xexp (m) ttheo (s) calculated from eq.4

xtheo (m) calculated from eq.2

g) Plot vx0 vs. xexp. (fig.1).

h) From the graph (question g), find the relationship between vx0 and xexp


Make a conclusion.


31 / 34

Experiment: HOOKE’S LAW

Objective: The purpose of this experiment is to verify Hooke’s law and to determine

the acceleration of gravity.

Apparatus: A spiral spring, a set of weights, a balance, a stop watch, and a meter stick

Theory: An ideal spring is remarkable in the

sense that it is a system where the generated

force is linearly dependent on how far it is

stretched. This behavior is described by

Hooke's law. Hooke’s Law states that to

extend a string by an amount x from its

previous position, one needs a force F which is

determined by F = k x. Here k is the spring

constant.

In our case the external force is determined by attaching a mass M to the end of the

spring. The mass will of course be acted upon by gravity, so the force exerted

downward on the spring will be:

gMF)1(

The distance h (the spring is stretched) and the masse M are proportional (linearly

dependence), and that the constant of proportionality is km:

Mkh)2( m

If the mass M is pulled so that the spring is stretched beyond its equilibrium (resting)

position, the restoring force of the spring will cause an acceleration back toward the

equilibrium position of the spring, and the mass will oscillate in simple harmonic

motion. The period of vibration, T, is defined as the amount of time it takes for one

complete oscillation, and for the system described above is:


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g)mM(k2T)3( m

Where (M+m) is the equivalent mass of the system, that is, the sum of the mass, M,

which hangs from the spring and the spring's equivalent mass, m.

Squaring both sides of the equation 3 yields:

g)mM(k4T)4( m22

Expanding the equation 4:

mg

k4Mg

k4T)5( m2m22

Therefore, if we perform an experiment in which the mass hanging at the end of the

spring (the independent variable) is varied and measure the period squared (T2 ; the

dependent variable), we can plot the data and fit it linearly. Comparing equation 5 to

the equation for a straight line (y = ax + b), we see that the slope and y-intercept,

respectively, of the linear fit is:

Slope = g

k4 m2 and y- intercept = mg

k4 m2

Procedure:

6. Part I: Determine the constant km.

A mass is added to a

vertically hanging rubber

band and the displacement

h is measured with the

addition of each mass.


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s) Record the data in the following format:

M(g) M(Kg) h(m) 50 100 150 200 250 300 350 400

Table 1: The displacement h vs. Mass M

t) Plot h vs. M. (fig.1)

u) From the graph (question b), determine the constant km

7. Part II: Determine the acceleration of gravity g

A mass is attached to the other end of the string and pulled back and let go, so that it

executes (approximately) simple harmonic motion. The time required to complete 20

cycles (t) is measured with a stopwatch and recorded. To improve accuracy, three trials

are completed for each measurement. The average of the recorded values of t for the

three trials is then divided by 20 to obtain the period (T) of the motion. In this part of

the experiment, different masses are attached to the end of spring to determine the

dependence of the period on mass. All times are measured in seconds. The mass is

measured in grams, and then converted to kilograms.


M(g) M(Kg) t1(s) t2(s) t3(s) tavg (s) T(s) T2(s2) 200 250 300 350 400

Table 2: Period T2 vs. Mass M

b) Plot T2 vs. M. (fig.2).


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c) From the graph (question b), the relationship between T2 and M is linear.

Deduce from the slop the value of the acceleration of gravity g and from the

y- intercept the value of the mass m. km is determined in part I.

d) Given that greal= 9.8 m.s2, calculate the absolute and the percentage error.


Make a conclusion.

Physics Laboratory Manual

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