Units and Measurement,Dimensions, Errors, Significant figures

 Units and Measurement,Dimensions, Errors, Significant figures

Units and Measurements

   Physics is a quantitative science, where we measure various physical quantities during experiments.

•A measurement always involves a comparison with a standard measuring unit which is internationally accepted. eg. measuring the mass of a given fruit we need standard mass units of 1kg, 500g,etc.

• “The standard measure of any quantity is called the unit of that quantity. ’’

Requirements of Good Units: The unit must be

1. Invariable

2. Reproducible

3. Universally accepted

      There are large number of physical quantities which we have to study in physics, these quantities are broadly divided into two types.

1 Fundamental Physical quantities

2 Derived Physical quantities

Fundamental Quantities and Units:

•Fundamental Quantity: A quantity which does not depend on any other physical quantity is   called as fundamental quantity.

 e.g. length, mass, time, temperature etc.

•There are seven  fundamental quantities.

•Fundamental Unit: The units used to measure fundamental quantities are called fundamental units.

Derived Quantities and Units:

•Derived Quantity: A quantity which depends on fundamental quantity is called as Derived quantity.

 e.g. Density, Velocity, Force , Work , Pressure etc.

•Derived Unit: The units used to measure Derived quantities are called Derived units.

Ø A set of units for physical quantities is called System of Units

Ø FPS:-       Foot, Pound, Second system

Ø CGS:-       centimetre ,gram ,Second system

Ø  MKS:-     metre, kilogram, Second system

Ø SI:-           Systeme International

Seven SI fundamental units:

Quantity	Unit	Symbol
Length	metre	m
Mass	kilogram	kg
Time	second	s
Electric current	ampere	A
Temperature	kelvin	K
Quantity of substance	mole	mol
Luminous Intensity	candela	cd

Two  Supplementary Units:  

Plane angle	radian	rad
Solid angle	steradian	sr

Conventions for the use of SI units:

1  Full name of unit always starts with small letter even  if  named after a person.

      eg.  newton, joule and not Newton, Joule

2  Symbol for unit named after a person should be in

       capital letters. 

      eg. N for newton, J for joule, A for ampere etc.

3  Symbols for other units are written in small letters .

     eg. m for metre, s for second etc.

4     Symbol of units are not to be expressed in plural form .

      eg. 25m and not 25ms

5       Full stop and other punctuation mark should not be   written after the symbols. 

      eg. kg and not kg. or N and not N.

6       Unit of every physical quantity should be represented by its symbol.

 Dimensions:

 The powers to which fundamental units are raised to obtain the units of that  quantity are called as Dimensions.

    Dimensions are always written in square bracket in the form of symbols of the quantities involved in formula and their corresponding powers.

  eg :  M for mass, L for length,  T for time ,   [ M^a L^b T^c ]

Let us consider steps to find dimensions of velocity

 1 Write the formula-  

        velocity = displacement / time

 2 Write in symbol form    v = L / T

 3  Express in dimension formula  

 [ v ] = [ M⁰ L¹ T^-1 ]

 

Sr no	Physical quantity	Formula	SI Unit	Symbols	Dimensions
1	Velocity	Velocity=displacement/time	m/s	v = L / T	[ M0 L1 T-1 ]
2	Volume	Volume=cube of length	m3	V = L3	[ M0 L-3 T0 ]
3	Density	Density=mass/volume	Kg/m3	ρ=M/ L3	[ M1 L-3 T0 ]
4	Acceleration	Acceleration=velocity/time	m/s2	a = L/T2	[ M0 L1 T-2 ]
5	Momentum	Momentum= Mass x Velocity	kg m/s	P = ML / T	[ M1 L1 T-1 ]
6	Force	Force= mass x acceleration	N	F= ML /T2	[ M1 L1 T-2 ]
7	Work or Energy	Work =Force x Displacement	J	W= (ML/T2)*L	[ M1 L2 T-2 ]
8	Power	Power=Work/time	W	P=ML2T-3	[ M1 L2 T-3 ]
9	Torque	τ =rFsinϴ	Nm or J	τ =L*ML/ T2	[ M1 L2 T-2 ]
10	Moment of Inertia	M.I = mr2	kg m2	I =ML2	[ M1 L2 T0 ]
11	Frequency	n =cycles/second	Hz	n = 1/ T	[ M0 L0 T-1 ]
12	Wavelength	λ = Length	m	λ = L	[ M0 L1 T0 ]

Use of Dimensional Analysis:

1.  To verify the correctness of physical equation,

2.     Conversion of one system of unit into another system of unit.

3.   To derive the relation between different physical quantities

Limitations of Dimensional Analysis:

1.The value of dimensionless constant can be obtained with the help of experiments only.

2. Dimensional analysis cannot be used to derive relations involving trigonometric, exponential and logarithmic functions as these quantities are  dimensionless.

3. This method is not useful if constant of proportionality is not a  dimensionless quantity.

4. If the correct equation contains some more terms of the same dimension, it  is not possible to know about their presence using dimensional equation.

Errors in Measurements:

We perform experiments and find out result by calculating its value with the help of given formula, this result may be right or wrong.

Causes of wrong Results of experiment:

1 Mistakes    2  Errors

1 Mistakes are mostly due to observer which can be avoided with carefulness.

2 Error is uncertainty in the measurement of physical quantity

Errors cannot be avoided but can be minimized.

Types of Errors:

1. Instrumental Error:

•      These errors are caused due to faulty construction of instruments.   

    eg. Thermometer not graduated properly.

2. Systematic Error:

•      This error occurs due to defective setting of an instrument.

    eg. If pointer of an ammeter is not pivoted exactly at the zero of the scale it will show correct reading.

3. Personal Error:

•      These errors are introduced due to fault of an observer while taking readings.

 eg. Due to non-removal of paralax between pointer and image.

4. Random Error:

•      Even after taking proper care and minimizing above types of errors there are  chances to occur error due to unavoidable circumstances such as change in  temperature, pressure or fluctuation in voltage while experiment is carried out . 

Minimizing  Errors:

Following rules are adopted to minimize errors:

1. Taking large magnitude of measurement of a quantity.

2. Taking large number of readings and calculate mean value.

3. Using instrument with smallest possible least c

Estimation of Error:

1.   Absolute error:

   The difference between the measured value of a quantity and its actual value , given by is called the absolute error.

If a₁ , a₂, a₃ ,…, a_n are the measured values of any quantity a in an experiment performed n times, then the arithmetic mean of these values is called the true value (a_m) of the quantity.

a_m = (a₁ + a₂ + a₃ + ------ a_n )/  n

The absolute error in measured values is given by

Δa₁ = a_m – a₁
Δa₂ = a_m – a₁

………….

Δa_m = Δa_m – Δa_n

2. Mean Absolute Error:

The arithmetic mean of the magnitude of absolute errors in all the measurement is called mean absolute error.

Δa_mean = ( | Δa₁  | + | Δa₂  | +  | Δa₃  | + -----+ | Δa_n  | ) / n

3. Relative Error:

 The ratio of mean absolute error to the mean value is called relative error

Relative error   = ( Mean absolute error ) / Mean Value = Δa_mean / a_m 

4. Percentage Error :

 The relative error expressed in percentage is called percentage error.

Percentage error  = ( Δa_mean / a_m  ) x 100 %

Propagation of Error:

1. Error in Addition or Subtraction Let A = a + b or A = a – b

If the measured values of two quantities a and b are (a ± Δa and (b ± Δb), then maximum absolute error in their addition or subtraction.

ΔA = ±(Δa + Δb)

2.   Error in Multiplication or Division Let P = a x b or P = (a/b).
If the measured values of a and b are (a ± Δa) and (b ± Δb), then maximum relative error

ΔP /P  = ± [ Δa/a + Δb / b ]

Significant Figures:

 In the measured value of a physical quantity, the number of digits about the correctness of which we are sure plus the next doubtful digit, are called the significant figures.

Rules for Finding Significant Figures:

1.    1. All non-zeros digits are significant figures, e.g., 8372 m has 4 significant figures.

2.    2. All zeros occurring between non-zero digits are significant figures, e.g., 3006 has     4 significant figures.

3.   3.  All zeros to the right of the last non-zero digit are not significant, e.g., 5450 has only 3 significant figures.

4.   4.  In a digit less than one, all zeros to the right of the decimal point and to the left of a non-zero digit are not significant, e.g., 0.00375 has only 3 significant figures.

5.     5.All zeros to the right of a non-zero digit in the decimal part are significant, e.g., 1.9550 has 5 significant figures.

Significant Figures in Algebraic Operations:

1.   In Addition or Subtraction In addition or subtraction of the numerical values the final result should retain the least decimal place as in the various numerical values. e.g.,

If l₁= 3.326 m and l₂ = 4.50 m

Then, l₁ + l₂ = (3.326 + 4.50) m = 7.826 m

As l₂ has measured up to two decimal places, therefore

l₁ + l₂ = 7.83 m

2.    In Multiplication or Division In multiplication or division of the numerical values, the final result should retain the least significant figures as the various numerical values. e.g., If length 1= 2.50 m and breadth b = 5.125 m.

Then, area A = l x b = 2.50 x 5.125 = 12.8125 m²

As l has only 3 significant figures, therefore

A= 12.8 m²

Rules of Rounding Off Significant Figures:

1. 1.    If the digit to be dropped is less than 5, then the preceding digit is left unchanged. e.g., 3.54 is rounded off to 3.5.

2.    2. If the digit to be dropped is greater than 5, then the preceding digit is raised by one. e.g., 1.49 is rounded off to 1.5.

3.  3.   If the digit to be dropped is 5 followed by digit other than zero, then the preceding digit is raised by one. e.g., 7.55 is rounded off to 7.6.

4.   4.  If the digit to be dropped is 5 or 5 followed by zeros, then the preceding digit is raised by one, if it is odd and left unchanged if it is even. e.g., 9.750 is rounded off to 9.8 and 3.650 is rounded off to 3.6.