GENERAL PHYSICS - INTRODUCTION
TO MEASUREMENT AND THE
PHYSICAL QUANTITIES
Study Notes
Measurement is the basis of all experiments and research. Physical quantities that
can be measured directly or indirectly are called physical quantities. Examples:
length, mass, time, velocity, volume, etc.
Units
To measure a physical quantity, its value is compared with a standard of the
same kind, called a unit.
Examples: meter (m) is the unit of length, kilogram (kg) is the unit of mass,
second (s) is the unit of time.
In measurement, two things are involved: a number and a unit.
Characteristics of a Good Unit A good unit should be:
1. Well-defined
2. Easily accessible
3. Invariable
4. Easily reproducible
Fundamental and Derived Units
A set of independent physical quantities are known as fundamental
quantities.
The units of fundamental quantities are called fundamental units (meter,
kilogram, second).
Derived quantities can be expressed in terms of fundamental quantities.
Their units are called derived units.
Examples: Volume (m^3), Speed (m/s)
Rules for Writing Units
1. Symbols of units not named after a person are written in lowercase (m, kg).
2. Symbols of units named after a person have a capital initial letter (N, K).
3. Full names of units are written with a lowercase initial letter (newton, kelvin).
4. Compound units are written with a dot or space between symbols (N.m or N
m).
5. Units in short form are never written in plural.
Systems of Units
A complete set of units for all physical quantities with particular basic units is
called a system of units.
Commonly used systems: CGS, FPS, MKS, SI units (adopted in 1960 by the
International Committee for Weights and Measures).
SI Units
SI system has seven fundamental units and two supplementary units.
Fundamental units: mass (kg), length (m), time (s), temperature (K), electric
current (A), luminous intensity (cd), amount of substance (mol).
Supplementary units: angle (rad), solid angle (sr).
Multiples and Submultiples of Units
To express large or small magnitudes, prefixes are used with the unit.
Examples: kilo (k), milli (m), micro (μ), nano (n), etc.
Dimensions
Physical quantities can be represented in terms of fundamental quantities
(length, mass, time).
Dimensions are the powers to which the fundamental units must be raised
to represent a quantity.
Example: Area = M^0 L^2 T^0, Velocity = M^0 L T^-1
Dimensional equation indicates the units of a physical quantity in terms of
fundamental units.
Principle of Homogeneity
An equation representing a physical quantity will be correct if the dimensions
of each term on both sides are the same.
Uses of Dimensional Analysis
1. To check the correctness of an equation.
2. To derive the correct relationship between physical quantities.
Limitations of Dimensional Analysis
1. Cannot provide information about dimensionless constants.
2. Cannot be used if the physical quantity depends on more than three other
quantities.
3. Cannot be used if the left-hand side of the equation contains more than one
term.
4. Difficult to guess the parameters on which the physical quantity depends.
Significant Figures
Significant figures give the number of meaningful digits in a number, based
on the accuracy of the instrument.
More significant figures indicate higher accuracy.
Rules for Determining Significant Figures
1. All non-zero digits are counted.
2. In measurements involving decimals, the position of the decimal is
disregarded.
3. All zeros between non-zero digits are counted.
4. Zeros preceding the first non-zero digit are not counted.
Rules for Rounding Off
1. If the digit to be dropped is less than 5, the preceding digit remains
unchanged.
2. If the digit to be dropped is more than 5, the preceding digit is increased by
1.
3. If the digit to be dropped is 5, the preceding digit is made even.
Significant Figures in Calculations
In multiplication and division, the result can have no more significant figures
than the least significant figures in the original numbers.
In addition and subtraction, the least significant digit of the sum or difference
occupies the same relative position as the least significant digit of the
quantities being added or subtracted.
Accuracy and Errors in Measurements
Accuracy refers to the correctness of a measurement.
Least count is the smallest measurement that can be made accurately with
an instrument.
Error is the deviation between the measured value and the true value.
Types of Errors
1. Constant error: Error remains the same in a series of readings.
2. Systematic errors: Errors due to known causes that act according to a
definite law.
oInstrumental errors, personal errors, imperfection errors, external
cause errors.
3. Random errors: Errors that occur irregularly and at random in magnitude
and direction.
4. Gross errors: Errors caused by carelessness.
5. Absolute error: Magnitude of the difference between the true value and the
measured value.
6. Mean absolute error: Arithmetic mean of the absolute errors.
7. Relative and percentage errors: Ratio of the mean absolute error to the
true value, expressed as a percentage.
Combination of Errors
The error in the final result depends on individual measurements and the
type of mathematical operations performed.
Rules are provided for calculating errors in sums/differences,
products/quotients, and when a quantity is raised to a power.
Motion in a Straight Line
Motion is the change in position of an object with respect to time, relative to
an observer.
Types of motion: Translatory, Rotatory, Oscillatory.
Concept of Point Object or Particle
If the distance traveled by an object is very large compared to its size, it can
be considered a point object or particle.
Distance and Displacement
Distance is the length of the path covered by an object.
Displacement is the change in position in a definite direction.
Speed and Velocity
Speed is the distance traveled by an object in one second or the rate of
change of position.
Velocity is the rate of change of displacement, with both magnitude and
direction (a vector quantity).
Uniform Velocity
An object moves with uniform velocity if it makes equal displacements in
equal intervals of time, no matter how small the intervals.
Position-Time Graph
A graph connecting position (y-axis) and time (x-axis) of a particle.
For a stationary object, the graph is a straight line parallel to the time axis.
For uniform motion, the graph is a straight line inclined to the time axis
(slope = velocity).
Velocity-Time Graph
For uniform motion, the velocity-time graph is a straight line parallel to the
time axis.
The area under the velocity-time graph is equal to the displacement of the
particle.
Acceleration
Acceleration is the rate of change of velocity of an object.
If velocity is increasing, acceleration is positive; if decreasing, acceleration is
negative (retardation or deceleration).
Uniform Acceleration
An object has uniform acceleration if its velocity changes by equal amounts
in equal intervals of time.
Equations of Uniformly Accelerated Motion
1. Velocity-time relation: v = u + at
2. Position-time relation: s = ut + (1/2)at^2
3. Position-velocity relation: v^2 = u^2 + 2as
Acceleration due to Gravity
A freely falling body near the Earth's surface moves vertically downward with
uniform acceleration called acceleration due to gravity (g), approximately
9.8 m/s^2.
Relative Velocity
Relative velocity of a body with respect to another body is the rate of
displacement of the first body relative to the second body.
It is equal to the vector difference between the velocities of the two bodies as
measured by an observer.
Projectile Motion
A particle projected with an initial velocity makes a curved path called a
trajectory.
Horizontal and vertical components of the initial velocity are treated
separately.
Equations are provided for maximum height, time of flight, and horizontal
range.
