The document provides instructions for an experiment to measure the Young's modulus of brass using beam bending. Students are asked to:
1) Take preliminary measurements of a beam's deflection under different loads to estimate the Young's modulus.
2) Precisely measure the beam's deflection under varying loads and use the slope of a deflection vs. load graph to calculate Young's modulus.
3) Use error analysis to plan further measurements of beam dimensions to achieve a desired accuracy for calculating Young's modulus.
Dealing with Notations and conventions in tensor analysis-Einstein's summation convention covariant and contravariant and mixed tensors-algebraic operations in tensor symmetric and skew symmetric tensors-tensor calculus-Christoffel symbols-kinematics in Riemann space-Riemann-Christoffel tensor.
Dealing with Notations and conventions in tensor analysis-Einstein's summation convention covariant and contravariant and mixed tensors-algebraic operations in tensor symmetric and skew symmetric tensors-tensor calculus-Christoffel symbols-kinematics in Riemann space-Riemann-Christoffel tensor.
shooting method with Range kutta methodSetuThacker
how to solve the differential equation by using shooting method & range Kutta method.there is one problem statement, which is related to the temperature distribution problem over an iron rod. there is two graphs of the 2nd order and 4th order Range Kutta. there is one combine graph of the both the method which gives correct & closest answer.
Thank You...!!!
CHAPTER 5 Wave Properties of Matter and Quantum Mechanics I
5.1 X-Ray Scattering (review and some more material)
5.2 De Broglie Waves
5.3 Electron Scattering / Transmission electron microscopy
5.4 Wave Motion
5.5 Waves or Particles?
5.6 Uncertainty Principle
5.7 Probability, Wave Functions, and the Copenhagen Interpretation
5.8 Particle in a Box
Response of dynamic systems to harmonic excitation is discussed. Single degree of freedom systems are considered. For general damped multi degree of freedom systems, see my book Structural Dynamic Analysis with Generalized Damping Models: Analysis (e.g., in Amazon http://buff.ly/NqwHEE)
To Determine the Charge to Mass Ratio for Electron by JJ-Thomson’s MethodSachin Motwani
This demonstrates the practical method of determining charge to mass ratio. This experiment is generally undertaken in the first year of an engineering degree program.
Introduction to transient Heat conduction, Lamped System Analysis, Approxiamate Analytical and graphical method and Numerical method for one and two dimensional heat conduction by using Explicit and Implicit method
SSL8 Mass & Energy Analysis of Control SystemsKeith Vaugh
Conservation of mass
Mass and volume flow rates
Mass balance for a steady flow process
Mass balance for incompressible flow
Flow work and the energy of a flowing fluid
Energy transport by mass
Energy analysis of steady flow systems
Steady flow engineering devices
Nozzles and diffusers
Turbines and compressors
Throttling valves
Mixing chambers and heat exchangers
Pipe and duct flow
Energy analysis of unsteady flow processes
Determination of DensityRequired materials provided in tmackulaytoni
Determination of Density
Required materials provided in the Home Science Tools chemistry kit:
100mL graduated cylinder, balance (scale)
Required materials
not
provided in the Home Science Tools chemistry kit:
cell phone (with camera), metric ruler, 25-30 pennies, graph paper
Objectives:
to find the density of regular-shaped and irregular-shaped substances including graphing techniques
Introduction:
Density
is the intensive property of matter defined as the ratio of an object’s mass to its volume. In simpler words, density is the mass of an object divided by the volume which the object occupies. The term
intensive property
means that it is
independent
of the amount of the substance. The density of any substance remains the same, no matter the shape and size of the sample. The density of water at 4°C is 1.000 g/mL regardless if the sample size is 1 cup or 1 swimming pool. Thus, density is one of the characteristic properties which allows us to identify substances; it is fixed and has a unit of g/mL. As such, it is a useful tool to identify an unknown metal. One can calculate the density of an unknown metal and can match the value against a standard density table for its identification.
The density of a substance does change with a change in temperature. This change in density is
inversely proportional
to the change in temperature. This is to say, if the temperature rises, then the density decreases, and if the temperature falls, then the density increases. Cooling a substance causes its molecules to occupy a smaller volume, resulting in an increase in density. Hot water is less dense and will float on room-temperature water. Cold water is denser and will sink in room-temperature water.
Densities of various substances can be identified differently. For
regular
(shaped)
solids
, calculating the density is straightforward: simply weigh the solid and measure its dimensions, using a simple formula to calculate the volume. The density is calculated by dividing the mass by the volume. Each regular solid has its own formula for calculating its volume depending on the shape of the solid. The volume of a
rectangular
solid equals length times width times height. Note: 1 mL = 1 cm3. For
irregular
(shaped)
solids
, those that do
not
have a standard formula for calculating their volume, the volume can be determined by measuring the volume of liquid that the solid displaces. To do this, the solid is submerged in a liquid and the volume displaced is measured. This is done by taking an initial reading and a final reading and calculating the difference in volume. The mass of the object is then divided by this volume, and the density is determined.
Measuring the density of a liquid is very similar. Although the volume cannot be measured with a ruler, it can be determined using volumetric glassware, for instance, a graduated cylinder. The liquid’s mass is determined when this measured volume is weighed. Knowing the ...
11 - 3
Experiment 11
Simple Harmonic Motion
Questions
How are swinging pendulums and masses on springs related? Why are these types of
problems so important in Physics? What is a spring’s force constant and how can you measure
it? What is linear regression? How do you use graphs to ascertain physical meaning from
equations? Again, how do you compare two numbers, which have errors?
Note: This week all students must write a very brief lab report during the lab period. It is
due at the end of the period. The explanation of the equations used, the introduction and the
conclusion are not necessary this week. The discussion section can be as little as three sentences
commenting on whether the two measurements of the spring constant are equivalent given the
propagated errors. This mini-lab report will be graded out of 50 points
Concept
When an object (of mass m) is suspended from the end of a spring, the spring will stretch
a distance x and the mass will come to equilibrium when the tension F in the spring balances the
weight of the body, when F = - kx = mg. This is known as Hooke's Law. k is the force constant
of the spring, and its units are Newtons / meter. This is the basis for Part 1.
In Part 2 the object hanging from the spring is allowed to oscillate after being displaced
down from its equilibrium position a distance -x. In this situation, Newton's Second Law gives
for the acceleration of the mass:
Fnet = m a or
The force of gravity can be omitted from this analysis because it only serves to move the
equilibrium position and doesn’t affect the oscillations. Acceleration is the second time-
derivative of x, so this last equation is a differential equation.
To solve: we make an educated guess:
Here A and w are constants yet to be determined. At t = 0 this solution gives x(t=0) = A,
which indicates that A is the initial distance the spring stretches before it oscillates. If friction is
negligible, the mass will continue to oscillate with amplitude A. Now, does this guess actually
solve the (differential) equation? A second time-derivative gives:
Comparing this equation to the original differential equation, the correct solution was
chosen if w2 = k / m. To understand w, consider the first derivative of the solution:
−kx = ma
a = −
k
m
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
x
d 2x
dt 2
= −
k
m
x x(t) = A cos(ωt)
d 2x(t)
dt 2
= −Aω2 cos(ωt) = −ω2x(t)
James Gering
Florida Institute of Technology
11 - 4
Integrating gives
We assume the object completes one oscillation in a certain period of time, T. This helps
set the limits of integration. Initially, we pull the object a distance A from equilibrium and
release it. So at t = 0 and x = A. (one.
shooting method with Range kutta methodSetuThacker
how to solve the differential equation by using shooting method & range Kutta method.there is one problem statement, which is related to the temperature distribution problem over an iron rod. there is two graphs of the 2nd order and 4th order Range Kutta. there is one combine graph of the both the method which gives correct & closest answer.
Thank You...!!!
CHAPTER 5 Wave Properties of Matter and Quantum Mechanics I
5.1 X-Ray Scattering (review and some more material)
5.2 De Broglie Waves
5.3 Electron Scattering / Transmission electron microscopy
5.4 Wave Motion
5.5 Waves or Particles?
5.6 Uncertainty Principle
5.7 Probability, Wave Functions, and the Copenhagen Interpretation
5.8 Particle in a Box
Response of dynamic systems to harmonic excitation is discussed. Single degree of freedom systems are considered. For general damped multi degree of freedom systems, see my book Structural Dynamic Analysis with Generalized Damping Models: Analysis (e.g., in Amazon http://buff.ly/NqwHEE)
To Determine the Charge to Mass Ratio for Electron by JJ-Thomson’s MethodSachin Motwani
This demonstrates the practical method of determining charge to mass ratio. This experiment is generally undertaken in the first year of an engineering degree program.
Introduction to transient Heat conduction, Lamped System Analysis, Approxiamate Analytical and graphical method and Numerical method for one and two dimensional heat conduction by using Explicit and Implicit method
SSL8 Mass & Energy Analysis of Control SystemsKeith Vaugh
Conservation of mass
Mass and volume flow rates
Mass balance for a steady flow process
Mass balance for incompressible flow
Flow work and the energy of a flowing fluid
Energy transport by mass
Energy analysis of steady flow systems
Steady flow engineering devices
Nozzles and diffusers
Turbines and compressors
Throttling valves
Mixing chambers and heat exchangers
Pipe and duct flow
Energy analysis of unsteady flow processes
Determination of DensityRequired materials provided in tmackulaytoni
Determination of Density
Required materials provided in the Home Science Tools chemistry kit:
100mL graduated cylinder, balance (scale)
Required materials
not
provided in the Home Science Tools chemistry kit:
cell phone (with camera), metric ruler, 25-30 pennies, graph paper
Objectives:
to find the density of regular-shaped and irregular-shaped substances including graphing techniques
Introduction:
Density
is the intensive property of matter defined as the ratio of an object’s mass to its volume. In simpler words, density is the mass of an object divided by the volume which the object occupies. The term
intensive property
means that it is
independent
of the amount of the substance. The density of any substance remains the same, no matter the shape and size of the sample. The density of water at 4°C is 1.000 g/mL regardless if the sample size is 1 cup or 1 swimming pool. Thus, density is one of the characteristic properties which allows us to identify substances; it is fixed and has a unit of g/mL. As such, it is a useful tool to identify an unknown metal. One can calculate the density of an unknown metal and can match the value against a standard density table for its identification.
The density of a substance does change with a change in temperature. This change in density is
inversely proportional
to the change in temperature. This is to say, if the temperature rises, then the density decreases, and if the temperature falls, then the density increases. Cooling a substance causes its molecules to occupy a smaller volume, resulting in an increase in density. Hot water is less dense and will float on room-temperature water. Cold water is denser and will sink in room-temperature water.
Densities of various substances can be identified differently. For
regular
(shaped)
solids
, calculating the density is straightforward: simply weigh the solid and measure its dimensions, using a simple formula to calculate the volume. The density is calculated by dividing the mass by the volume. Each regular solid has its own formula for calculating its volume depending on the shape of the solid. The volume of a
rectangular
solid equals length times width times height. Note: 1 mL = 1 cm3. For
irregular
(shaped)
solids
, those that do
not
have a standard formula for calculating their volume, the volume can be determined by measuring the volume of liquid that the solid displaces. To do this, the solid is submerged in a liquid and the volume displaced is measured. This is done by taking an initial reading and a final reading and calculating the difference in volume. The mass of the object is then divided by this volume, and the density is determined.
Measuring the density of a liquid is very similar. Although the volume cannot be measured with a ruler, it can be determined using volumetric glassware, for instance, a graduated cylinder. The liquid’s mass is determined when this measured volume is weighed. Knowing the ...
11 - 3
Experiment 11
Simple Harmonic Motion
Questions
How are swinging pendulums and masses on springs related? Why are these types of
problems so important in Physics? What is a spring’s force constant and how can you measure
it? What is linear regression? How do you use graphs to ascertain physical meaning from
equations? Again, how do you compare two numbers, which have errors?
Note: This week all students must write a very brief lab report during the lab period. It is
due at the end of the period. The explanation of the equations used, the introduction and the
conclusion are not necessary this week. The discussion section can be as little as three sentences
commenting on whether the two measurements of the spring constant are equivalent given the
propagated errors. This mini-lab report will be graded out of 50 points
Concept
When an object (of mass m) is suspended from the end of a spring, the spring will stretch
a distance x and the mass will come to equilibrium when the tension F in the spring balances the
weight of the body, when F = - kx = mg. This is known as Hooke's Law. k is the force constant
of the spring, and its units are Newtons / meter. This is the basis for Part 1.
In Part 2 the object hanging from the spring is allowed to oscillate after being displaced
down from its equilibrium position a distance -x. In this situation, Newton's Second Law gives
for the acceleration of the mass:
Fnet = m a or
The force of gravity can be omitted from this analysis because it only serves to move the
equilibrium position and doesn’t affect the oscillations. Acceleration is the second time-
derivative of x, so this last equation is a differential equation.
To solve: we make an educated guess:
Here A and w are constants yet to be determined. At t = 0 this solution gives x(t=0) = A,
which indicates that A is the initial distance the spring stretches before it oscillates. If friction is
negligible, the mass will continue to oscillate with amplitude A. Now, does this guess actually
solve the (differential) equation? A second time-derivative gives:
Comparing this equation to the original differential equation, the correct solution was
chosen if w2 = k / m. To understand w, consider the first derivative of the solution:
−kx = ma
a = −
k
m
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
x
d 2x
dt 2
= −
k
m
x x(t) = A cos(ωt)
d 2x(t)
dt 2
= −Aω2 cos(ωt) = −ω2x(t)
James Gering
Florida Institute of Technology
11 - 4
Integrating gives
We assume the object completes one oscillation in a certain period of time, T. This helps
set the limits of integration. Initially, we pull the object a distance A from equilibrium and
release it. So at t = 0 and x = A. (one.
BIOEN 4250 BIOMECHANICS I Laboratory 4 – Principle Stres.docxtarifarmarie
BIOEN 4250: BIOMECHANICS I
Laboratory 4 – Principle Stress and Strain
November 13– 16, 2018
TAs: Allen Lin ([email protected]), Kelly Smith ([email protected])
Lab Quiz: A 10-point lab quiz, accounting for 10% of the lap report grade, will be given at the beginning of
class. Be familiar with the entire protocol.
Objective: The objective of this experiment is to measure the strains along three different axes surrounding
a point on a cantilever beam, calculate the principal strains and stresses, and compare the result
with the stress calculated from the flexure formula for such a beam.
Background: The ability to measure strain is critical to materials testing as well as many other applications in
engineering. However, strain gages that adhere to a surface can alter the local strain environment
if the material (or tissue) of interest is less stiff than the gage itself. For this reason, contact strain
gages (or strain gages that attach directly to a surface) are not typically used for the testing of soft
tissues such as ligament, arteries, or skin. However, when the material is on the stiffer side, or
when the absolute value of the strain is less important than the detection of the mere presence of
strain itself, contact strain gages are very useful. An example of a stiffer biological material would
be bone. However, due to the porous nature of bone, one needs to be extremely careful that the
strain gage is properly adhered to the material’s surface. Other applications range from real world
stress analysis of a structure (e.g., a wing of an aircraft during flight) to strain gages incorporated
into medical equipment to ensure proper function (e.g., gages wrapped around the tubing in a
hospital infusion pump to detect blockages in the line – since the tube swells more than it should
when the fluid path is occluded).
One common engineering loading case that involves a planar stress field (i.e., the only non-zero
stresses are in the same plane), is that of beam bending. Beam bending will be covered in greater
detail during lecture. However, in order to ensure you know the basics of what is going on in this
lab, we will cover some fundamental topics. The simplest case of beam loading is that of a
cantilever beam that is completely anchored at one end and loaded at a point along its length
(Fig. 1). In Figure 1, 𝑃 is the applied load, ℎ is the thickness of the beam (with 𝑐 as the half-
thickness), 𝑥 is the distance from the fixed wall to the location where we want to measure stress
and strain (point 𝑎), and 𝐿 is the length of the beam. There are a couple key points to know about
this loading scenario:
1. As the beam bends downward, the material above the midline (the dashed line) is in
tension and the material below that line is in compression.
2. At the top and bottom free surfaces, there is only axial stress, and zero shear stress.
3. At the midline (dashed line, also referred to as neutral axis)
Lagrange's Mean Value Theorem, also known as the Mean Value Theorem (MVT), is a fundamental result in calculus that describes the relationship between the slope of a tangent to a function's graph and the average rate of change of the function over an interval. It is a crucial tool in analyzing the behavior of functions and has wide-ranging applications in various areas of mathematics and science. The Mean Value Theorem states that if a function f satisfies the following conditions: 1. Establish inequalities: By comparing the slope of the tangent to the average rate of change, the Mean Value Theorem can be used to establish inequalities involving function values.
2. Prove Rolle's Theorem: Rolle's Theorem is a special case of the Mean Value Theorem that applies to functions that have zero values at the endpoints of an interval. 3. Analyze Rolle's Theorem: The Mean Value Theorem can be used to analyze the conditions for Rolle's Theorem and understand the geometric implications of the theorem.
In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their fundamental dimensions (such as length, mass, time, and electric charge) and units of measure (such as miles vs. kilometers, or pounds vs. kilograms vs. grams) and tracking these dimensions as calculations or comparisons are performed.
Physics 161Static Equilibrium and Rotational Balance Intro.docxrandymartin91030
Physics 161
Static Equilibrium and Rotational Balance
Introduction
In Part I of this lab, you will observe static equilibrium for a meter stick suspended horizontally. In Part II, you will observe the rotational balance of a cylinder on an incline. You will vary the mass hanging from the side of the cylinder for different angles.
Reference
Young and Freedman, University Physics, 12th Edition: Chapter 11, section 3
Theory
Part I: When forces act on an extended body, rotations about axes on the body can result as well as translational motion from unbalanced forces. Static equilibrium occurs when the net force and the net torque are both equal to zero. We will examine a special case where forces are only acting in the vertical direction and can therefore be summed simply without breaking them into components:
(1)
Torques may be calculated about the axis of your choosing:
(2)
where torque is specified by the equation:
(3)
where d is the lever arm (or moment arm) for the force. The lever arm is the perpendicular distance from the line of force to the axis about which you are calculating the torque.
Normally, up is "+" and down is "-" for forces. For torques, it is convenient to define clockwise as "-" and counterclockwise as "+". Whatever you decide to do, be consistent with your signs and make sure you understand what a "+" or "-" value for your force or torque means directionally.
Part II: Any round object when placed on an incline has tendency of rotating towards the bottom of an incline. If the downward force that causes the object to accelerate down the slope is canceled by another force, the object will remain stationary on the incline. Figure 1 shows the configuration of the setup. In order to have the rubber cylinder in static equilibrium we should satisfy the following conditions:
(4)
Figure 1: Experimental setup for Part II
The condition that the net force along the x-axis (which is conveniently taken along the incline) must be zero yields the relationship. (Prove this!)
Without static friction the cylinder would slide down the incline; the presence of friction causes a torque in clockwise (negative) direction. In order to have static equilibrium we need to balance that torque with a torque in counterclockwise direction. This is achieved by hanging the appropriate mass m.
Applying the last condition to the center of the cylinder will result in:
where r, the radius of the small cylinder (PVC fitting), is the moment arm for the mass m and R, the radius of the rubber cylinder, is the moment arm for the frictional force which accounts for M and m. Combining this equation with the equation for Ffr from above will result in:
(5)
(6)
By adjusting the mass m, we can observe how the equilibrium can be achieved.
Procedure
Part I: Static Equilibrium
Figure 2: Diagram of Torque Experiment Setup
1. Weigh the meter stick you use, including the metal hangers.
2. Attach .
What is greenhouse gasses and how many gasses are there to affect the Earth.moosaasad1975
What are greenhouse gasses how they affect the earth and its environment what is the future of the environment and earth how the weather and the climate effects.
Professional air quality monitoring systems provide immediate, on-site data for analysis, compliance, and decision-making.
Monitor common gases, weather parameters, particulates.
Observation of Io’s Resurfacing via Plume Deposition Using Ground-based Adapt...Sérgio Sacani
Since volcanic activity was first discovered on Io from Voyager images in 1979, changes
on Io’s surface have been monitored from both spacecraft and ground-based telescopes.
Here, we present the highest spatial resolution images of Io ever obtained from a groundbased telescope. These images, acquired by the SHARK-VIS instrument on the Large
Binocular Telescope, show evidence of a major resurfacing event on Io’s trailing hemisphere. When compared to the most recent spacecraft images, the SHARK-VIS images
show that a plume deposit from a powerful eruption at Pillan Patera has covered part
of the long-lived Pele plume deposit. Although this type of resurfacing event may be common on Io, few have been detected due to the rarity of spacecraft visits and the previously low spatial resolution available from Earth-based telescopes. The SHARK-VIS instrument ushers in a new era of high resolution imaging of Io’s surface using adaptive
optics at visible wavelengths.
Earliest Galaxies in the JADES Origins Field: Luminosity Function and Cosmic ...Sérgio Sacani
We characterize the earliest galaxy population in the JADES Origins Field (JOF), the deepest
imaging field observed with JWST. We make use of the ancillary Hubble optical images (5 filters
spanning 0.4−0.9µm) and novel JWST images with 14 filters spanning 0.8−5µm, including 7 mediumband filters, and reaching total exposure times of up to 46 hours per filter. We combine all our data
at > 2.3µm to construct an ultradeep image, reaching as deep as ≈ 31.4 AB mag in the stack and
30.3-31.0 AB mag (5σ, r = 0.1” circular aperture) in individual filters. We measure photometric
redshifts and use robust selection criteria to identify a sample of eight galaxy candidates at redshifts
z = 11.5 − 15. These objects show compact half-light radii of R1/2 ∼ 50 − 200pc, stellar masses of
M⋆ ∼ 107−108M⊙, and star-formation rates of SFR ∼ 0.1−1 M⊙ yr−1
. Our search finds no candidates
at 15 < z < 20, placing upper limits at these redshifts. We develop a forward modeling approach to
infer the properties of the evolving luminosity function without binning in redshift or luminosity that
marginalizes over the photometric redshift uncertainty of our candidate galaxies and incorporates the
impact of non-detections. We find a z = 12 luminosity function in good agreement with prior results,
and that the luminosity function normalization and UV luminosity density decline by a factor of ∼ 2.5
from z = 12 to z = 14. We discuss the possible implications of our results in the context of theoretical
models for evolution of the dark matter halo mass function.
This presentation explores a brief idea about the structural and functional attributes of nucleotides, the structure and function of genetic materials along with the impact of UV rays and pH upon them.
Nutraceutical market, scope and growth: Herbal drug technologyLokesh Patil
As consumer awareness of health and wellness rises, the nutraceutical market—which includes goods like functional meals, drinks, and dietary supplements that provide health advantages beyond basic nutrition—is growing significantly. As healthcare expenses rise, the population ages, and people want natural and preventative health solutions more and more, this industry is increasing quickly. Further driving market expansion are product formulation innovations and the use of cutting-edge technology for customized nutrition. With its worldwide reach, the nutraceutical industry is expected to keep growing and provide significant chances for research and investment in a number of categories, including vitamins, minerals, probiotics, and herbal supplements.
Comparing Evolved Extractive Text Summary Scores of Bidirectional Encoder Rep...University of Maribor
Slides from:
11th International Conference on Electrical, Electronics and Computer Engineering (IcETRAN), Niš, 3-6 June 2024
Track: Artificial Intelligence
https://www.etran.rs/2024/en/home-english/
THE IMPORTANCE OF MARTIAN ATMOSPHERE SAMPLE RETURN.Sérgio Sacani
The return of a sample of near-surface atmosphere from Mars would facilitate answers to several first-order science questions surrounding the formation and evolution of the planet. One of the important aspects of terrestrial planet formation in general is the role that primary atmospheres played in influencing the chemistry and structure of the planets and their antecedents. Studies of the martian atmosphere can be used to investigate the role of a primary atmosphere in its history. Atmosphere samples would also inform our understanding of the near-surface chemistry of the planet, and ultimately the prospects for life. High-precision isotopic analyses of constituent gases are needed to address these questions, requiring that the analyses are made on returned samples rather than in situ.
2. 2
Elasticity Measurements: Young Modulus Of Brass
1 Aims of the Experiment
The aim of this experiment is to measure the elastic modulus with as high precision as
possible. You will also find this experiment a valuable practice of error analysis and to
planning an experiment.
2 Skills Checklist
At the end of the experiment, you should have mastered and understood the following main
features:
• Basic error analysis
• Use of error analysis in planning experiments
• Selection of appropriate measuring instruments
• Measurements of quantities to the appropriate degree of accuracy.
3 Introduction
This experiment uses the bending of a beam
under an applied load to measure the Young
modulus of brass. Concentrate on the
experimental set-up, measurements and error
estimates, rather than on the detailed theory
of the experiment. You should, however,
know what the Young modulus is, and how
it enters into a bending situation.
For the experimental set-up shown in Figure
1, the Young modulus is given by
E
Mg
x
ac
bd
gac
mbd
= =
3
4
3
4
2
3
2
3
(1)
where a and c are as in Figure 1, b and d are the breadth and depth of the beam respectively, x
is the vertical deflection or elevation of the centre of the beam when a mass M is suspended,
and m is the average value of x/M, found from the slope of the graph of x plotted against M.
Shallow grooves have been cut in the upper surface of the bar to hold the knife edges from
which the load is suspended. The centre of the bar is marked, and scribed lines show suitable
positions for the supporting knife-edges. Please do not make any other marks on the bar.
4 Experiments
4.1 Preliminary Measurements (30 min)
Before proceeding with accurate measurements, measure quickly the displacement x for one or
two values of M to obtain a rough value for m = x/M. Make estimates or rough measurements
of a, b, c, and d. Calculate the Young modulus from equation 1; you should get a value near
1010
- 1011
Nm-2
.
4.2 Main Measurements (2 hr 30 min)
Mount the dial gauge with its pin resting on the mid-point of the bar and record the zero
Mg
½Mg½Mg ca a
x
Stirrup
Knife
Edge
Pillar
Figure 1 Experimental arrangement
3. 3
reading. Hang the mass holder from the mid-point of the bar passing through the lower ends
of the stirrups which hang from the beam. Measure the resulting vertical deflection x. Take a
series of measurements of x for masses M up to 6 kg. Then take readings while the bar is
unloaded to check for reproducibility. What should you do if the readings do not reproduce
exactly? Check the positions of the knife edges frequently, and that they remain square.
Plot a graph of x against M on mm graph paper whilst you are taking the results, not
afterwards: this will enable you to spot "rogue" results immediately they arise, and you can
check them at once. Use the Mathcad least squares fitting spreadsheet to find the best value of
m and its error.
You will find that the experimental points do not all lie exactly on the straight line. To
investigate these deviations, measure the deflection x for a mass of about 2-3 kg. Make
several measurements at this mass, unloading and replacing the mass between each reading.
Use Excel to calculate the mean value of x, and to estimate the standard deviation of the
observations about the mean, σ(x), NOT the standard deviation (standard error) of the mean,
u(x).
Since each point on the graph was obtained by measuring one value of x for each value of M,
so the error in x for any one of these points should be the error in x when only one reading of x
is made, and this is σ(x). Put errors bars equal to ±σ(x) on each point of your graph. About
2/3 of the error bars should intersect the best-fit straight line; is this true for your results?
What can you say if all the error bars intersect the line? What can you say if only a few of the
error bars intersect the line? What is your value of χ2
and how does that relate to the above
observations?
Another (very rough) check on your results can be made. The computer will give a value for
the error in m, σ(m). However, assuming that the error in M is negligible, and that there are p
points on your graph, then the fractional error in m, σ(m)/m, is roughly 1/√p times the
fractional error in x, σ(x)/x. Use both methods to estimate σ(m); the two values should agree
to within a factor of 2 or so.
The checks described above are examples of consistency checks. If the different methods do
not agree, then something is wrong somewhere. Unfortunately, agreement does not guarantee
that nothing is wrong!
4.3 Completing the Experiment (1 hr)
The object of this part is to use the theory of propagation of errors as an aid in planning the
rest of the experiment, and then to use these results in measuring the remaining variables to an
appropriate accuracy using an appropriate instrument which you have selected on the basis of
your results.
The fractional error in E in terms of the fractional errors in m, a, b, c and d is
( ) ( ) ( ) ( ) ( ) ( )σ σ σ σ σ σE
E
m
m
a
a
b
b
c
c
d
d
=
+
+
+
+
2 2 2 2 2 2
2 3 (2)
For maximum efficiency the percent errors in m, a, b, c², and d3
should be equal, i.e. the five
terms on the RHS of eq. (2) should be the same. But you have a value for one of these terms!
So knowing the percent error in m, use equation (2) to estimate the errors σ(a) etc. you want
in a, b, c, and d. What error in E would you expect to get at the end of the experiment?
Given that the resolution of a wood rule is 1mm, that of a good steel rule is 0.1mm, that of
micrometer screw gauge is 0.001mm (1µm),Vernier callipers is 0.02 mm and that of a
4. 4
select the appropriate instrument to make accurate measurements of a, b, c, and d
Make accurate measurements of a, b, c, and d, and find the errors in these values. The
following points should be noted.
1. Make and record repeated measurements. Use Excel to analyse your results. The errors
you want here are the standard errors of the means, NOT the standard deviations. The
estimates of errors that you have made before are targets, and it may not be possible to
reach them with the equipment you have available. Do not try to better any target unless
you can improve on all of them with little effort! Calculate the error in c² and d3
from the
errors in c and d.
2. When measuring b and d, check the beam for uniformity. Does it matter if the beam is
non-uniform outside the supports?
3. When you use a micrometer, use the ratchet mechanism to tighten the jaws; do not twist the
main barrel. Unless you are a skilled operator, using the barrel will give erratic results, and
you may damage the object being measured if it is squashed too fiercely. Remember it is
possible to read a micrometer to 0.1 of the smallest graduation on its barrel.
4. Use equation (1) to calculate the Young modulus of the brass. Use equation (2) to find the
error in your value of E, using the actual errors you have just found, not the target estimates
from the earlier work!
5 Discussion
1. The beam provided is made of brass. Compare your results with values of E for brass
given in books of tables. Take the experimental uncertainty in your value of E into
account.
2. Sketch a bent beam and mark the regions of tensile and compressive strains. Show the
stresses producing these strains; do they form a couple? Use your sketch to explain why
bending a beam allows the measurement of the tensile Young modulus.
3. Consider the advantages and disadvantages of this method compared with the direct
stretching of the beam or a wire of the same material. Can you think of any other ways of
measuring the Young modulus?
4. What are the main sources of error in E? Is the limit to your accuracy variation due to e.g.
non-uniformity or to the precision of your measuring equipment? Is it feasible, bearing in
mind the time and equipment available, to reduce these errors? Is it really essential to
reduce the errors?
5. Equation (1) relates the elevation per unit mass x/M to the lengths a and c. It is clear that
x/M=0 for a=0 or c=0, and therefore for a beam of fixed length (c+2a) there must be a
value of c for which x/M is a maximum. If you have time, show that this is achieved when
c=4a. Why should you choose c and a to satisfy this relation? Does it matter if c is not
exactly equal to 4a?
6 References
Newman F.H., Searle V.H.L., The General Properties of Matter, 5th edition, (Edward
Arnold, London) 1957.
5. 5
Sprackling M.T., Liquids and Solids, (Routledge and Kegan Paul, London) 1985.
7 Appendix 1
These notes summarise the general forms of the response of a solid body to applied forces.
Further details can be found in Sprackling, chapter 2, or Newman and Searle, chapter 5.
7.1 Elastic Moduli
The moduli of elasticity of a material are measures of its resistance to a change of size or
shape under the influence of a set of applied forces. The applied forces constitute stresses,
expressed as a force per unit area, and the resulting deformation is described as a strain, which
is the ratio of the change in some dimension to an original dimension. If the strain returns to
zero when the stress is removed, the deformation is said to be elastic. In many materials for
small elastic strains, the deformation obeys Hooke's law which states that the stress is
proportional to the strain. The constant of proportionality is the elastic modulus, so that
Modulus = stress/strain
Area A
F
F
F
F
ϕ
δll
Cross-sectional
area A
Figure 2 Shear modulus and Young modulus
Materials can be deformed in several different ways, corresponding to different moduli of
elasticity. For isotropic materials (i.e. those whose properties are the same in all directions)
there are three moduli of particular importance.
1. The bulk modulus, K, corresponds to a change of volume without change of shape. This
applies to deformation under a uniform hydrostatic pressure, The stress is the pressure p,
and the strain is the change in volume -δV (negative because increase in pressure produces
a decrease in volume) divided by the initial volume V. The bulk modulus is given by K =
p/(-δV/V) .
2. The shear modulus, n, also known as the modulus of rigidity, corresponds to a change of
shape at constant volume. The stress is the tangential force F acting over a surface of area
A divided by A, and the strain is represented by the angle of shear, φ. The modulus is given
by n = (F/A)/φ.
3. Young modulus, E, is used when a change in length of a sample is produced when a
tensile or compressive stress is applied, with no external forces applied to the side surfaces
of the specimen. The stress is the tensile force divided by the cross-sectional area of the
sample, and the strain is the change in length δl divided by the original length l, The
modulus is given by E = (F/A)/(δl/l).
At the same time, there is also a contraction or expansion at right angles to the tensile or
compressive stress. The ratio of the magnitude of this transverse strain to the principal strain
is called the Poisson ratio, σ.
6. 6
These various moduli are related by the equations
( ){ }
K
E
=
−3 1 σ
and
( ){ }
n
E
=
+2 1 σ
.
All types of elastic deformation of isotropic media can be described in terms of any two of
these moduli. Note however that an elastic modulus only has meaning if Hooke's law is
obeyed; the ratio stress/strain is not constant for a non-linear material, even if it is perfectly
elastic.
7.2 Theory of A Bending Beam
When a beam is bent into an arc of radius R, the material in one part of the beam is stretched
and under tension, and the material in another part is under compression. These tensile and
compressive strains are produced by related stresses, which combine to form a couple called
the bending moment G. R and G are obviously related to each other, and a little thought
should convince you that the Young modulus of the material, E, and the shape of the bar are
also involved. The full theory of the elastic bending of a beam or cantilever is given in
Newman and Searle (section 5.9). There it is shown (eq. 5.12) that E is given by
E
GR
I
= (A1)
where I, called the second moment of area, is the factor that takes account of the shape of the
bar. (Newman and Searle use the symbol Y for the Young Modulus. They call the second
moment of area I the geometrical moment of inertia, and they use the combination Ak² for
this.) In this experiment, the bending moment is constant along the length AB of the beam,
and its value is
G Mga=
1
2
(A2)
From the geometry of a circle, it can be shown that the radius R is related to the elevation x of
the mid-point by
R
c
x
=
8
(A3)
provided that x is much smaller than R
For a rectangular beam of thickness d and width b, the second moment of area I about its
neutral axis is
I b y y
bd
d
d
= =
−
∫ 2
2
2 3
12
d
/
/
(A4)
Substituting expressions (2), (3) and (4) into (1) gives an expression for the Young modulus in
terms of easily measurable quantities:
E
Mgac
xbd
gac
bd m
= =
3
4
3
4
2
3
2
3 (A5)
where m is the average value of x/M. This is equation (1) of the main text.