Physics I Laboratory Manual
King Mongkut’s Institute of Technology Ladkrabang
School of Engineering
SIIE
Authors : Supun Dissanayaka
: Ashan Eranga
Course: 01006702/01006724
Version(Draft)
Contents
Title Page . . . . .
Table of Contents
List of Figures . .
List of Tables . . .
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Introduction
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Semester Lab Schedule and Rules and Regulations
1.3 General usage policies of the Lab . . . . . . . . . . . .
1.4 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Before the Lab . . . . . . . . . . . . . . . . . . . . . . . .
1.6 During the Lab . . . . . . . . . . . . . . . . . . . . . . .
1.7 Lab Report . . . . . . . . . . . . . . . . . . . . . . . . . .
1.8 Grading . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Experiments
2.1 Experiment 1: Basic Measurement equipment and Measurement Uncertainty . . . . . . . . . . . . . . . . . . . . . . .
2.2 Experiment 2: Introduction to Vectors . . . . . . . . . . .
2.3 Experiment 3: Linear & Planar motion . . . . . . . . . . . .
2.4 Experiment 4: Newton’s laws of motion . . . . . . . . . . .
2.5 Experiment 5: Friction . . . . . . . . . . . . . . . . . . . . . .
2.6 Experiment 6: Investigating the behaviors of the Spring
systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7 Experiment 7: Simple Harmonic Motion . . . . . . . . . . .
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List of Figures
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
2.14
2.15
2.16
2.17
2.18
2.20
2.21
2.22
2.23
2.24
2.25
2.26
2.27
2.28
2.29
2.30
2.31
Fun with Measurements . . . . . . . . . . . . . . .
Parts of a Vernier Caliper . . . . . . . . . . . . . .
Example of Finding the LC . . . . . . . . . . . . .
Complete reading of a Vernier Caliper . . . . . . .
Parts of Micrometer . . . . . . . . . . . . . . . . .
Measurement reading of a Micrometer . . . . . . .
Dial Height Gauge . . . . . . . . . . . . . . . . . .
Spherometer . . . . . . . . . . . . . . . . . . . . . .
Direction impact on Vectors . . . . . . . . . . . . .
Direction impact on Vectors . . . . . . . . . . . . .
Direction impact on Vectors . . . . . . . . . . . . .
Components of a vector . . . . . . . . . . . . . . .
Vector addition . . . . . . . . . . . . . . . . . . . .
Direction impact on Vectors . . . . . . . . . . . . .
Force table . . . . . . . . . . . . . . . . . . . . . . .
Ticker . . . . . . . . . . . . . . . . . . . . . . . . .
Tic . . . . . . . . . . . . . . . . . . . . . . . . . . .
Tic . . . . . . . . . . . . . . . . . . . . . . . . . . .
Experiment setup . . . . . . . . . . . . . . . . . . .
Direction of the friction force . . . . . . . . . . . .
A certain mass pulley system . . . . . . . . . . . .
A skier moving down a slope . . . . . . . . . . . . .
Spring reaction after adding a mass . . . . . . . . .
Hooke’s law apparatus . . . . . . . . . . . . . . . .
Spring Systems . . . . . . . . . . . . . . . . . . . .
Position plot showing sinusoidal motion of an object
Five Key points of a mass oscillating on a spring . .
Familiar Spring Mass System . . . . . . . . . . . .
Familiar Spring Mass System . . . . . . . . . . . .
Simple Pendulum . . . . . . . . . . . . . . . . . . .
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List of Tables
2.1
2.2
2.3
2.4
2.5
Final results . . . . . . . . . . . . . . . .
Results table . . . . . . . . . . . . . . .
Vibrating mass variation & time period.
Mass of the bob variation & time period.
Length variation & time period. . . . . .
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Chapter 1
Introduction
Purpose
”Life is like riding a bicycle. You must keep moving to keep your balance. So, let us
Keep moving with the physics towards enlightenment”
1.1
Introduction
This lab manual has prepared, referring to your main physics course syllabus. Our labs
would typically emphasize thorough
• preparation,
• an underlying mathematical model of nature,
• good experimental technique and
• analysis of data(including the significance of error).
There is no special prerequisites required other than your creativity and the curiosity
towards the science. You would find here the “instructions” that are not part of real
experiments(where the methods and/or outcomes are not known in advance). However
you would give recommendations and your opinions throughout this course. Eventually
you would be familiar with the concepts of the general physics and be able to apply
them in your engineering applications.
1.1.1
The Goals of our Lab session
• Perform basic measurements and recognize the associated limitations.
• Practice the methods which allow you to determine how uncertainties in measured
quantities propagate to produce uncertainties in calculated quantities.
• Perform certain experiments that illustrate the foundations of Newton’s mechanics.
• Practice the process of verifying a mathematical model, including data collection,
data display, and data analysis (particularly graphical data analysis with curve
fitting).
• Practice the process of keeping an adequate lab notebook.
1
• Gain experience by fiddling with the experiments and finally find ways to get it
work.
• Ability of applying the knowledge that gained from the previous lab sessions.
1.2
Semester Lab Schedule and Rules and Regulations
• Lab class Schedule would be announced in the class.
• Attendance count as a marks.
• Note that some of the lab sessions may not be scheduled in order.
• All the students must read and understand the lab manual.
• All the students must attend every lab session without excuses.
• If a student unable to attend a lab session, would inform the Lab instructor before
the particular lab session.
• However the lab session would be rescheduled or recovered by the home work
exercises under the supervision of the instructor if a student has a valid reason
regarding his/her absence.
• In the case of reschedule lab session, absentee would only receive the 70% of
marks of the particular lab
• If a student unable to attend a lab session without informing the instructor would
consider as an absent.
• All the students must submit a flow chart of the experimental steps before start
the lab session to the instructor to the teaching assistant. Students who are
unable to show the flow chart aren’t allowed to attend the lab session.
• All the students must upload the lab report to Canvas after a period of week.
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1.3
General usage policies of the Lab
1. Any kind of Food or Drinks are not allowed at any time in the lab premises.
2. No Furniture or Teaching Equipment is to be removed from this lab room.
3. All the Furniture and Equipment here are for teaching / learning use only.
4. Return back the equipment or the furniture in the proper position after using it.
5. Vandalism and the writing on the computer table is prohibited.
6. Playing or dancing or any kind of disruptive behavior inside the lab is prohibited.
7. Students are not allowed to use the lab room for the personnel use such as party
or any kind of personnel gatherings. However, students are allowed to use the
lab for the experiment after the class time with permission by the lab supervisor.
8. Please always lock the door and turn off the lights and air-conditioning after the
class.
9. Please make sure to inform to the lab supervisor any kind of damage caused
during the experiments.
10. Keep clean the environment around the lab premises.
11. Students must attend the lab session not later than 10 minutes.
Violation of the above code of ethics can results, depending on the
severity of the case, to the following:
a.
b.
c.
d.
1.4
Warning from Lab supervisor
Disciplinary measure from the UG coordinator.
Ban from Physics Labs.
Suspension or Dismissal from the UG program as per decision by the dean.
Materials
You are required to bring the following to each lab:
1. Lab manual (this document)
2. A Calculator preferably scientific
3. A metric ruler (example 20cm)
4. A pen and few A4 sheets or a scratch book
5. Protractor, Compass and other equipment
3
1.5
Before the Lab
There may be one or more quizzes sometime during the semester to test how well you
prepared for the lab. These quizzes, as well as the pre-lab assignments, will be collected
during the FIRST Thirty MINUTES of lab, so it is wise to show up on time or early.
1.6
During the Lab
Note the condition of your lab station when you start so that you can re arrange it to
that state when you leave. Check the apparatus assigned to you. Be sure you know the
function of each piece of equipment and that all the required pieces are present. Your
are free to ask questions from your instructor or from the teaching assistants regarding
the experiment.
You can always refer manual and your physics theory books for further understanding.
Each an every one of the group mate must take part in the experiment.
Check your data table and graph, and make sample calculations. Make sure if everything looks satisfactory before going on to the next step. Most physical quantities
will appear to vary continuously and thus yield a smooth curve. If your data looks
questionable (e.g., a jagged, discontinuous “curve”) you should take some more data
near the points in question. Check with the instructor or the teaching assistant if you
have any doubts. If you make repeated calculations of any quantity, you need only
show one sample calculation.
Complete the analysis of data in your lab report (during the class or after the class
since you have given a week to submit the final report) and indicate your final results
clearly. Often a spreadsheet will be used to make repeated calculations. If you have to
make repeated calculations of any quantity, you are required to show only one sample
calculation. Answer all questions that were asked in the Lab Manual.
CAUTION: for your protection and for the good of the equipment, please check with
the instructor before turning on any electrical devices.
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1.7
Lab Report
You will be provided an example lab report during the introduction session. All students must submit a lab report prior to the next lab session within a period of week.
Nevertheless follow the below instruction when making a lab report.
TITLE PAGE - should be included the Experiment name, Course name and the
number, Student name and Student ID.
DATE - The date the experiment was performed.
PURPOSE - Brief statement of the objectives the experiment
THEORY - Brief description base on your understanding (you could refer lab manual
theory part)
PROCEDURE - You could use the created flow chart prior to the lab
OBSERVATIONS and CALCULATIONS - All the experiment data and the required calculations should be included.
RESULTS and CONCLUSIONS - show your results graphs, figures and final conclusion.
DISCUSSION - If any. however considered to be extra marks in the case of grading.
1.8
Grading
• 17 - 20 points: A
• 16 - 14 points: B+
• 13 - 10 points: B
• 09 - 07 points: C+
• 06 - 4.5 points: C
• < 4.5 points: D
• If absent without a valid reason - F
5
Chapter 2
Experiments
2.1
Experiment 1: Basic Measurement equipment and Measurement Uncertainty
2.1.1
Introduction
Suppose you want to find the mass of a gold ring that you would like to sell to a friend.
You do not want to jeopardize your friendship,Therefore you would like to find the
accurate mass of the ring in order to give a fair market price to your friend.
You found an electronic balance which gives a mass reading of 17.43 grams. Since the
digital display of the balance is limited to 2 decimal places, you could measure the
mass as m = 17.43 ± 0.01 g. Suppose you use the same electronic balance and obtain
several more readings:
• 17.46 g,
• 17.42 g,
• 17.44 g.
so that the average mass appears to be in the range of 17.44 ± 0.02 g.
By now you may feel confident that you know the mass of this ring to the nearest
hundredth of a gram, but how do you know that the true value definitely lies between
17.41 g and 17.45 g?
Since you want to be honest, you decide to use another balance which gives a reading
of 17.22 g.
This value is clearly below the range of values found on the first balance, and under
normal circumstances, you might not care, but you want to be fair to your friend. So
what do you do now?
To answer these questions, we should first define the terms accuracy and precision
Accuracy is the closeness of agreement between a measured value and a true or accepted value. Measurement error is the amount of inaccuracy.
Precision is a measure of how well a result can be determined (without reference
to a theoretical or true value). It is the degree of consistency and agreement among
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independent measurements of the same quantity; also the reliability or reproduciblity
of the result.
The statement of uncertainty associated with a measurement should include factors
that affect both the accuracy and precision of the measurement.
Measurement = Best estimate ± Uncertainty
(Equation 2.1)
Notice that in order to determine the accuracy of a particular measurement, we have
to know the ideal, true value.
Hence it is required to have a standard to compare with.
If the standard measurement has the same uncertainty as your preferred measurement,
the accuracy of the desired measurement is with in the rage.
However in the previous example there is no such standard to compare the accuracy
of the mass measurement of the Gold ring.
Precision is often reported quantitatively by using relative or fractional uncertainty:
Relative Uncertainty =
Uncertainty
Measured Quantity
(Equation 2.2)
Following the example of the Gold Ring; Suppose the Mass of Gold Ring m =17.43 ±
0.01 g has a fractional uncertainty of
0.01
= 0.057 = 5.73%
17.43
Accuracy is often reported quantitatively by using Relative Error:
Relative Error =
Measured Value − Expected Value
Expected Value
(Equation 2.3)
if the expected value from m = 17.68 g then the relative error:
17.43 − 17.68
= −0.014 = 1.4%
17.45
note that the minus sign indicates that the measured value is less than expected value.
In most cases, we do not know the actual value of a measurement. For an example,
gravitational acceleration or free fall acceleration has conventional standard 9.8 ms−2 .
7
But this value varies depending on the altitude and latitude. Another, measured value
may differ from this standard value that suggests an uncertainty. At a juncture such
as this, we term the conventional standard value as accepted value. Then we compare
the measured value with this accepted value to determine a percentage difference
between the measured value and the standard.
Also, it is essential to understand about Uncertainty Propagation. This term is
defined when we do calculations based on our measurements. If you do a calculation
without understanding its uncertainty factor, the calculation itself is useless. Therefore, a simplest way of determining the Uncertainty propagation in terms of addition,
subtraction, multiplication, division and raised to a power is given below.
Assume, we are adding,subtracting,multiplying and dividing two measurements x and
y, each with uncertainty, δx and δy.
(x ± δx) + (y ± δy) = x + y ± (δx + δy)
(Equation 2.4)
(x ± δx) − (y ± δy) = x − y ± (δx + δy)
(Equation 2.5)
δx δy
+
(x ± δx) (y ± δy) = xy ± xy
x
y
!
δx δy
(x ± δx) ÷ (y ± δy) = x ÷ y ± (x ÷ y)
+
x
y
(Equation 2.6)
!
(Equation 2.7)
If we want to raise a measurement x to power, n with an uncertainty of δx,
δx
(x ± δx) = (x ± δx) ± nx
x
n
2.1.2
!
n
Theory
Several items will be discussed during the lab session as follows:
• How to find uncertainty will be briefed in the class.
• 5ME concept will be briefed in the class.
• Measurement co-relation concept will be briefed in the class.
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(Equation 2.8)
Figure 2.1: Fun with Measurements.
2.1.3
Type of Errors
Measurement errors may be classified as either
1. Random or,
2. Systematic
depending on how the measurement was obtained (an instrument could cause a random
error in one situation and a systematic error in another).
2.1.3.1
Random Errors
Statistical fluctuations (in either direction) in the measured data due to the precision
limitations of the measurement device.
Also cause by the (human error??) due to improper measuring methods.
These Random deviations can be reduced by averaging over a large number of quantitative observations, after adjusting the experiment setup base on 5ME concept.
2.1.3.2
Systematic Errors
Reproducible inaccuracies that are consistently in the same direction.
These errors are difficult to detect.
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If a systematic error is identified when calibrating against a standard, the bias can be
reduced by applying a correction or correction factor to compensate for the effect.
Unlike random errors, systematic errors cannot be detected or reduced by increasing
the number of observations.
In this Lab we will be using three different types of measuring equipment to understand
the concept of relative errors and measurement uncertainties. They are :
• Vernier Caliper
• Micrometer
• Height gauge with a dial
2.1.4
Lab Exercises
1. Two spheres are cut from a certain uniform rock. One has radius 4.50 cm. The
mass of the other is five times greater. Find its radius.
2. Newton’s law of universal gravitation is represented by
F =
GM m
r2
Here F is the gravitational force, M and m are masses, and r is a length. Force
has the SI units kgm/s2 . What are the SI units of the proportionality constant
G?
3. A useful fact is that there are about π× 107 s in one year. Find the percentage
error in this approximation, where ”percentage error” is defined as
4. Identify the type of error using the common sources of errors given below
• Incomplete definition - Measurement is not defined properly.
• Failure to account for a factor - Failure to control or account for all possible
factors except the one independent variable that is being analyzed.
• Environmental factors - Vibrations,changes in temperature, Electronic noises
etc.
• Instrument resolution - All instruments have finite precision that limits the
ability to resolve small measurement differences.For instance, a meter stick
cannot distinguish distances to a precision much better than about half of
its smallest scale division (0.5 mm in this case.
• Failure to calibrate or check zero of instrument.
• Physical variations - It is always wise to obtain multiple measurements over
the entire range being investigated. Doing so often reveals variations that
might otherwise go undetected.
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• Parallax - This error can occur whenever there is some distance between the
measuring scale and the indicator used to obtain a measurement.
• Lag time and hysteresis - Some measuring devices require time to reach
equilibrium, and taking a measurement before the instrument is stable will
result in a measurement that is generally too low.
• Instrument drift - Most electronic instruments have readings that drift over
time.
• Human Error - come from carelessness, poor technique, or bias on the part
of the experimenter. The experimenter may measure incorrectly, or may
use poor technique in taking a measurement, or may introduce a bias into
measurements by expecting (and inadvertently forcing) the results to agree
with the expected outcome.
2.1.5
Vernier Caliper
Figure 2.2: Parts of a Vernier Caliper.
In order to take a measurement from Vernier Caliper we must know the least count
of the vernier caliper.
Smallest value that can be measured by a measuring device is called the Least Count
of that measuring instrument.
Least count of the vernier Caliper =
Least Count of Main Scale
Total number of divisions of the vernier scale
(Equation 2.9)
Let’s take look of complete measurement by vernier caliper.
In the above example the Least count of the Vernier Caliper is 0.02mm 50 Vernier
Scale divisions.
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Figure 2.3: Example of Finding the LC ()
Figure 2.4: Complete reading of a Vernier Caliper.
Now you would be able to use Vernier Caliper in the experiment section.
Hint: Make sure to check the 0 error of your vernier caliper.
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2.1.6
Micrometer Screw Gauge
Figure 2.5: Parts of Micrometer.
Taking a measurement from a Micrometer is similar to the Vernier Caliper reading.
Figure 2.6: Measurement reading of a Micrometer.
• To obtain the first part of the measurement: Look at the image above, you will
see a number 5 to the immediate left of the thimble.
This means 5.0 mm. Notice that there is an extra line below the datum line,
this represents an additional 0.5 mm. So the first part of the measurement is
5.0+0.5=5.5mm.
• To obtain the second part of the measurement: Look at the image above,
the number 28 on the rotating vernier scale coincides with the datum line on the
sleeve. Hence, 0.28 mm is the second part of the measurement.
• Similar to the Vernier Caliper you have to consider the Least Count.
Also make sure to check the zero error always before start.
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2.1.7
Height gauges
Dial Height Gage
Figure 2.7: Dial Height Gauge.
Height gauge is a measuring instrument in which a slider with a measuring stylus
moves relative to a measuring scale on a beam and in which this motion is along a single
vertical axis nominally perpendicular to a reference plane on the instrument base. This
type of a height gauge is also called ”Dual Post Height Gauges”. Height gauges are
normally used on a granite surface plate to measure and/or scribe part features from
a datum plane. Generally a scriber, test indicator, touch probe or CMM-style probe
is used to locate the measured feature. Height gauges are an indispensable instrument
for quality control. The uses of this device:
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• To measure the distance from a reference surface to a specific feature of a part
to verify that it meets specifications and tolerances
• To scribe a part with accurate vertical dimensions or features from a datum plane
so that additional machining can be done
• To perform 2D measurements of part features
• To verify center-to-center dimensions
• To measure flatness
• To measure straightness/squareness or perpendicularity of parts
The resolution of this device specific device is 0.01 mm. There are accurate digital
height gauges popular in the industry as well. However, it is best to understand the
mechanism of height gauges since you are able to verify the measurement in any given
scenario.
Please refer the link to learn how to use dial height gauge
https://youtu.be/b2yxdUVmFXQ
2.1.8
Spherometer
Figure 2.8: Spherometer.
A Spherometer is an instrument for measuring the curvature of a surface and very
small thickness of a flat material.
A spherometer consists of a metallic tripod framework supported on three fixed legs of
equal lengths. A screw passes through the center of the tripod frame, parallel to the
15
three legs. A large circular disc graduated with 100 equal parts is attached to the top
of the screw. A small vertical scale known as the Pitch scale (P) with the scale reading
divided into millimeters is fixed at one end of the tripod frame. This small scale also
divide in to two parts in order to measure Convex and concave surfaces.
2.1.9
Example measurement : Thickness of a coin
Refer the link below to understand the Spherometer and the measurement of a thickness
of a coin.
Thickness of a coin represents by
t = (n × p) + (X × L.C)
(Equation 2.10)
Where, ’n’ is the number of complete rotation made by the circular disc.
’P’ is the pitch, which is the distance moved by the middle screw per revolution.
’x’ is the number of additional circular scale divisions in excess of complete rotations.
’L.C’ is the least count of spherometer.
If there are N divisions in the circular scale on the circumference of the disc then;
LeastCount =
2.1.10
1
P
=
mm = 0.01mm
N
100
(Equation 2.11)
Experiments
There are four experiments in this lab. you are expected to write
1. Measure the given objects using Vernier Caliper
2. Measure the given objects using Micro Meter
3. Measure the given objects using Height Gauge
4. Measure the given objects using Spherometer
Furthermore you must write a lab report following the format as given below.
• Your purpose of doing this lab,
16
• summary of theory that how you understood,
• pre-lab exercises,
• procedures for each experiment setup and how to take measurements
• both qualitative and quantitative observations,
• necessary calculations required to obtain results,
• result tables,
• conclusion
• and discussion.
Use the give example lab report for your reference.
2.1.10.1
Apparatus
• Vernier Caliper
• Micrometer
• Height gauge
• Spherometer
• Acrylic disk
• PVC Pipe standard dia = 18 mm
• thin ruler piece
• Metal Sphere
2.1.10.2
Experimental Procedure
Please refer the below videos to further understand Vernier Caliper and Micrometer
Measurements
• https://youtu.be/i-5gEN0N0pk
• https://youtu.be/2-cm_ocn9p4
1. Write a Standard Procedure of experiment of setup each device before taking a
measurement, base on your understanding.
2. Measure the given objects using appropriate measurement devices and record the
dimensions in the below table.
17
Equipment
Measuring Object
Relative Uncertainty
Relative Error
Final Measurement +/- uncertainty
Measurement (mm)
VC MM HG
SM VC MM HG
SM VC MM HG
thickness
diameter
radius of curvature
thickness
diameter
radius of curvature
thickness
diameter
radius of curvature
thickness
diameter
radius of curvature
Table 2.1: Final results
Please refer the acronyms as follows;
• VC - Vernier Caliper
• MM - Micrometer
• HG - Height Gauge
• SM - Spherometer
18
SM VC MM HG
SM
2.2
2.2.1
Experiment 2: Introduction to Vectors
Introduction
A vector is a quantity that has both magnitude (numerical size) and direction. This is
the opposite of a scalar, which is a quantity that only has magnitude and no direction.
Figure 2.9: Direction impact on Vectors (?, ?)
Figure 2.10: Direction impact on Vectors
Figure 2.11: Application of Vectors
19
2.2.2
Theory
2.2.2.1
Vectors and Forces concepts
• Scalar- a quantity that is measured by magnitude only.
• Vector- a quantity defined by both magnitude and direction.
2.2.2.2
Relationship of components to a vector
Figure 2.12: Components of a vector
The lengths of the components of the vector can be related to the length (magnitude) of the vector by the trigonometric functions. In the Figure 2.12 at right
showing vector A,if the angle θ is measured with respect to the x-axis of the
coordinate system, where θ is positive when measured counterclockwise from the
x-axis, the components of the vector can be calculated using the trigonometric
functions:
Ax = Acosθ , Ay = Asinθ
These relationships are valid for a vector in any quadrant as long as θ is measured
with respect to the x-axis.
• Scalar addition - the algebraic sum of two or more quantities.
• Vector addition - If two vectors are parallel, being in the same (opposite) direction, their magnitudes can be added (subtracted) to obtain the magnitude of the
Resultant Vector.
• If the two vectors are not parallel, adding them requires establishing an x-y coordinate system, then breaking down each vector into its “x” and “y” components
before algebraically adding these vector components together to yield the Resultant Vector’s “x” and “y” components.
20
• Resultant Vector’s magnitude is then calculated as the hypotenuse of the x-y
vector triangle. The angle of the Resultant Vector from a designated coordinate
axis uses the Tangent function of the x-y Resultant Vector components.
Figure 2.13: Vector addition (?, ?)
• Weight - a force vector (magnitude w = mg) which is in the direction of gravitational acceleration (g - down,toward the center of the Earth).
• Net Force - the resultant vector that is the sum of all forces being applied to an
object.
• Equilibrant Force - one that is equal in magnitude and opposite in direction
to the Net Force.The Equilibrant Force balances the Net Force causing static
equilibrium.
Figure 2.14: Geometry of Vectors (?, ?)
21
2.2.2.3
Parallelogram Law of Vectors
If two vectors acting simultaneously on a particle are represented in magnitude and
direction by the two adjacent sides of a parallelogram drawn from a point,
then their resultant is completely represented in magnitude and direction by the diagonal of that parallelogram drawn from that point.
Let two vectors P and Q
• act simultaneously on a particle O at an angle θ,
• they are represented in magnitude and direction by the adjacent sides OA and
OB of a parallelogram OACB drawn from a point O,
• the diagonal OC passing through O, will represent the resultant R in magnitude
and direction.
|R|=
q
P2 + Q2 + 2PQcosθ
tan−1 α =
QSinθ
P + Qcosθ
22
(Equation 2.12)
(Equation 2.13)
2.2.2.4
Head to tail method
2.2.2.5
Lab exercises
1. ABCD is a quadrilateral. Simplify the following:
23
2. Ashan and Kasun Pulling a Box, Ashan pulls with 200N force at 60◦ angle and
Kasun pulls 120N force at 45◦ as shown in the figure below.
What is the combine force and the direction?
3. Identify the followings are Scalers or a Vectors
• Airplane Cabin Pressure
• Weight of a Planet
• Length of the Milky Way
• Torque of a car engine
• Internal temperature of the sun
• Gravitational force between two planets.
• wind speed of 10 mph to the North East direction
4. Imagine a situation that you are being a slack wire walker. Calculate the resultant vector with appropriate units. Assume you are walking on a uniform Rope.
Suppose your weight is 90kg
24
2.2.3
Experiments
There are two experiments in this lab session.
2.2.3.1
Experiment 1 : Gravesand’s apparatus
Apparatus
• Gravesand’s board
• Triple beam balance
• Ring weights
• pulley magnets
• hooks
• metal pin
• Strings
• Protractor
• A4/A3 sheet
25
2.2.3.2
Experimental Procedure - equipment setup
1. Fix the A3 paper with pulley magnets on the board, set in a vertical plane such
that it should be parallel to the edge of board as in the Figure ??.
2. Pass one thread over the pulleys carrying a hook at its each end. Take a second
thread and pass it over the pulley.
3. Attach weight rings in the hooks in a way that the metal ring center align metal
pin center.
4. Displace slightly the weight rings from their position of equilibrium and note if
they come to their original position of rest. This will ensure the free movement
of the pulleys.
5. Mark lines of forces represented by thread without disturbing the equilibrium of
the system and write the magnitude of forces i.e Added weight (assume the mass
of the hook is negligible)
6. Remove the graph paper and draw vectors in the direction of the strings, with
lengths directly proportional to the appropriate masses.(Eg: if you choose a mass
40g you can draw 4cm line along with the free body diagram. therefore the Scale
will be 10g = 1cm) Use notation force P, Q, R and create the parallelogram AB,
BC, and CA.
7. Video reference https://youtu.be/fowRMegIFvw
8. Now Calculate the Force vector that balancing the Ring by using the vector
parallelogram law.
9. Find the relative error(don’t forget we have a better estimate of the weights of
the rings measured by the triple beam balance).
26
2.2.3.3
Experiment 2 : Force Table (Optional)
Figure 2.15: Force table
Apparatus
• Force table
• strings
• weights and weight hangers
1. Implement three vector forces addition for two different types of force table setup.
2. Write down your observations about the behavior of the center ring when you
change the masses and the angles of the resultant (in observation section).
27
2.3
Experiment 3: Linear & Planar motion
2.3.1
Introduction
Imagine a takeoff of an airplane, it takes a certain speed to generate the lift. Then,if
you throw a tennis ball straight up in the air and the time takes for you to catch it
again.Another scenario would be when a train requires to stop in a particular station,
it needs to start decelerating from a particular distance. These are few incidents
we encounter in our life and we are going to interpret these in the world of physics.
Therefore, in this lab session we will be attempting to determine the motion of an
object in a linear path in terms of displacement,velocity and acceleration/deceleration.
However, it is confined in to a two dimensional platform.
Go through the following for more information:
https://physics.info/motion-graphs/
https://www.youtube.com/watch?v=4dCrkp8qgLU
2.3.2
2.3.2.1
Theory
Equation of motions
The equations of motion, also commonly termed as SUVAT equations, are used with
the assumptions that the acceleration, a, is constant. They are known as SUVAT
equations because of the following variables:
• s - distance
• u - initial velocity
• v - velocity at time t
• a - acceleration
• t - time
However, each equation does not contain all these variables. Therefore, in order to
solve some questions it might be necessary to use one or more of them. The equations
are as follows:
v = u + at
(Equation 2.14)
(v + u)
t
2
(Equation 2.15)
v 2 = u2 + 2as
(Equation 2.16)
s=
28
1
s = ut + at2
2
2.3.2.2
(Equation 2.17)
Planar projectile motion
Projectile motion is a form of motion where an object moves in parabolic path; the
path that the object follows is called its trajectory. Projectile motion only occurs when
there is one force applied at the beginning on the trajectory. After that it continues in
it’s trajectory by the interference of gravity.
In order to understand the projectile motion we have to know key components as
follows
1. Initial Velocity - Which can be express as it’s vertical and horizontal components.
ux = ucosθ
uy = usinθ
θ = direction(angle) of the projectile
2. Time of Flight - The time of flight of a projectile motion is total time the object
takes to reaches the surface from its initial position. As we discussed previously,
T depends on the initial velocity magnitude and the angle of the projectile:
Most common equation of to find the time is
1
S = ut + at2
2
Where s = horizontal displacement, u = initial velocity, t = time of
flight, a = gravitational acceleration
3. Acceleration - Only acceleration contribute to the projectile is gravitational
acceleration. Hence there is no horizontal acceleration.
4. Velocity - Horizontal velocity of a Projectile is constant where the vertical velocity varies linearly.
vx = vcosθ
vy = usinθ − gt
29
θ = direction(angle) of the projectile
5. Displacement - At time t you can find vertical or horizontal displacement
x = vtcosθ
1
y = usinθt − gt2
2
Where x = horizontal displacement and the y = vertical displacement
6. Maximum Height - The maximum height is reached when vy =0
7. Range - The horizontal displacement of the motion.
Law of Energy Conservation
30
2.3.3
Lab Exercises
1. The graph below shows velocity as a function of time for some unknown object.
• What can we say about the motion of this object?
• Plot the corresponding graph of acceleration as a function of time.
• Plot the corresponding graph of displacement as a function of time.
2. In 2 s, a car increases its speed from 60 km/h to 65 km/h while a bicycle goes
from rest to 5 km/h. Which undergoes the greater acceleration?
3. A point moves in a straight line so that its displacement x metre at time t second
is given by x2 = 1 + t2 find the acceleration in ms−2 at time t seconds.
31
4. A stone is dropped from the top of a tower 400 m high and at the same time
another stone is projected upward vertically from the ground with a velocity of
100 ms−1 . Find where and when the two stones will meet.
5. Two trains are moving in opposite directions. Train A moves east with a speed
of 10 ms−1 and train B moves west with a speed of 20 ms−1 . What is the
• relative velocity of B w.r.t A.
• the relative velocity of ground w.r.t B.
• A dog is running on the roof of train A against its motion with a velocity of
5 ms−1 w.r.t train A. What is the velocity of the dog as observed by a man
standing on the ground?
2.3.4
2.3.4.1
Experiments
Experiment 1: Motion on a straight line
In this experiment you would be experimenting the Constant velocity and the acceleration with the help of a Ticker Tape timer. Please refer following video links for your
reference and understanding.
• https://www.youtube.com/watch?v=MZabnUdYr9g
• https://www.youtube.com/watch?v=RP4b2nCJrts
• https://www.youtube.com/watch?v=ztZevUk1dvw
You are given the following apparatus:
• Ticker Tape Timer and Tapes
• Low Voltage AC power Supply
• Trolley
• Metric Ruler
1. Setup the Apparatus referring diagram however on the flat table.(May or May
not require an incline plane)
2. Connect the ticker timer to the power supply.
3. Give trolley a push to move it forward or let the gravity take its course.
32
Figure 2.16: Ticker Tape Timer set up
Figure 2.17: A Ticker Tape
4. Note up to 10-20 dot spaces as in the figure below (including starting point).
5. Measure the lengths of each adjacent spaces (up to 10 or 20) and record them in
a table (displacement & time).
6. Note down the time for the incline angle to indicates a constant acceleration in
ticker tape.
7. Repeat these steps for 5 times (trials for uncertainty).
8. Now use the displacement-time table to draw a graph and then use it to obtain
the acceleration of the trolley.
9. hint: Plot the displacement-time, velocity-time, and acceleration-time graphs.
2.3.4.2
Experiment 2: Motion in a plane
In this experiment you will get to use a projectile launcher. However the given launcher
has several issues that you can observe.
Quick question : Observe the apparatus and find the problem/problems. Then
explain how can you improve or redesign the launcher to improve the problems that
you found.
33
You are given the following apparatus:
• Steel ball
• Projectile equipment
• Measuring tape or ruler
• Timer
Figure 2.18: Projectile launcher
34
1. A particle is projected from point O with speed of u at an angle of elevation
α above the horizontal and moves freely under gravity. When the particle has
moved a horizontal distance x, height of the moving particle above O is y:
• Draw the free body diagram.
• show that
gx2
y = (xtanα) −
2u2 cos2 α
2. Set the inclination angle to be 15, 30, 45, 60 degrees and place it on the ground
or table with a length of 1m (enough free space).
3. setup the level of the releasing mechanism of the launcher base on your experiment
environment horizontal range.
4. Release the steel ball from the projectile device.
5. At first observe the motion of the steel ball. (Recommended to take a slow motion
video) Explain the motion of the ball until it hits the ground. Draw a diagram
to explain.
6. Then, release the ball from the projectile device and start counting the time at
the point where the ball just releases from the device.
7. Repeat the above step several time in order to get the average time of the projectile motion.Also make sure to mark the place that ball hits the ground in each
trial.
8. Now you have to measure the horizontal ranges where the ball landed and obtain
the average horizontal range
9. Make sure that you consider the measurement uncertainty in each measurement
you take.
10. calculate the releasing velocity of the motion, use uncertainty propagation(combined
uncertainty).
11. Find calculate the Y value for each given angle
12. Explain your observations of y and the x values
35
2.4
Experiment 4: Newton’s laws of motion
2.4.1
Introduction
”To any action there is always an opposite and equal reaction; in other words, the
actions of two bodies upon each other are always equal and always opposite in direction.” like wise, if you happen to choose not to study action, as reaction, grade F will
decorate your transcript.
adopted from Isaac Newton’s book of Principia Mathematica
2.4.2
2.4.2.1
Theory
Newton’s first law of motion
Newton’s First Law states that an object will remain at rest or in uniform motion in
a straight line unless acted upon by an external force on the body of the object under
the following conditions
• the system exists in an inertial reference frame.
• all the masses of the system considered to be regular shapes that will not deform
during a motion over a period of time.
36
However a motion of a massive object is harder to change than the motion of a lighter
object due to the phenomena called inertia.
Inertia is a term use to determine the ability of an object to resist a change in its
state of motion. An object with a lot of inertia(lot of mass) takes a lot of force to start
or stop; an object with a small amount of inertia requires a small amount of force to
start or stop.
What systems of a car helps to overcome the law of inertia or newton’s first law?
37
2.4.3
Newton’s Second law of motion
The acceleration of an object is equal to the force you apply divided by the mass of
the object under the conditions of
• the system exists in an inertial reference frame.
• all the masses of the system considered to be regular shapes that will not deform
during a motion over a period of time.
38
2.4.4
Newton’s third law of motion
Newton’s third law states that for every action force there is a reaction force that is
equal in strength and opposite in direction under the conditions of
• the system exists in an inertial reference frame.
• all the masses of the system considered to be regular shapes that will not deform
during a motion over a period of time.
39
2.4.5
2.4.5.1
Experiments
Experiment Part 1 : Newton’s first law of Motion
You have given a spring loaded trigger, acrylic plate attached device that is capable of
demonstrating the Newton’s first law of motion. There are
• Metal, rubber, marble balls having different masses and shapes
also provided with the equipment.
Think thoroughly about the first law of motion and design an experiment to demonstrate all possible conditions of the first law of motion.
• write a short procedure of experiment setup.
• identify the related observation.
• explain the reason behind observations.
• conclude your experiment.
40
2.4.5.2
Experiment Part 2 : Newton’s second law of Motion
In this experiment you have given;
• an incline plane
• pulleys
• rollers (300 ± 20g)
• mass holders
• masses
• Nylon threads
• newton meter
as apparatus.
1. Attach the pulley in the top of the incline plane.
2. Put the thread over the pulley that connected the roller mass and the aluminium
petri dish.
3. Set the incline plane angle to 20◦ and find the critical mass that maintain the
system at equilibrium as in theFigure 2.20.
4. What assumptions you would take before consider the masses are at a constant
acceleration?
5. what parameters will remain the same during this experiment?
6. Draw the free body diagram of the system.
41
Figure 2.20: Experiment setup
7. Determine an mathematical expression for the constructed system in order to
find acceleration a
8. Now increase the angle of the incline plane by every 2◦ increment till 38◦
(a) write your observations in the observations section and explain each an every
observation in the discussion section.
(b) Find the acceleration at every different angle. Use motion in straight line
knowledge.
(c) find both accelerations of roller mass and the hanging mass
(d) Are the accelerations are same or different? explain.
(e) record the values in a table α, sinα, acceleration of roller mass, acceleration
of hanging mass and calculated acceleration of the system
9. Arrange the constructed equation of the system to represent Y = mx+c
10. Now draw appropriate graphs to determine the acceleration of the system.
(a) Acceleration of the roller mass
(b) Acceleration of the hanging mass
(c) Calculated acceleration at each angle
11. What you could find by the slope of the graph? find the parameter value
12. What do you think about the value of the parameter? is there any relative error?
if then why?
42
2.5
2.5.1
Experiment 5: Friction
Introduction
Friction is the resistance to the motion of bodies being in contact with any kind of
surface. Friction can be mathematically expressed as the amount of force proportional
to the normal force applied by each surface. The frictional force always acts in the
direction opposite to the motion of the object. In general, there are two types of
friction. Friction that resist the object from rest to motion is called static friction µs .
Then the kinetic friction µk which resists the moving object. kinetic friction leads
to the skidding motion on a wheel.
Friction perhaps was firstly stated as the law by Newton and Leonardo Da Vinci.
There are three laws of friction commonly used at the present. The first two laws of
friction have not been indicated since Newton and Leonardo Da Vinci eras. Until 1699,
French engineer named Amontons had re-developed the concept of the first two laws
of friction. The last friction law was stated by Scientist named Coulumb [1]. These
three laws of friction are:
1. Friction is independent to the apparent area of contact
2. Friction is directly proportional to the normal load
3. Kinetic friction is independent to the sliding velocity
Figure 2.21: Direction of the friction force
43
2.5.2
Lab Exercises
1. Find the force (P ) where the system moves with a constant velocity and the
kinetic friction of all surfaces are 0.6
Figure 2.22: A certain mass pulley system
2. If a skier of 60 kg is skiing as shown in the below figure and the friction force is
known to be 45 N, what is the value of coefficient of kinetic friction?
Figure 2.23: A skier moving down a slope
44
2.5.3
2.5.3.1
Experiments
Experiment 1: Determining the static coefficient of friction
In this experiment you have been given;
• an incline plane set (Protractor attached)
• roller (300 ± 20g)
as apparatus.
1. Keep the inclined plane at a horizontal position (θ = 0).
2. Place the roller (no rolling: fixed) at a the center of the inclined plane.
3. Now lift the inclined plane until the roller start to move, note down the angle
when it just about to move.
4. Now what would be the coefficient of static friction between plastic on metal?
2.5.3.2
Experiment 2: Determining the kinetic coefficient of friction
In this experiment you have been given the following apparatus:
• an incline plane
• pulleys
• rollers (300 ± 20g)
45
• mass(150 ± 1g) & holders
1. Set the incline plane angle to be 20◦ (make sure the angle remains same during
the experiment)
2. Now fix the pulley on top of the incline plane (Or any reference point) as shown
in the figure above.
3. Draw the free body diagram of the system.
4. Write a general equation for the constructed system in order to find the acceleration (with variables).
5. Release the M2 mass (150 ± 1g) and find the acceleration (By distance and time
measurement or use tracker), repeat the measurement at least 5 times.
6. Now find the coefficient of kinetic friction.
46
2.6
Experiment 6: Investigating the behaviors of the Spring systems
2.6.1
Objectives
1. Find the Spring Constant by Experimental method.
2. Understand the difference between the given Springs.
3. Find the Work done by the Springs.
4. Observe the behavior of Parallel and Series Springs.
2.6.2
Theory
• When a spring is stretched by an applied force, a restoring force is produced.
• Due to the restoring force, simple harmonic motion take place in a straight line,
which the acceleration and the restoring force are directly proportional to the
displacement of the vibrating load from the equilibrium position.
• The relation between the force F and displacement x is
F = −kx
(Equation 2.18)
Restoration force is opposite in direction to the displacement
• Constant k is known as the force constant of the spring.
• This is a force, expressed in Newton, which will produce an elongation of one
meter in the spring.
The equation of energy of the spring is shown below.
1
W = ∆P E = − K(x21 − X02 )
2
Figure 2.24 shows an extended spring due to an applied load
47
(Equation 2.19)
Figure 2.24: Spring reaction after adding a mass
2.6.3
Experiments
2.6.3.1
Finding the Spring Constants of given springs
Apparatus
• Hooke’s law apparatus is shown in the Figure 2.25
• There are physically different springs that can be categorized as thick or thin and
stiff or flexible.
• Mass hanger is 50g and slotted weights.
• Helical Springs, specification as follows
Spring constant 2.5N/m 5N/m
Length
122mm
10N/m
145mm 150mm
15N/m
25N/m
147mm 142mm
Experiment Procedure
1. Setup the equipment as in the Figure 2.25 don’t forget to discuss the techniques
you have used to setup the equipment.
2. Use only mass hanger(50g) without additional weights and set the indicator to a
known reference.
48
Figure 2.25: Hooke’s law apparatus
3. Measure and record the lengths of the springs of spring constants 2.5N m−1 and
25N m−1 .
4. Hang the mass hanger on the thin spring and record the elongation.
5. Now add more appropriate slotted weights(discuss reasons why to choose the
your desired weights).Use the below Table 2.2 to record your results.
6. Now use the thickest spring to do the experiment using the appropriate slotted
weights (provide the reason)Use the below Table 2.2 to record your results.
7. Now Calculate the Spring Constant k of both the springs using your measurement
results.
8. It is impossible to forget the uncertainties of measurement in this experiment.
Explain what type of errors can occur using this experiment. Can we find Relative
error in this experiment? if not explain why
49
Spring Mass(Kg) Length of the Spring(m) Elongation(m) Force(N)
Calculated Spring Constant
k Avg
Thin
Thin
Thin
Thick
Thick
Thick
Table 2.2: Results table
9. Write your observations and opinion about the two springs you used to do the
experiment using your results.
10. Now calculate the Work done by the both springs.
50
2.6.3.2
Finding the equivalent Spring Constant using Thick & Thin Springs
connected in series & parallel
In this experiment we are going to observe the behavior of Series and Parallel Spring
systems.
• In a series spring system, the spring force remains same.
• In parallel connection Total Spring force equal to the addition of each individual
spring force.
Experiment Procedure
1. Connect the Springs in series and add appropriate masses.
2. Record the elongation.
3. Create a mathematical expression in order to find the equivalent Spring constant
in series.(Also refer to the PPT Slides)
4. Calculate the equivalent Spring constant k by substituting theoretical values.
5. Now find the equivalent spring constant by experimental method.
6. Calculate the work done by the Spring system. and Compare the results to work
done by a single thick spring. Discuss the results.
7. Write down your observations clearly and elaborate them in the discussion section
of the lab.
Figure 2.26: Spring Systems
51
2.7
2.7.1
Experiment 7: Simple Harmonic Motion
Introduction
The motion of the pendulum is a particular kind of repetitive or periodic motion called
simple harmonic motion. A pendulum is a mass hang from a pivot allowing a mass
to swing freely. Ideally, the pivot is assumed to be friction less. However, in practice
the pivot is not friction-less.Hence, the mass hang from a pivot will not keep swinging
infinitely like ideal case. Kinetic and potential energy are the main physical mechanism
that drives the mass to swing as a cycle. The position of the oscillating object varies
sinusoidal with time. Many objects oscillate back and forth.The motion of a child
on a swing can be approximated to be sinusoidal and can therefore be considered as
simple harmonic motion. Some complicated motions like turbulent water waves are not
considered simple harmonic motion.When an object is in simple harmonic motion,the
rate at which it oscillates back and forth as well as its position with respect to time
can be easily determined. In this lab, you will analyze a simple pendulum and a
spring-mass system, both of which exhibit simple harmonic motion.
2.7.2
Theory
A particle that vibrates vertically in simple harmonic motion(SHM) moves up and
down between two extremes y = ±A. The maximum displacement A is called the
amplitude. This motion is shown graphically in the position-versus-time plot in the
figure below.
Figure 2.27: Position plot showing sinusoidal motion of an object in SHM
One complete oscillation or cycle or vibration y = −A to y = +A and back again to
y = −A.
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The time interval T required to complete one oscillation is called the period.
A related quantity is the frequency f, which is the number of vibrations the system
makes per unit of time.
1
f=
(Equation 2.20)
T
If a particle is oscillating along the y-axis, its location on the y-axis at any given instant
of time t, measured from the start of the oscillation is given by the equation
y = ASin(2πft)
(Equation 2.21)
velocity of the object is the first derivative and the acceleration the second derivative
of the displacement function with respect to time. The velocity v and the acceleration
a of the particle at time t are given by
V = 2πfACos(2πft)
a = −(2πf )2 ASin(2πft)
(Equation 2.22)
(Equation 2.23)
Notice that the velocity and acceleration are also sinusoidal. However the velocity
function has a 900 or π/2 phase difference while the acceleration function has a 1800
or π phase difference relative to the displacement function. For example, when the
displacement is positive maximum, the velocity is zero and the acceleration is negative
maximum.
Spring Mass System
Figure 2.28: Five Key points of a mass oscillating on a spring
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The mass completes an entire cycle as it goes from position A to position E. A description of each position is as follows:
• Position A: The spring is compressed; the mass is above the equilibrium point at
y = A and is about to be released.
• Position B: The mass is in downward motion as it passes through the equilibrium
point.
• Position C: The mass is momentarily at rest at the lowest point before starting
on its upward motion.
• Position D: The mass is in upward motion as it passes through the equilibrium
point.
• Position E: The mass is momentarily at rest at the highest point before starting
back down again.
Spring Mass System
Figure 2.29: Familiar Spring Mass System
When the mass is motionless, its acceleration is zero. According to Newton’s second
law the net force must therefore be zero. There are two forces acting on the mass; the
downward gravitational force and the upward spring force. See the free-body diagram
in Figure 2.30 below.
Figure 2.30: Familiar Spring Mass System
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Using Newton’s second law we could achieve an equation for Spring mass system as
follows.
r
1 m
T=
(Equation 2.24)
2π k
Simple Pendulum
The simple pendulum consists of a mass m, called the pendulum bob, attached to the
end of a string. The length L of the simple pendulum is measured from the point of
suspension of the string to the center of the bob as shown in Figure 2.31 below.
Figure 2.31: Simple Pendulum
If the bob is moved away from the rest position through some angle of displacement
Îÿ as in Figure above, the restoring force will return the bob back to the equilibrium
position. The forces acting on the bob are the force of gravity and the tension force
of the string. The tension force of the string is balanced by the component of the
gravitational force that is in line with the string (i.e. perpendicular to the motion of
the bob). The restoring force here is the tangential component of the gravitational
force.
Using the newton’s second law and the angular displacement we could achieve the
period of oscillation as below.
s
L
T = 2π
(Equation 2.25)
g
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2.7.3
Exercises
Draw the SHM graph to express the position, velocity & acceleration of the spring mass
system or simple pendulum scenario. Also fill in the table below in terms of Positive
max, zero, Negative max where amplitude is 2 cm & time period is 5 seconds.
position
Velocity
Acceleration
Point A
Point B
Point C
Point D
Point E
2.7.4
Exercises
1. A 102g (grams) mass hung from a weak spring has stretched it by 3.00cm. Let
g=9.81ms2 and calculate (a) the load on the spring and (b) the spring constant
in N/m.If the mass-spring system is in static equilibrium and motionless, and the
mass is pushed up by +2.00cm and released, calculate (c) its angular frequency,
(d) its frequency, (e) its period, (f) the amplitude of oscillations, and (g) the
equation of motion of such oscillations.
2. A mass M at the end of a spring executes SHM with a period t1. While the
same mass execute SHM with a period t2 for another spring. T is the period of
oscillation when the two springs are connected in series and Mass M is attached
at the end. Find out the correct relation.
2.7.5
2.7.5.1
Experiments
Determine the Spring Constant by Simple Harmonic Motion
Apparatus
• Hooke’s Law apparatus with a Thickest spring (25N/m)
• Meter Stick
• Stop watch
• 50g mass hanger.
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In this experiment we have assumed the spring to be mass-less. However there will be
an affect to the period of oscillation.
1. Setup the experiment using the Thickest Spring provided.
2. use 50g mass hanger.
3. Pull the mass hanger down a short distance and let go to produce a steady up
and down motion without side-sway or twist. As the mass moves downward pass
the equilibrium point, start the clock and count ”zero.” Then count every time
the mass moves downward past the equilibrium point.
4. Repeat the above two more times and record the values for the three trials in
Data Table given below and determine an average time for 20 oscillations or as
you prefer (Not necessary to be 20 oscillations).
Time for 20 oscillations
Vibrating mass
Observed period
1 2 3
Average
Table 2.3: Vibrating mass variation & time period.
5. Determine the period from this average value and record this on the worksheet.
6. Repeat steps 2 through 4 for three other significantly different masses.You can
choose the value for the mass.
7. Use Excel or any other software to plot a graph of T 2 vs m
8. Use the plot fitting option in Excel or using ”Tracker” to determine the slope and
record this value on the worksheet.
9. Determine the spring constant k from the slope.
2.7.5.2
Determine the period of the Simple Harmonic Motion
Apparatus
• Meter Stick
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• Stop watch
• string with pendulum bob.
1. Adjust the pendulum to the longest length possible and firmly fasten the cord.
2. Measure the length of the pendulum using a metric Ruler.
3. Find the period of the pendulum by counting 10 oscillations.
4. Change the pendulum bobs (3 different masses) and repeat the first 3 steps to
determine the time period.
Time for 10 oscillations
Mass of the bob
Observed period
1 2
3 Average
Table 2.4: Mass of the bob variation & time period.
5. Keep the same pendulum bob and change the length (3 different including the
original length), repeat the first 3 steps to determine the time period.
Time for 10 oscillations
Length of the pendulum
Observed period
1 2 3
Average
Table 2.5: Length variation & time period.
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