The accuracy of an instrument is the extent to which the reading it gives might be wrong. Accuracy is often quoted as a percentage of the full-scale deflection (f.s.d.) of the instrument.
Measurements of Errors - Physics - An introduction by Arun Umraossuserd6b1fd
This note is calculus based error measurement. It explains how calculus can be used to find errors in functions and measurements. Best for quick revision before CBSE Board Examination.
The accuracy of an instrument is the extent to which the reading it gives might be wrong. Accuracy is often quoted as a percentage of the full-scale deflection (f.s.d.) of the instrument.
Measurements of Errors - Physics - An introduction by Arun Umraossuserd6b1fd
This note is calculus based error measurement. It explains how calculus can be used to find errors in functions and measurements. Best for quick revision before CBSE Board Examination.
The standard deviation is a measure of the spread of scores within a set of data. Usually, we are interested in the standard deviation of a population.
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Chapter 3: Describing, Exploring, and Comparing Data
3.1: Measures of Center
These notes are of chemistry class 11th first chapter which are strictly according to CBSE & state Board. This notes covers Some basics concepts of chemistry i.e. Branches of chemistry, classification of matter & many more..
The standard deviation is a measure of the spread of scores within a set of data. Usually, we are interested in the standard deviation of a population.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 3: Describing, Exploring, and Comparing Data
3.1: Measures of Center
These notes are of chemistry class 11th first chapter which are strictly according to CBSE & state Board. This notes covers Some basics concepts of chemistry i.e. Branches of chemistry, classification of matter & many more..
Complex Number,
Mathematical Requirement,
Geometrical Requirement,
Conventions,
Representation,
Modulus And Argument,
Real Vs Complex Numbers ,
Purely Real Complex Number ,
Purely Imaginary Complex Number ,
Equality Between Two Complex Numbers ,
Operation on Complex Number ,
Polar Form of Complex Number ,
About Other Than Origin ,
Properties of Complex Number ,
Logarithm of Complex Number ,
Parametric Conversion,
De Moivre’s Theorem ,
Properties of the Arguments ,
Roots of a Complex Number ,
Analytical Complex Numbers ,
Limit & Continuity,
Poles & Zeros,
Complex Derivative ,
Complex Integration ,
o level mathematics complete revision guide for quick and deep revision to solve questions with meticulous techniques and tricks with question to solve for clearance of concepts
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Functions
Function
Type of Function
Algebraic Function
Trigonometric Function
Logarithmic Function
Integral Function
Rational Fraction
Rational Function
Explicie & Implicit Function
Unique Values of Function
Odd & Even Function
Properties of Odd-Even Functions
Homogeneous Function
Linear Function
Inverse Function
Sampling of Function
Piece-wise Function
Sketch the Function
Straight Line
Domain & Range
Ordered Pairs
Ordered Pairs from Function
Function in Different Domains
Increasing-Decreasing Function
Modulo Function
Sub-equations of Modulo Function
Inequality
Inequalities
Properties of Inequalities
Transitivity
Addition and subtraction
Multiplication and Division
Additive inverse
Multiplicative Inverse
Interval Notion In Inequality
Answer Set
For Real Numbers
For Integers
Compound Statements
Linear Inequality
Quadratic Inequality
Quotients and Absolute Inequalities
Energy & Work, Measurement of Energy, Measurement of Energy, Kilo Watt Hour, Energy, Energy Exchange, Methods of Energy Exchange, Type of Energy Source, Energy Equilibrium, Heat Capacity, Specific Heat Capacity, Heat Exchange, Phases of Matter, Latent Energy, Work, Work In Linear Motion, Work Energy Relation, Work In Vertical Motion, Efficiency, Efficiency, Efficiency in Parallel Network, Efficiency in Series Network, Gain
Polynomial (poly), Polynomial Division (pdiv), Residue (residu), Roots of Polynomial (roots) Roots of Quadratic Equation Roots of Polynomials, Simplification (simp), Flip Matrix Dimension (flipdim), Permutation (permute), Matrix Replication (repmat), Cumulative Product (cumprod), Cumulative Summation (cumsum), Kronekar Product (kron), Product (prod), Summation (sum) . Matrices, Determinant, Transpose Matrix, Diagonal Matrix, Identity Matrix, Inverse of Matrix, Normalization of Matrix, Normalzation Factor (norm), Permutation & Transpositi . Interpolation, Lineary Interpolation (interp), Two Data Linear Interpolation (interpln), Linear Interpolation (linear interpn) . Symbolic, Symbolic Addition (addf), Symbolic Left Division (ldivf), Symbolic Multiplication (mulf), Symbolic Right Division (rdivf), Symbolie Subtraction (subf) Special Functions . Bessel Function, Bessel Function of Second Kind (besselj), Bessel Function of Second Kind (bessely), Bessel Function of First Kind (besseli), Hyperbolic Bessel Function of Second Kind (besselk), Bessel Function as Hankel Function (besselh) . Beta Function (beta) . Gamma Function (gamma)
, Logarithm of Gamma Function (gammaln) . Legendre Function (legendre) . Error Functions, Calculate Error (calerf), Error Function (erf), Complementary Error Function (erfc), Inverse Error Function (erfinv) Directory & Files . Directory Operation, Change Directory (chdir), Create Directory (createdir), List Current Directory (dir), Whether A Directory (isdir), List Current Directory (ls), Make Directory (mkdir), Present Working Directory (pwd), Remove Directory (removedir), Remove Directory (rmdir), Base Name of File (basename), Directory name (dirname), Extension Name (fileext), Parts of File (fileparts), Path Separator (filesep), Fill File Name (fullfile), Full File Path (fullpath), Drives In System (getdrives), Temporary Name (tempname) . file, Open a file (file), Close a file (file), Copy File (copyfile), Delete File (deletefile), File Information (fileinfo), Search Files (findfiles), Write Matrix to File (fprintfMat), Read Matrix from File (fscanfMat), Is It A File (isfile), Close a File Stream (mclose), Delete a File (mdelete), Check End of File (meof), Write Data to File (mfprintf), Read Data Stream (mfscanf), Read Line By Line (mgetl), Open File Stream (mopen), Move File (movefile), Write Bytes to Stream (mput), Current Position of Binary File (mseek), Length of Passed Data (mtell), Environment (getenv), Process ID (getpid), Stop Execution (halt), Prompt The Message (input), Read From a File (read), Set Environment (setenv), Write Formatted Output to File (write) Strings . String Operations, ASCII, Blanks, Convret To String (convstr), Empty String (emptystr), Evaluate (eval), Array Index (grep), Is Argument Alphabets or Numeric (isalphanum), Is Argument Digit (isdigit), Is Argument Letter (isletter),
Errors Contents,Errors, Measurements, Estimates and Errors, Accuracy & Precision, Discrepancy Acceptance & Measured Value Why Errors are Absolute? Errors, Calculation of Errors, Limitation of Errors, Error By Calculus, Application of Error Analysis, Rounding Off Errors, Error in Sample Measurements, Definitions
Motion, Body May Have Many Types of Energies, Linear Motion, Straight Path Motion Displacement, Velocity Distance Speed, Direction Plane Uniform Motion Average Speed, Average Velocity Non Uniform Motion Instantaneous Velocity And Instantaneous Speed Acceleration Single Variable Motion Motion In Cartesian Plane, Derivative of Unit Vector, Newton’s Law Of Motion, Net Force Equations of Motion In Horizontal Plane, Motion In Vertical Plane Effect Of Center of Mass In Vertical Motion, Velocity Curve, Linear Velocity Curve, Vertical Throw, Vertical Drop Relative Velocity Equations of Motions for Relative Motion Motion in River, First Type, Second Type Motion in Lift Lift is In Rest Lift Moving Upward Direction Lift Moving Downward Direction, Motion of Parachute Friction, Walking Man Coefficient of Friction, What is R?, Object at Rest, Object is About to Move, Object is Moving Why μk < μs?, Arun of Mo of Mo V ti V ti Umrao tion In H tion In H l Pl l Pl, MOTION, Angle of Friction Angle of Repose, Law of Dry Friction, Rolling & Sliding Wheal, Motion of Coupled Objects System Is In Rest, System is In Motion, Impulse Energy Kinetic Energy, Potential Energy Total Mechanical Energy, Potential Energy Graph, Momentum, Conservation of Momentum Rebounding of particle, Explosion of particle, Collision Oblique Collisions Rules of Elastic Collision Perfect Plastic Collision, Conservation of Momentum of Varying particle Mass, Momentum & Newton’s Law of Motion First Law, Second Law Third Law Two Dimensional Collisions, Scattering Angle After Collision
Expression, Index Expression, Reshape Elements (reshape), Is Element an Index (isindex) Arithmetic Operators Addition & Subtraction, Multiplication, Division Power, Unary Operations Left Division (ldivide) Matrix Left Division (mldivide), Subtraction (minus), Matrix Power (mpower), Matrix Right Division (mrdivide) Recursive Product (mtimes), Element-wise Recursive Product (times) Element-wise Right Division (rdivide), Addition of Elements (plus), Power (power) Unary Subtraction (uminus), Unary Addition (uplus), Comparison Operator Equals (eq) Greater Than or Equal (ge), Greater Than (gt) Is Arguments are Equal (isequal) Less Than or Equal (le), Less Than (lt) Not Equals (ne), Evaluation, Arithmetic, Absolute Value (abs), Ceiling (ceil), Truncate Fraction (fix) Geometry, Cartesian to Polar Conversion (cartpol), Polar to Cartesian Conversion (polcart), Spherical to Cartesian Conversion (sphcart), Cartesian to Spherical Conversion (cartsph), Logarithm, Natural Logarithm (log), Logarithm Base Ten (log), Unit Increment Logarithm (logp), Binary Base Logarithm (log) Exponential Base (e), Matrix, Transpose of Matrix (transpose) Complex Conjugate Transpose of Matrix (ctranspose) Dot Product (dot) Cross Product (cross) Determinant (det) Identity Matrix (eye), Eigenvalues (eig), Eigens (eigs), Inverse of Matrix (inv) Linear Equation Solver (linsolve) Type of Matrix (matrix type) Normalized Matrix (norm), Null Space Matrix (null), Orthogonal Basis (orth), Rank of Matrix (rank) Trace of Matrix (trace), Cholesky Matrix (chol), Inverse Cholesky Matrix (cholinv), Matrix Exponential (expm), Logarithmic Matrix (logm), Square Root of Matrix (sqrtm), Kronecker Product (kron) Diagonal Matrix (diag), Single Value Decomposition (svd) Lower Upper Decomposition (lu) Lower Upper Composition (qr), Length of Matrix (length) Special Functions, Bessel Function of First Kind (besselj), Bessel Function of Second Kind (bessely), Hyperbolic Bessel Function of First Kind (besseli) Hyperbolic Bessel Function of Second Kind (besselk), Bessel Function as Hankel Function (besselh), Beta Function (beta), Gamma Function (gamma), Error Function (erf), Complementary Error Function (erfc), Inverse Error Function (erfinv), Legendre Function (legendre) Differentiation Derivative (diff), Linear ODE Solver (lsode) Options For Linear ODE Solver (lsode options) Differential Algebraic System Solver (dassl) Differential Algebraic System Solver Options (dassl options) Differential Algebraic Equations (daspk) Differential Algebraic Equations Options (daspk oot Solver Options (dasrt options) Integration Quadratic Integration (quad) Vectorized Quadratic Integration (quadv), Quadratic Lobatto’s Integration (quadl) Quadratic Gauss-Kronrod Integration (quadgk) Quadratic Clenshaw-Curtis Integration
Expression, Index Expression, Reshape Elements (reshape), Is Element an Index (isindex) Arithmetic Operators Addition & Subtraction, Multiplication, Division Power, Unary Operations Left Division (ldivide) Matrix Left Division (mldivide), Subtraction (minus), Matrix Power (mpower), Matrix Right Division (mrdivide) Recursive Product (mtimes), Element-wise Recursive Product (times) Element-wise Right Division (rdivide), Addition of Elements (plus), Power (power) Unaryn or Equal (le), Less Than (lt) Not Equals (ne), Evaluation, Arithmetic, Absolute Value (abs), Ceiling (ceil), Truncate Fraction (fix) Geometry, Cartesian to Polar Conversion (cartpol), Polar to Cartesian Conversion (polcart), Spherical to Cartesian Conversion (sphcart), Cartesian to Spherical Conversion (cartsph), Logarithm, Natural Logarithm (log), Logarithm Base Ten (log), Unit Increment Logarithm (logp), Binary Base Logarithm (log) Exponential Base (e), Matrix, Transpose of Matrix (transpose) Complex Conjugate Transpose of Matrix (ctranspose) Dot Product (dot) Cross Product (cross) Determinant (det) Identity Matrix (eye), Eigenvalues (eig), Eigens (eigs), Inverse of Matrix (inv) Linear Equation Solver (linsolve) Type of Matrix (matrix type) Normalized Matrix (norm), Null Space Matrix (null), Orthogonal Basis (orth), Rank of Matrix (rank) Trace of Matrix (trace), Cholesky Matrix (chol), Inverse Cholesky Matrix (cholinv), Matrix Exponential (expm), Logarithmic Matrix (logm), Square Root of Matrix (sqrtm), Kronecker Product (kron) Diagonal Matrix (diag), Single Value Decomposition (svd) Lower Upper Decomposition (lu) Lower Upper Composition (qr), Length of Matrix (length) Special Functions, Bessel Function of First Kind (besselj), Bessel Function of Second Kind (bessely), Hyperbolic Bessel Function of First Kind (besseli) Hyperbolic Bessel Function of Second Kind (besselk), Bessel Function as Hankel Function (besselh), Beta Function (beta), Gamma Function (gamma), Error Function (erf), Complementary Error Function (erfc), Inverse Error Function (erfinv), Legendre Function (legendre) Differentiation Derivative (diff), Linear ODE Solver (lsode) Options For Linear ODE Solver (lsode options) Differential Algebraic System Solver (dassl) Differential Algebraic System Solver Options (dassl options) Differential Algebraic Equations (daspk) Differential Algebraic Equations Options (daspk options) Differential Algebraic System Root Solver (dasrt), Differential Algebraic System Root Solver Options (dasrt options) Integration Quadratic Integration (quad) Vectorized Quadratic Integration (quadv), Quadratic Lobatto’s Integration (quadl) Quadratic Gauss-Kronrod Integration (quadgk) Quadratic Clenshaw-Curtis Integration
Force and its application for k12 studentsArun Umrao
Force changes the state of body. If body is in rest and a force is applied on it, body came in motion. Similarly, a force bring a body to rest from its motion if applied force is in opposite direction to the direction of momentum of the body. Unit of force is Kg m/s2 . Second unit of force is Newton represented by N, honoring to Sir James Newton. Mass of a body is m and force F is applied on it then mass force relation is F = ma (1) While we discuss the physics’ rule, we always take ideal conditions not real one. For ex- ample, in Newton’s force law, “body” means tiny, round, symmetrical particle of sufficient large mass but not too much small in size. Its center of mass lies at its center. As the “size” of body increases, the environmental phenomenon shall affect the motion of body in several ways, by means of frictional or drag force.
Electric Field Contents 1 Electric Field 3 1.1 Electric Field . 3 1.1.1 Electric Field (Quantitatively Approach) 3 1.1.2 Electric Dipole 6 1.1.3 Electric Field Due To Electric Dipole 6 1.1.4 3-Dimensional Electric Field Problems . 16 1.2 Numerical Computation . 20 1.2.1 Electric Field . 20 1.2.2 Electric Potential . 24 1.3 Displacement Field 26 1.4 Electric Force . 27 1.4.1 Electric Dipole in Electric Field . 27 1.4.2 Oil Drop Experiment . 32 1.5 Charge Density 34 1.6 Motion of Charged Particle in Electric Field 34 1.6.1 Motion of Charge in Uniform Electric Field 34 1.7 Relative Permittivity . 37 1.8 Electric Field Due To Charge Distribution . 37 1.8.1 Charged Rod At Axial Position . 38 1.8.2 Charged Rod On Equatorial Position 39 1.8.3 Charged Rod On Un-Symmetrical Position . 42 1.8.4 Charged Ring At Position On Its Axis . 45 1.8.5 Charged Disk On Its Axis 46 1.8.6 Cavity in a Non-Conducting Sphere . 49 1.9 Electric Force on Surface of Conductor
What is function? 1. A function f relates with each element of x of a set, say Df , with exactly one element y of another set, say Rf . 2. Df is called domain of function f. 3. Rf is called range of function f. 4. x is independent variable. 5. y is called dependent variable.
Angular Motion Contents 1 Angular Motion 3 1.1 Rigid Body 3 1.1.1 Axis of Rotation . 3 1.1.2 Moment of Force . 4 1.1.3 Equilibrium & Stability . 7 1.1.4 Equilibrium Point 8 1.1.5 Equilibrium of Moment of Force 9 1.1.6 Center of Mass 10 Reduction Method 11 Vector Form . 14 Cartesian Method 15 Calculus Method . 18 Sign Conventions . 19 Objects of Non-Homogeneous Form 20 Homogeneous Object With Varying Density 24 1.1.7 Radius of Gyration 24 1.1.8 Center of Gravity . 27 Reduced Mass 28 1.2 Angular Motion 28 1.2.1 Angular Velocity . 29 1.2.2 Linear Velocity & Angular Velocity . 29 1.2.3 Linear & Angular Accelerations . 30 1.2.4 Angular Momentum - Moment of Momentum . 30 1.2.5 Rolling of Object . 35 1.2.6 Angular Motion In Inclined Plane . 36 1.2.7 Power in Rotational Motion . 38 1.3 Inertia . 38 1.3.1 Moment of Inertia 39 1.3.2 Law of Inertia . 39 Law of Parallel Axis . 39 Law of Perpendicular Axis 42 1.3.3 Relation Between Torque and Inertia 43 1.3.4 Moment of Inertia of Rod 45 1.3.5 Moment of Inertia of Rectangular Body 49 1.3.6 Moment of Inertia of Ring 50 1.3.7 Moment of Inertia of Disk 50 1.3.8 Moment of Inertia of Solid Sphere . 53 1.3.9 Moment of Inertia of Cylinder . 56 1.4 Energy . 66 1.4.1 Angular Kinetic Energy .
(May 29th, 2024) Advancements in Intravital Microscopy- Insights for Preclini...Scintica Instrumentation
Intravital microscopy (IVM) is a powerful tool utilized to study cellular behavior over time and space in vivo. Much of our understanding of cell biology has been accomplished using various in vitro and ex vivo methods; however, these studies do not necessarily reflect the natural dynamics of biological processes. Unlike traditional cell culture or fixed tissue imaging, IVM allows for the ultra-fast high-resolution imaging of cellular processes over time and space and were studied in its natural environment. Real-time visualization of biological processes in the context of an intact organism helps maintain physiological relevance and provide insights into the progression of disease, response to treatments or developmental processes.
In this webinar we give an overview of advanced applications of the IVM system in preclinical research. IVIM technology is a provider of all-in-one intravital microscopy systems and solutions optimized for in vivo imaging of live animal models at sub-micron resolution. The system’s unique features and user-friendly software enables researchers to probe fast dynamic biological processes such as immune cell tracking, cell-cell interaction as well as vascularization and tumor metastasis with exceptional detail. This webinar will also give an overview of IVM being utilized in drug development, offering a view into the intricate interaction between drugs/nanoparticles and tissues in vivo and allows for the evaluation of therapeutic intervention in a variety of tissues and organs. This interdisciplinary collaboration continues to drive the advancements of novel therapeutic strategies.
A brief information about the SCOP protein database used in bioinformatics.
The Structural Classification of Proteins (SCOP) database is a comprehensive and authoritative resource for the structural and evolutionary relationships of proteins. It provides a detailed and curated classification of protein structures, grouping them into families, superfamilies, and folds based on their structural and sequence similarities.
Multi-source connectivity as the driver of solar wind variability in the heli...Sérgio Sacani
The ambient solar wind that flls the heliosphere originates from multiple
sources in the solar corona and is highly structured. It is often described
as high-speed, relatively homogeneous, plasma streams from coronal
holes and slow-speed, highly variable, streams whose source regions are
under debate. A key goal of ESA/NASA’s Solar Orbiter mission is to identify
solar wind sources and understand what drives the complexity seen in the
heliosphere. By combining magnetic feld modelling and spectroscopic
techniques with high-resolution observations and measurements, we show
that the solar wind variability detected in situ by Solar Orbiter in March
2022 is driven by spatio-temporal changes in the magnetic connectivity to
multiple sources in the solar atmosphere. The magnetic feld footpoints
connected to the spacecraft moved from the boundaries of a coronal hole
to one active region (12961) and then across to another region (12957). This
is refected in the in situ measurements, which show the transition from fast
to highly Alfvénic then to slow solar wind that is disrupted by the arrival of
a coronal mass ejection. Our results describe solar wind variability at 0.5 au
but are applicable to near-Earth observatories.
Seminar of U.V. Spectroscopy by SAMIR PANDASAMIR PANDA
Spectroscopy is a branch of science dealing the study of interaction of electromagnetic radiation with matter.
Ultraviolet-visible spectroscopy refers to absorption spectroscopy or reflect spectroscopy in the UV-VIS spectral region.
Ultraviolet-visible spectroscopy is an analytical method that can measure the amount of light received by the analyte.
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.
Richard's aventures in two entangled wonderlandsRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
Deep Behavioral Phenotyping in Systems Neuroscience for Functional Atlasing a...Ana Luísa Pinho
Functional Magnetic Resonance Imaging (fMRI) provides means to characterize brain activations in response to behavior. However, cognitive neuroscience has been limited to group-level effects referring to the performance of specific tasks. To obtain the functional profile of elementary cognitive mechanisms, the combination of brain responses to many tasks is required. Yet, to date, both structural atlases and parcellation-based activations do not fully account for cognitive function and still present several limitations. Further, they do not adapt overall to individual characteristics. In this talk, I will give an account of deep-behavioral phenotyping strategies, namely data-driven methods in large task-fMRI datasets, to optimize functional brain-data collection and improve inference of effects-of-interest related to mental processes. Key to this approach is the employment of fast multi-functional paradigms rich on features that can be well parametrized and, consequently, facilitate the creation of psycho-physiological constructs to be modelled with imaging data. Particular emphasis will be given to music stimuli when studying high-order cognitive mechanisms, due to their ecological nature and quality to enable complex behavior compounded by discrete entities. I will also discuss how deep-behavioral phenotyping and individualized models applied to neuroimaging data can better account for the subject-specific organization of domain-general cognitive systems in the human brain. Finally, the accumulation of functional brain signatures brings the possibility to clarify relationships among tasks and create a univocal link between brain systems and mental functions through: (1) the development of ontologies proposing an organization of cognitive processes; and (2) brain-network taxonomies describing functional specialization. To this end, tools to improve commensurability in cognitive science are necessary, such as public repositories, ontology-based platforms and automated meta-analysis tools. I will thus discuss some brain-atlasing resources currently under development, and their applicability in cognitive as well as clinical neuroscience.
3. 1.1. PHYSICAL UNITS 3
1Units
1.1 Physical Units
Physical quantities are measured in comparison of some standard parameters.
These parameters are called units. There are seven fundamental units which
are adapted as International Standards (SI Units) and other units are the
derivative of these fundamental units.
Base Unit Symbol Dimension
Mass kilogram kg [M]
Length metre m [L]
Time second s [T]
Temperature Kelvin K [θ]
Current Ampere A
Molecular Mass mole mol
Illumination Intensity candela cd
Except above seven fundamental units there are two units of angle.
1.1.1 Plane Angle
The unit of plane angle is radian and it is related with degree by
1 radian =
180
π
0
θ 450 π/4
Figure 1.1: Angle notation, angle in degree and angle in radian are shown in
first, second and third part of the figure respectively.
4. 4 Units
1.1.2 Solid Angle
This is the angle swept by curved surface at the center. Maximum solid angle
swept by any surface is 4π. The solid angle is given by
Solid Angle =
Surface Area
Square of distance of surface from center
r r̂
ˆ
dS
dS
dΩ
Figure 1.2: Solid Angle.
dΩ =
dS
r2
1.2 Significant Figures
The significant figures of a number are the digits necessary to take in cal-
culations. The significant figures must be carefully chosen in calculations to
find the accurate result. A digit in a figure is significant if presence of it
rejects the claim that the measured value is equal to the true value. This
is the probability of mistakenly rejecting equal values. For example, while
measuring length of pen by a scale precise upto millimeter, we find that it
is 8.6 centimeter long. The actual length of pen is 8.6 centimeter long. If
a student measures it 8.65 centimeter then digit 5 has no significance as we
do not reject the claim that pen is 8.65 centimeter long by using the same
scale. Due to precision upto millimeter, digit 5 is removed. But if another
student claim that length of pen is 8.7 centimeter then it shall be rejected as
8.6 centimeter (true value) is not equal to 8.7 centimeter (measured value).
So, a significant figure is that digit, whose presence rejects our claim that
measured value is equal to actual/true value.
5. 1.3. SCIENTIFIC NOTATION 5
1.3 Scientific Notation
A number n is written in scientific notation as
n = ±y.yy × 10±xxx
Here x, y are two arbitrary numbers. y.yy is called mantissa and xxx is
exponent. Mantissa is always written in decimal form and its modulus is
always greater than 1 and less than 10. For example 12.4533 = 1.24533×101
is correct format where 12.4533 = 0.124533 × 102
is wrong format.
1.3.1 Significant Figure
The digits of mantissa of a number expressed in scientific notation are called
significant figures or significant digits. A zero is always significant if it is
right of the non zero digits or right of the decimal. For example
Number Scientific Notation Significant Figures
132.00 1.3200 × 102
5
0.0123540 1.23540 × 10−2
6
-0.125 −1.25 × 10−1
3
25250 2.5250 × 104
5
A point is to be remember that 2.5250×104
and 2.525×104
are same values
but first has five significant figures while second has four significant figures.
In calculation first number is more accurate because it give the precision
value upto fourth place of decimal while second number gives precision upto
third place of decimal.
1.3.2 Identification of Significant Figures
There are no strict rules for identification of significant figures but some basic
rules should be followed during identification significant figures.
1. All non-zero digits are considered significant. For example 23.221 has
five significant figures and 2001.22 has six significant figures.
2. Zeroes between two non-zero digits are significant. For example 20010001
has eight significant figures and 400001 has six significant figures.
6. 6 Units
3. Leading zeros are not significant. For example 0014114 has five signif-
icant figures and 000014114 has five significant figures.
4. Trailing zeros in a number containing decimal point are significant. For
example 2.300 has four significant digits and 2.30 has three significant
digits. Both numbers are same in arithmetic but first one is more
accurate than second.
5. A bar may be placed at the last significant digit; any trailing zeros
following bar digit are not significant and can leave in calculation.
12.001̄0000 has only five significant digits.
6. All values betweenn zero and one, and values larger than one and with-
out decimal are first transform into scientific notations before finding
significant figure.
7. All values larger than one and with decimal, decimal part is also sig-
nificant. A number ends with decimal has only significant figures equal
to the digits which are before decimal.
1.4 Significant Figures in Measurement
There are two similar mathematical operations in unit. (i) Adding and sub-
traction and (ii) Multiplication and Division.
1.4.1 Significant Figures in Sum & Difference
If two physical quantities are in addition or in subtraction then the significant
figures in result are taken upto the decimal place which is equal to the least
decimal place in the physical quantities. For example, two physical quantities
are 1.230m and 2.30m and their addition is 3.530m. But 2.30m has least
decimal places (2 decimal places). Hence the result is 3.53m.
Solved Problem 1.1 Two physical quantities 2.31 and 4.591 are added to get
the result. Find the result with maximum precision.
Solution The sum of two given quantities is 2.31 + 4.591 = 6.901. The
final result should have same number of decimal digits which are in the
addend which have minimum decimal digits. So, the final answer will be
6.90.
7. 1.5. ROUNDING OFF 7
Solved Problem 1.2 Find the best result of the sum of of two numbers 2.01
and 2.0120.
Solution The sum of two given quantities is 2.01 + 2.0120 = 4.0220. The
best final result should have same number of decimal digits which are in the
addend which have minimum decimal digits. So, the best result will be 2.02.
Solved Problem 1.3 Find 2.45/0.2935.
Solution The result of division 2.45/0.293 is 8.3475. In significant rules,
in division, the result will have same number of significant digits that is in the
physical quantity having least number of significant digits. In this problem
least number of significant digits is 3. So, the result is 8.34.
1.4.2 Significant Figures in Multiplication & Division
If two physical quantities are in product or in division then the result has the
same number of significant digits that is in the physical quantity having least
number of significant digits. For example, if quantity 1.2482m is divided by
2.2s then the result is 0.5673m/s. This result is rounded off for two significant
digits as the least significant digits are two in the quantity 2.2s, one of the
both physical quantities. The rounded off result is 0.56m/s.
Solved Problem 1.4 Find the division of 2.459 by 1.2.
Solution The division of 2.459 by 1.2 is 2.459/1.2 = 2.0492. In significant
rules, in division, the result will have same number of significant digits that
is in the physical quantity having least number of significant digits. In this
problem least number of significant digits is 2. So, the result is 2.0
1.5 Rounding Off
Let a number is to be round off upto n significant digits as
1. If non-significant digit is less than 5 and is to be rounded off, the previ-
ous digit does not change and non-significant digit is simply dropped.
For example 2.301 is significant upto the second place of the decimal
then digit 1 is simply dropped and number is rounded off to 2.30. Sim-
ilarly 2.304 is rounded off upto two place of decimal as 2.30.
8. 8 Units
2. If non-significant digit is greater than 5 and is to be rounded off, the
previous digit (left side digit) is raised by 1 and non-significant digit
is dropped. For example 2.306 is significant upto the second place of
the decimal then digit 6 is simply dropped and previous digit (left side
digit) 0 is raised by 1 and number is rounded off to 2.31.
3. If the rounding off digit is greater than 5, rounding off digit is simply
dropped when the previous digit (left side digit) is even. For example,
when 2.3546 is rounded off, result is 2.354.
4. If the rounding off digit is greater than 5, rounding off digit is dropped
and previous digit (left side digit) is raised by 1 when the previous digit
(left side digit) is odd. For example, when 2.3576 is rounded off, result
is 2.358.
5. In mathematics and specially in pure science, the large numbers are
mostly written in scientific notation form. Hence these numbers are
rounded-off upto significant digits. To round-off a large number upto n
digits, the rounding-off process is always started from leftmost non-zero
digit. For example start process from 2 in 2500 and from 3 in 0.0325.
i Keep n significant digits and replace rest of digits with zeros.
ii Write number in scientific notation, i.e. a × 10±b
form, where a and
b are real and significant numbers.
iii Round off mantissa upto the last significant digit. For example
0.2336 can be written as 2.336 ×10−1
in scientific notation and if there
are only three significant digits then third significant digit becomes 4
and number becomes 2.34 × 10−1
. Rest of digits are dropped.
1.5.1 Effect of Rounding Off
Rounding off a fraction number is required to limit the observed value to
the nearest actual value. For example 2.505 rupees is rounded to 2.50 to
represent rupees in standard form. Sometime rounding off creates havoc
in calculations if rounding a number off is not taken properly, specially in
inverse functions. For example, take the function
f(t) =
1
n1 − n2
9. 1.6. DIMENSIONS 9
Assuming n1 = 300, n2 and f(t) = 20 are finite numbers. Comparatively
f(t) and n1 are not negligible. Now the value of n2 is
20 =
1
300 − n2
On simplification,
6000 − 20n2 = 1
Here comparatively 1 is negligible to 6000. If 1 is neglected then n2 becomes
300 and again function will tends to infinity. Hence 1 can not be neglected
here. Now value of n2 from above relation is ‘299.955’. If the same question
is modified as; n1 = 300, n2 = 299.955 and f(t) are finite numbers then what
will be the value of function f(t). Here n2 is slightly different to n1. If n2
is upper rounded off then value of function f(t) tends to infinity and if n2 is
taken as it is then function value is
f(t) =
1
n1 − n2
=
1
300.000 − 299.955
Or, f(t) = 22.222. Hence rounding off the number is not permitted in these
type of cases.
1.6 Dimensions
Dimensional analysis is a tool to find or check relations among physical quan-
tities by using their dimensions. The dimension of a physical quantity is the
combination of the basic physical dimensions (usually mass, length, time,
electric charge, and temperature) which describe it. Dimensional analysis is
based on the fact that a physical law must be independent of the units used
to measure the physical variables. A straightforward practical consequence
is that any meaningful equation (and any inequality and in-equation) must
have the same dimensions in the left and right sides. Checking this is the
basic way of performing dimensional analysis. Dimension method is used for
unification of various units of a physical quantity.
10. 10 Units
bcb
bcb
Pressure
bcb
Tension
Figure 1.3: Equality in dimensions.
Solved Problem 1.5 Height, h, of water in a leaky tank is time dependent
function and the rate of water leakage is directly proportional to the instan-
taneous height of the water in the tank. Mathematically, it is given by
dh
dt
∝ −h
Or
dh
dt
= −kh
Find the dimension of constant k.
Solution The given relation is
dh
dt
= −kh
From the property of the dimension, dimension units must be equal in left
and right hand side of the above relation. So, left hand unit is m/s and right
hand side is m. Including the dimension of constant k, to make the unit
both side equal, k must has unit of 1/τ (where τ represents to the time and
known as time constant). Now,
k =
1
τ
This is unit of k.
1.6.1 Mathematical properties
The dimensions that can be formed from a given collection of basic physical
dimensions, such as M, L, and T, form a group. The identity of dimension
11. 1.6. DIMENSIONS 11
is written as 1. Zero power of a dimension is one ie L0
= 1 and the inverse
to dimension is dimension−1
ie 1
L
= L−1
. If two similar dimensions are
in product then Lp
× Lq
= Lp+q
, ie the powers of similar dimensions are
additive and if two dimensions are in quotient then powers are subtraction.
Product of unlike dimensions takes place in algebraic form. Dimensions
obey algebraic operations. Addition or subtraction takes place among
like dimensions and multiplication or division takes place among like or unlike
dimensions.
1.6.2 Deriving Physical Relations
Dimensional analysis is used in the finding physical relations by using math-
ematical dimensional properties.
Force Relation
To calculate the force relation F = ma we use the second statement of the
newton’s force law. It states that the force on a body is the product of mass
having power p and acceleration having power q. Where p and q are arbitrary
constants. Now
F = mp
× aq
(1.1)
Now substituting the units of force, mass and accelerations in SI fundamental
units.
kg ·
m
s2
= kgp
m
s2
q
Substituting the dimension value of fundamental units. And
[M][L]
[T]2
= [M]p
[L]
[T]2
q
Using arithmetic operations of dimensions
[MLT−2
] = [Mp
Lq
T−2q
]
Comparing the powers of same fundamental units, and
p = 1 q = 1
Substituting the value of p and q in relation (1.1) the force relation is
F = ma
This is the required relation.
12. 12 Units
Time Period of Pendulum
Assume that the time period of a pendulum is depends on the mass of bob,
length of pendulum and gravitational acceleration. Hence
T = kma
lb
gc
(1.2)
Substituting the dimensions of time period, mass, length and gravity. Time
period of pendulum becomes
[T] = k[Ma
][Lb
][Lc
T−2c
]
On simplification
[T] = k[Ma
Lb+c
T−2c
]
On comparing of power of same bases in both side
a = 0; −2c = 1; b + c = 0;
On solving these equations the values of a, b and c are
a = 0; c =
−1
2
; b =
1
2
;
Substituting the values of power in equation (1.2), the time period of pen-
dulum is
T = kl
1
2 g
−1
2
T = k
s
l
g
(1.3)
Solved Problem 1.6 The charge on a conductor is defined by q = at2
+ bt,
where a, b are constants. Find the dimension of constants in form of current.
Solution We know that the current in a circuit is given by i = dq
dt
1
I =
d
dt
(at2
+ bt)
= 2at + b
1
Remember that any relative result in physics would be find by using derivative method
in current and next chapters.
13. 1.6. DIMENSIONS 13
We know that a relation is consists if dimensions in both sides of relation are
same. Again if two are more physical quantities are in addition then each
physical quantity has same dimension as its resultant. Hence the dimensions
of two physical quantities given above in right hand side are
2at = [I] b = [I]
Now substituting the dimension of time in left hand side of first physical
quantity and leaving 2 as constant without dimension. So
a[T] = [I] b = [I]
a =
[I]
[T]
b = [I]
a = [IT−1
] b = [I]
These are the dimensions of constants a and b.
Solved Problem 1.7 A physical quantity A is linearly depend on two other
physical quantities, i.e. C and D. The units of C and D are N/m2
and m/s2
.
If unit of A is s then find the relation between A, C and D.
Solution From the given problem, A is linearly depend C and D. So,
A = αC + βD (1.4)
Substituting the dimensions of A, C and D as given in the problem.
[T] = α[ML−1
T−2
] + β[LT−2
]
If this relation is possible, then each term at right hand side should have
dimension equal to [T]. So,
[T] = α[ML−1
T−2
] [T] = β[LT−2
]
α = [M−1
LT3
] β = [L−1
T3
]
These are dimensions of α and β. On substituting these values in equation
1.4, we shall get the exact relation between A, C and D.
14. 14 Units
1.6.3 Dimension in Trigonometric Functions
Consider a trigonometric function, say tan θ, that has two parts. One an
argument and other an operator. Output of trigonometric functions is always
a dimensionless pure numeric value. For example,
tan 45◦
= 1
θ
p
b
Again, trigonometric function with its argument is equal to ratio of ap-
propriate sides of a right angle triangle, i.e.
tan θ =
p
b
Therefore,
θ = tan−1
p
b
As p/b is dimensionless quantity, hence θ shall also be a dimensionless quan-
tity.
Solved Problem 1.8 Find the dimension of physical quantity C if it is related
to velocity as C
v
= sin C
v
.
Solution As output of trigonometric function is a pure numeric value,
hence right hand side is dimensionless quantity. Therefore left side term
should also be a dimensionless quantity. So, dimension of C should be equal
to dimension of v. Now, the dimension of C is [LT−1
].
1.6.4 Dimension in Logarithmic Functions
1.7 Measurement Instruments
Here vernier caliper and Screw Gauge will be discussed as the measuring in-
struments. Important definitions related to vernier caliper and Screw Gauge
are given below:
15. 1.7. MEASUREMENT INSTRUMENTS 15
Least Count of Instrument This is the minimum possible measurement
by an instrument. It is also called resolution of the instrument. The quantity
measured by the instrument is always integer multiple of its least count.
Least Count of Main Scale It is the length between two consecutive
marks on the main scale of vernier caliper.
Least Count of Vernier Scale It is the length between two consecutive
marks on the vernier scale of vernier caliper.
Divisions These are number of marks between two fixed ends or between
two major distant marks. For example, in centimeter and millimeter scale,
there are ten divisions in one centimeter length.
1.7.1 Vernier Caliper
A vernier scale is a device that lets the user measure more precisely than could
be done by reading a uniformly-divided straight or circular measurement
scale. It is scale that indicates where the measurement lies in between two of
the marks on the main scale. It has two scales, (a) main scale or fixed scale
and (b) vernier scale. Main scale and vernier scale has partitions of equal
size with labels. Main scale is used to measure the main reading of the scale.
It is an estimated value and further precise value is measured with help of
vernier scale. Vernier scale has a leas count value. Least count depends on
the number of partitions in vernier scale. If there are n partitions in the
vernier scale then least count ℓ of the vernier caliper is given as the ratio of
difference of length of ‘n’ partitions of main scale dn
ms and vernier scale dn
vs
to the ‘n’.
ℓ =
dn
ms − dn
vs
n
Second method of finding least count is
ℓ = d1
ms − d1
vs
Where d1
ms is length of one partition in main scale and d1
vs is length of one
partition in vernier scale.
16. 16 Units
0 1
0 10
ℓ 5ℓ
There are two ways to scale vernier scale; (i) by marking more divisions
in the vernier scale than the number of divisions in main scale in equal
length, for example, within 1cm, there are n divisions in main scale and
m n divisions in vernier scale. Or (ii) marking equal number of divisions
in vernier scale whose length is less than by one division to the length of
main scale for the same number of divisions, for example, 10 divisions are
made in 1cm of main scale and 0.9cm of vernier scale each. First case is not
acceptable as the precision measurement passes to the least count value of
main scale at the minimum count of vernier divisions. For example, in main
scale there are ten divisions, hence two consecutive divisions has value 0.1cm.
There are 20 divisions in vernier scale in 1cm. Least count of vernier scale
is 1/20 = 0.05. When second tick of the vernier scale is perfectly coincide
with the main scale tick then precision value is 0.05 × 2 = 0.1. It is equal to
the least count of the main scale. In this case other ticks of vernier scale are
unused as they give values larger than the least count of main scale. Second
case is best suited as highest positioned tick gives value less than the least
count of the main scale.
0 1
0 10
5ℓ
Thus between two consecutive divisions of main scale distance is 1/10 cm
while distance between two consecutive divisions of vernier scale is 0.9/10
cm. Now, the least count of this Vernier caliper (i.e. instrument) is
ℓ =
1
10
−
0.9
10
=
0.1
10
= 0.01cm
There are restricted number of divisions in vernier scale. It means, For equal
number of division of main scale and vernier scale, the difference between
17. 1.7. MEASUREMENT INSTRUMENTS 17
distances should be equal to the least count of the main scale. Again, lm is
least count of main scale and lv is least count of the vernier scale. There are
m and n divisions in main scale and vernier scale respectively. Now, relation
between these two least count is
lm × m − lv × n = lm
0 1
0 20
In short form, the vernier precision value for any tick of the vernier scale
can not be larger than the least count of the main scale. This is main principle
of design of the vernier caliper.
Solved Problem 1.9 A vernier caliper has 20 divisions in 1 cm of main scale.
The vernier scale of the same caliper has 20 divisions in 0.95 cm. Find the
least count of the caliper.
Solution The least count of the main scale is
lm =
1
20
= 0.05cm
Similarly least count of the vernier scale is
lv =
0.95
20
= 0.0475cm
Now, the least count of the vernier caliper is
ℓ = lm − lv = 0.05 − 0.0475 = 0.0025cm
Solved Problem 1.10 A vernier caliper has main scale of least count 0.05cm.
There are 50 divisions in 2.45 cm of vernier scale. Find the least count of
the caliper.
18. 18 Units
Solution The least count of the main scale is 0.05cm. Least count of the
vernier scale is
lv =
2.45
50
= 0.049cm
Now, the least count of the vernier caliper is
ℓ = lm − lv = 0.05 − 0.049 = 0.001cm
Measurement of Size
To measure the length or size of an object by Vernier Caliper, it is placed
between two heads of vernier caliper and heads are gently tightened. Now
two observations are taken
0 1 2 3 4
0 10
Figure 1.4: Vernier Calipers.
Principal Measurement Firstly, observe the length on main scale that is
passed by zero marker of the Vernier scale. Generally zero marker lies
in between two consecutive equally spaced partition markers in the
main scale. It is principal measurement. Assume it is m and denoted
by p.
Precision Measurement Secondly, observe the partition number from zero
marker of Vernier scale which is perfectly coincide with any of the par-
tition mark of the main scale. Assume it is nth
marker of vernier scale.
Now multiply it by least count (ℓ). The value obtained is precision
measurement.
∆p = n × ℓ
Now the actual dimension of the object is sum of principal measurement (p)
and precision measurement (∆p).
d = m + n × ℓ
19. 1.7. MEASUREMENT INSTRUMENTS 19
Solved Problem 1.11 In a Vernier Caliper, the distance between two consec-
utive divisions of main scale is 0.05cm. In vernier scale, there are 40 divisions
in a span of 1.95cm. Find the least count of the Vernier Caliper.
Solution
0 1 2 3 4
0 20 40
The least count of vernier caliper is difference of distances between two
consecutive divisions of main scale and vernier scale. Distance between two
consecutive divisions of main scale is 0.05cm. Similarly, distance between
two consecutive divisions of vernier scale is given by 1.95/40 = 0.04875cm.
Now, least count of the vernier caliper is 0.05 − 0.04875 = 0.00125cm.
Solved Problem 1.12
Solution
Measurement Errors
There are following errors observed in the measurement by using Vernier
Calipers.
Zero Error When two forks are closed, zero markers of both, main scale and
Vernier scale must be coincide to each other. If they are not coincide
then there is a deviation in the measured value from actual value. This
deviation of measured value from actual value is called zero error. It is
sub classified as (i) positive zero error and (ii) negative zero error.
Positive Zero Error When two jaws of the Vernier calipers are closed
and zero marker of Vernier scale leads to the zero marker of main
scale then it adds additional value to the actual value of measure-
ment. This error is taken as positive value. For exact measure-
ments, this error is subtracted from the final result.
20. 20 Units
0 1 2
0 10
Figure 1.5: Vernier Calipers : Positive Error.
Considering the positive error, the actual measured value is given
by t = d − e+
Negative Zero Error When two jaws of the vernier caliper are closed
and zero marker of Vernier scale trails to the zero marker of main
scale then it measures less value to the actual value of measure-
ment. This error is taken as negative value. For exact measure-
ments, this error is subtracted to the final result.
0 1 2
0 10
Figure 1.6: Vernier Calipers : Negative Error.
Considering the negative error, the actual measured value is given
by t = d − e−
Mechanical Error If heads are clamped tightly over the object then mea-
sured value is lesser than the actual value.
Environmental Errors In hot and cold environment, scales are expanded
or contracted and cause the errors in measurements.
Design Error If scales are faulty designed then measured value shows de-
viation from the actual value.
Solved Problem 1.13 The diameter of a cylinder is measured using a Vernier
Calipers with no zero errors. It is found that the zero of the Vernier scale
lies between 5.10cm and 5.15cm of the main scale. The Vernier scale has 50
divisions equivalent to 2.45cm. The 24th
division of the Vernier scale exactly
coincides with one of the main scale divisions. Calculate the diameter of the
cylinder.
21. 1.7. MEASUREMENT INSTRUMENTS 21
Solution The zero marker of the Vernier scale lies between 5.10cm and
5.15cm hence the principal measurement is
dp = 5.10cm
0 1 2 3 4 5 6 7 8
0 10 20 30 40 50
Now from the least count relation
ℓ =
dn
ms − dn
vs
n
• Number of partitions in Vernier scale = 50
• Length of one partition in main scale = 0.05cm
• Length of 50 partitions of main scale = 50 × 0.05 = 2.50cm
• Length of 50 partitions of Vernier scale = 2.45cm (given)
Now,
ℓ =
2.50 − 2.45
50
= 0.001
Now precision measurement is 0.001 × 24 = 0.024cm. Now the diameter of
the cylinder is
Dcyl = 5.100cm + 0.024cm = 5.124cm
This is required result.
1.7.2 Screw Gauge
A screw gauge is also known as micro-meter. Its structure is shown in the
figure 1.7.2. It composed of:
Frame The C-shaped body made of thick material of larger coefficient
of thermal expansion that holds the anvil and barrel.
Anvil The shiny part fixed at the one end of C-shape body. Spindle
moves toward it when it is rotated. Samples are rests against it.
22. 22 Units
Sleeve It is stationary cylindrical component with the linear scale on a
base line.
Screw It is the heart of the micrometer. It is inside the the barrel.
Distance between two consecutive grooves of the screw are called pitch.
Spindle This is cylindrical component that is revolved with help of thim-
ble. It moves towards the anvil.
Thimble It is used to rotate the screw by help of thumb. It has graduated
50-100 markings.
Ratchet Stop It is device on the end of handle that limits applied pressure
by slipping at a calibrated torque.
0.0 0.5
10
15
20
Measurement of Thickness
We can measured the value with help of screw gauge by using following steps.
Least Count: First we calculate the least count of the device by using re-
lation
ℓ =
p
N
Here p is pitch and N is number of marks in round scale. Pitch (p) is the
distance moved by the screw leftward or rightward when it is rotated
by one complete revolution. Note that, the pitch is always equal to the
length between two consecutive division on main scale.
Principal Measurement: Further we check the main scale and find which
main scale mark is visible correctly. This is main scale value (say m).
It is principal value.
Precision Measurement: Again we see the round scale to get the division
which is perfectly coincide with main scale lien. Let it is the nth
di-
vision. This value is multiplied by the least count ℓ. This measured
value obtained is called precision value (p).
23. 1.7. MEASUREMENT INSTRUMENTS 23
Measured Value: The final measured value is the sum of principal value
and precision value.
d = m + n × ℓ
Taking the considerations of the positive or negative error, actual measured
value is given by
t = d − e±
In screw gauge the pitch of the round scale is always equal to the least count
of the main scale. If not so, then screw gauge has a design error/flaws.
Zero Errors
When we put close the two heads of the screw gauge, 0-mark of round scale
should be coincide perfectly to the base line of main scale. If it is not, then
there is a zero error. There are two types of measurement errors in screw
gauge.
Positive Zero Error
0.0
0
5
10
Initially, by rotating thimble, two jaws of the screw gauge are placed in
contact. If zero line of the round scale is not aligned with the main scale line
then it is said that the screw gauge has measurement error. Now if we can
see zero of main scale clearly then the error is positive error. Positive error
is obtained as
e+ = n × ℓ
Here n is nth
division aligned to the main line and ℓ is least count of the
screw gauge. This error is always taken as positive value. It adds extra value
to the actual dimension of substrate. If positive error is larger than the least
count of the main scale then fixed jaw is adjusted with help of screw driver
to align the zero mark of round scale to main scale line.
Negative Zero Error
24. 24 Units
0.0
30
35
40
Initially, by rotating thimble, two jaws of the screw gauge are brought
in contact. If zero line of the round scale is not aligned with the main scale
line then it is said that the screw gauge has measurement error. Now if we
can hardly saw the zero of main scale clearly then the error is negative error.
Negative error is obtained as
e− = −(N − n) × ℓ
Here n is nth
division aligned to the main scale line and ℓ is least count of the
screw gauge. N is total numbers of divisions in main scale. It subtracts an
amount of value from the actual dimension of substrate. This error is always
taken as negative value. If negative error is larger than the least count of the
main scale then fixed jaw is adjusted with help of screw driver to align the
zero mark of round scale to main scale line.
Solved Problem 1.14 A screw gauge with a pitch of 0.5mm and a circular
scale with 50 divisions is used to measure the thickness of a thin sheet of
Aluminium. Before starting the measurement, it is found that when the two
jaws of the screw gauge are brought in contact, the 45th
division coincides
with the main scale line and that the zero of the main scale is barely visible.
What is the thickness of the sheet if the main scale reading is 0.5mm and
the 25th
division coincides with the main scale line?
Solution When round scale is rotated anticlockwise direction, it moves
away from fixed jaw of screw gauge. Round scale has graduated marks in re-
ducing order when it rotates anticlockwise (i.e. when jaws move away). This
is why when two jaws are brought constant and main scale line is coincide
with 45th division, then it is said negative error. The total divisions on round
scale passed to main scale line are (50-45)=5. Now, the least count of the
screw gauge is 0.5/50 = 0.01mm. Negative error is −5 × 0.01 = −0.05mm.
Using the standard measurements of screw gauge, the measured value of
sheet is
t = 0.5 + 25 × 0.01 = 0.75mm
25. 1.8. LAND AREA UNITS 25
Including the error in measurement, the true thickness of the sheet is 0.75 −
(−0.05) = 0.80mm.
1.8 Land Area Units
The following units are useful for measuring the land area.
1. One international acre is equal to 4,046.8564224 square meters and
0.40468564224 hectare.
2. One hectare is equal to a square with 100m × 100m area.
3. One acre is equal to 43,560 square feet.
4. One chain is equals to 66 feet or 22 yards or 4 rods or 100 links.
5. A furlong is equals to 220 yards or ten chains.
6. One square mile is equal to 640 acres.