Unit 1 - Units and Measurement (PDF)

# Units and Measurements ## Physical Quantities A quantity that can be measured and by which various physical happenings can be explained and expressed in the form of laws is called a physical quantity. Examples: Length, mass, time, force, acceleration, temperature, pressure, etc. ## Fundamental...

# Units and Measurements ## Physical Quantities A quantity that can be measured and by which various physical happenings can be explained and expressed in the form of laws is called a physical quantity. Examples: Length, mass, time, force, acceleration, temperature, pressure, etc. ## Fundamental Physical Quantities Out of a large number of physical quantities that exist in nature, there are seven quantities that are independent of all other quantities and do not require the help of any other physical quantity for their definition. Therefore, these are called absolute quantities. These quantities are also called fundamental or base quantities. | Fundamental quantity | Base unit | Symbol | |---|---|---| | Length | meter | m | | Mass | Kilogram | kg | | Time | Second | s | | Electric Current | Ampere | A | | Thermodynamic Temperature | Kelvin | K | | Luminous Intensity | Candela | cd | | Quantity of matter | Mole | mol | ## Derived Physical Quantities Apart from the seven fundamental quantities, all other physical quantities can be derived by fundamental quantities. Therefore, these are called derived physical quantities. Examples: - Velocity - m/s - Acceleration - $m/s^2$ - Force - $kg \ m/s^2$ - Area - $m^2$ - Volume - $m^3$ - Pressure - $kg/m \ s^2$ ## Characteristics of Units - The measure of a unit should be definite and unambiguous. - The unit should be such that its measure does not change with time. - Unit is defined with the help of some phenomenon that phenomenon must be permanent. - Unit should be easily reproducible. - The replica of a unit should be easily available. ## Requirement of Unit To express any physical quantities from a unit. 1. Fundamental Physical Quantities 2. Derived Physical Quantities ## Some Important Practical Units - 1 Angstrom (A) = $10^{-10} \ m$ - 1 AU or astronomical unit: $1.496 \times 10^{11} \ m$ - 1 light year = $9.46 \times 10^{15} \ m$ - 1 micron: 1 $\mu m = 10^{-6} \ m$ - 1 nm = $10^{-9} \ m$ - 1 nanometre = $10^{-9} \ m$ - 1 fermi or femtometer (F) = $10^{-15} \ m$ - a.m.u. = $1.66 \times 10^{-27} \ kg$. ## Supplementary Units ### Radian (rad) One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius of the circle. - If an arc of length *ds* subtendes an angle *dθ* at the centre *O* of a circle of radius *r* $dθ = \frac{ds}{r} \ radian$ ### Steradian (sr) One steradian is the solid angle subtended at the centre of a sphere by that surface of the sphere which is equal in area to the square of radius of the sphere. - If an area *dA* of a special surface subtunds a solid angle *dΩ* at the centre of the sphere of radius *r*. $dΩ = \frac{dA}{r^2} \ steradian$ ## Curie Disintegration of $3.70 \times 10^{10}$ atoms of radioactive elements per second is called 1 curie. - 1 Ci = $3.70 \times 10^{10}$ disintegration/second. - 1 μCi = $3.70 \times 10^{4}$ disintegration/second ## Interconversion of Units - MKS to CGS and vice versa - $1 \ m^2 = 10^4 \ cm^2$ - $1 \ kg/m^3 = 10^{-3} \ g/cm^3$ - $9.8 \ m/s^2 = 980 \ cm/s^2$ - $1 \ N = 10^5 \ dyne$ - $36 \ km/h = 10 \ m/s$ ## Vernier Calipers French mathematician Pierre Vernier invented the Vernier Caliper. ##### Principle: It works on least count. ##### Least Count: The smallest fraction of a unit length which can be measured by Vernier Calipers is known as least count of Vernier Calipers. ##### Diagram: A diagram showing the main scale, vernier scale, jaws, above jaws, below jaws, and screw. ##### Construction: 1. **Main Scale:** The upper scale on the main scale is in inches and the lower scale is in inches and the lower scale is in centimeters. This scale is called the main scale. 2. **Vernier Scale:** A small sliding scale arranged on the main scale is called the Vernier scale. 3. **Jaws:** There are two jaws namely A and C on the left side of the main scale. The edge of both jaws A and C are in the same line. The jaws A, B, C, and D are used to hold the object which is to be measured. ##### Formula of Least Count: $Least \ count\ = \frac{the \ value \ of \ the \ smallest \ division \ on \ main \ scale}{total \ divisions \ on \ vernier \ scale}$ $Least \ count \ = \frac{0.1 \ cm}{10} = 0.01 \ cm$ ##### Error: $Error = 1 \ matching \ divisions \times least \ count$ ##### Zero Error: A diagram showing zero error of Vernier calipers. There is no gap between the jaws of vernier calipers AC or BD, then zero of vernier scale coincindes with zero of main scale. This equipment causes zero error. ##### Positive Error: A diagram showing positive error of Vernier calipers. There is no gap between the jaws of vernier calipers AC or BD, then zero of vernier scale stands right side of zero of main scale. This equipment causes positive error. $Positive \ Error = + matching \ division \times least \ count$ $Positive \ Error = +2 \times 0.01 \ cm$ $Positive \ Error = +0.02 \ cm$ $Correction \ for \ positive \ error = -0.02 \ cm$ ##### Negative Error: A diagram showing negative error of Vernier calipers. There is no gap between the jaws of vernier calipers AC or BD, then zero of vernier scale stands left side of main scale. This equipment causes negative error. $Negative \ Error = - matching \ division \times least \ count$ $Negative \ Error = -2 \times 0.01 \ cm$ $Negative \ Error = -0.02 \ cm$ $Correction \ for \ negative \ error = +0.02 \ cm$ ##### Working of Vernier Calipers: A diagram showing a Vernier caliper with a screw, main scale, vernier scale, above jaws, below jaws, and object. 1. $Least\ count = \frac{The \ value \ of \ smallest \ division \ on \ main \ scale}{total \ divisions \ on \ vernier \ scale}$ $Least \ count = \frac{0.1 \ cm}{10} = 0.01 \ cm$ 2. Check out error of given vernier calipers Error (c') = zero error. 3. To find out diameter of object placed between below jaws of Vernier calipers and note down the readings of main scale. 4. Main scale reading = 5.6 cm 5. To find out Vernier scale reading coincide mark multiply into least count. Vernier Scale reading = coinciding mark x least count = 9 x 0.01 = 0.09 cm 6. Outer diameter of given object D = M.S.R + v.S.R + c' = 5.6 + 0.09 + 0 = 5.69 cm ## Applications of Vernier Calipers - It is used to measure inner diameter of hollow object. - It is used to find outer diameter of solid object. - It is used to measure volume of block. - It is used to measure depth or height of liquid filled in the vessels which have small diameters. ## Micrometer Screw Gauge ##### Principle: It works on least count. ##### Least Count: The smallest fraction of a unit length which can be measured by micrometer screw gauge is known as least count. ##### Diagram: An illustration of a micrometer screw gauge consisting of a hollow tube, circular scale, base line, anvils, circular cylinder, semicircula plate, and a metal sheet. ##### Construction: There are two main parts of the micrometer screw. It's one part is a semicircular steel plate (frame) whose both ends are in the form of a hollow cylinder. The screw $S_2$ is inseted in the left side hollow cylinder. A hollow tube N is attached to the right side of hollow cylinder. On this hollow cylinder, a straight line XY, called a base line, is drawn. A scale in mm is marked on this line. It is called main scale. Second part of the micrometer screw $S_1$, whose left end is flat and its right end is fixed with the hollow tube. The circular edge of the hollow tube is divided into 50 or 100 equal parts. This scale is called circular scale. ##### Pitch: Pitch of the screw is the distance between two consecutive thread of the screw. ##### Formula: $Least\ count = \frac{pitch\ distance}{total\ divisions\ on\ circular\ scale}$ $Least \ count = \frac{1 \ mm}{100} = 0.01 \ mm$ ##### Error: $Error = 1\ matching\ division \times least\ count$ ##### Zero Error: A diagram showing zero error of micrometer screw gauge. There is no gap between the anvils of the micrometer screw gauge, then the zero of the circular scale coincindes with the base line. This equipment causes zero error. ##### Positive Error: A diagram showing a positive error of a micrometer screw gauge. There is no gap between the anvils of the micrometer screw gauge, then the zero of the circular scale lies/stands below the base line. This equipment causes positive error. $Positive \ Error = + matching\ division \times least \ count$ $Positive \ Error = +7\times0.01\ mm$ $Positive \ Error = +0.07\ mm$ $Correction \ for \ positive \ Error = -0.07\ mm$ ##### Negative Error: A diagram showing negative error of micrometer screw gauge. There is no gap between the anvils of the micrometer screw gauge, then the zero of circular scale stands above the base line. This equipment causes negative error. $Negative \ Error = - matching\ division \times least\ count$ $Negative \ Error = -7\times0.01\ mm$ $Negative \ Error = -0.07\ mm$ $Correction \ for \ negative \ Error = +0.07\ mm$ ##### Working of Micrometer Screw Gauge: A diagram showing a micrometer screw gauge consisting of a hollow tube, circular scale, base line, anvils, semicircular plate, and a metal sheet. 1. To find out the least count of given micrometer screw gauge. $Least \ count = \frac{pitch \ distance}{total \ divisions \ on \ circular \ scale}$ $Least \ count = \frac{1 \ mm}{100} = 0.01 \ mm$ 2. Check out the error of given micrometer screw gauge. Error (c') = zero error. 3. To find out thickness of metallic sheet placed between two anvils of micrometer screw gauge and note down the main scale reading. 4. Main scale = 6mm 5. To find out circular scale reading coinciding mark multiply with least count. Circular scale reading = coinciding mark x least count = 99 x 0.01 mm = 0.99mm 6. Thickness of metallic sheet = M.S.R + C.S.R + c' = 6 + 0.99 + 0 = 6.99 mm ## Types of Error 1. **Systematic Error:** During experiments systematic error is either positive or negative. These errors cannot be both positive and negative. Reasons for systematic error can be known. So these errors can be reduced. - **Instrumental Error:** This error is due to defect in the instrument or defect in the calibration of scale, e.g., One end of meter scale is broken, then systematic error is created. - **Errors due to imperfection in experimental technic or procedure.** - This type of error is created due to procedure of experiment e.g. Temperature of body is taken by putting thermometer in armpit, which is less than true temperature of the body. - **Personal Error:** This error is created due to procedure of taking reading by an individual, unproper arrangement of instruments carelessness. - **Error due external reasons:** Temperature, pressure, moisture in air, velocity of air, etc., can create error in taking reading in experiment. 2. **Random Error:** Even after using fault free, correctly calibrated instruments, taking every care while taking observations, some error occur in the measurements. These errors are called random errors. The random error can be estimated by taking large numbers of observation and using statistical methods. ## Estimation of Errors ### True Value Physical quantity be measured *n* times and the measured values be *a₁, a₂, a₃, ... a_n*. The arithmetic mean of these values is true value. *a̅ = (a₁ + a₂ + ... + a_n) / n* ### Absolute Error The difference between the true value and the individual measured value of quantity is called absolute value. - Δa₁ = a̅ - a₁ - Δa₂ = a̅ - a₂ - Δa₃ = a̅ - a₃ - Δa₄ = a̅ - a₄ - Δa₅ = a̅ - a₅ ### Mean Absolute Error Mean absolute error is defined as mean of positive values (mode) of absolute errors of all the observations. A_cd = ( |Δa₁| + |Δa₂| +...+ |Δa_n| ) / n ### Relative Error or Fractional error: The relative error or fractional error of measurement is defined as the mean absolute error to the mean value of the quantity measured. δa = A_cd / a̅ ### Percentage Error When the relative/fractional error is expressed in percentage is called percentage error. Percentage Error = δa x 100% ## Significant Figures The digits that are known reliably plus the first uncertain digit are known as significant digits or significant figures. ### Rules for counting/Identification of Significant Figures: 1. **All non-zero digits are significant.** Example: 1, 2, 3, 4, 5, 6, 7, 8 and 9 2. **All zeros occurring between two non-zero digits are significant no matter where the decimal point is.** Example: 1007 has four significant figures. 1.0809 has five significant figures 3. **If the number is less than one, the zero(s) on the decimal point and to the left of the first non-zero digit are not significant.** Example: 0.005784 has four significant figures. 4. **In a number without a decimal point, the terminal or trailing zeros are not significant.** Example: 3210 has three significant figures. 5. **In a number with decimal point, the trailing zeros are significant.** Example: 3.500 has four significant figures 0.0069000 has five significant figures. 6. **Change of units does not change the number of significant figures in a measurement.** Example: 5.608 has four significant digits. 7. **If the digit to be dropped is less than 5, then the preceding digit is left unchanged.** Example: 7.82 rounded off to 7.8. 8. **If the digit to be dropped is more than 5, then the preceding digit is raised by one.** Example: 6.87 rounded off to 6.9. ## Definition: ### Physical Quantities A quantity that can be measured and by which various physical happenings can be explained and expressed in the form of laws is called a physical quantity. Example: length, mass, time, force, etc. ### Unit A unit is a value, quantity or magnitude in terms of which other values, quantities or magnitudes are expressed. Generally, a unit is independent of physical and environment conditions and is fixed by definition. ### Metre: It is defined in terms of the standard wavelength of light and is equal to 1650763.73 wavelengths in vacuum of radiation corresponding to transition between the energy levels 2p¹⁰ and 5d⁵ of krypton 86 atom. ### Kilogram: The mass of 1 litre pure water at 4 °C is known as kilogram. ### Second: It is defined as 9192631770 periods of radiation corresponding to the unpereturbed transition between the hyper fine levels of the ground state of the atom of caesium (¹³³Cs) atom. ### Ampere: Ampere is defined as the current which when flowing through two parallel conductors of infinitely long and negligible cross-section area placed in vacuum one meter part, conductors experiences a force of 2.0 x 10^-7 newton per meter. ### Kelvin: On thermodynamic scale 1/273.16th part of temperature of triple point of water is defined to be 1'k. ### Candela: The luminous flux radiated by 1/6,00,000 sq meter surface of a perfectly black body having 1.01325 x 10⁵ N/m² and temperature corresponding to melting point of platinum is known as candela. ### Mole One mole contains exactly 6.02214076 x 10²³ elementary entities. This number is the fixed numerical value of the Avogadro constant N_A, when expressed in the unit mol-¹, and is called the Avogadro number. An elementary entity may be an atom, a molecule, an ion, an electron, any other particle, or a specified group of particles. ### Least Count The smallest fraction of the unit of length which can be measured with vernier calipers is called it's least count. ### Pitch Pitch of the screw is the distance between two consecutive thread of the screw. ### Accuracy The accuracy of a measurement is a measure of how much close the measured value is to the true value of the quantity. ### Precision Precision means to what resolution or limit the quantity is measured by a measuring instrument. ### Vector Quantities Those physical quantities which have magnitude and direction both these are called Vector quantities, e.g., velocity, acceleration etc. ### Scalar Quantities Those physical quantities which possess only magnitude and no direction in space is known as scalar quantities, e.g., temperature, mass, etc. ### Error Difference between true value and measured value of any physical quantity is known as error. ### Write the difference between MKS and CGS system. | MKS | CGS | |---|---| | MKS stands for Meter - Kilogram - Second | CGS stands for Centimeter - Gram - Second | | In MKS, length's unit is meter | In CGS, length's unit is centimeter | | In MKS, mass' unit is kilogram | In CGS, mass' unit is gram | ## Write the difference between accuracy and precision. | Accuracy | Precision | |---|---| | The accuracy of a measurement is a measure of how close the measured value is to the true value of the quantity | Precision means to what resolution or limit the quantity is measured by a measuring instrument | | When we shoot on a target with a rifle all shoots nearer to target, so, here we can see accuracy | When we shoot on a target with a rifle all shoots are far away from the target but are nearer to each other. So, here we can see precision | ## Conversion from MKS to CGS or vice Versa 1. $0.046 N/m^2 = 0.46 \ dyne/cm^2$ $0.046 \ N/m^2 = 0.046 \times 10^5 \ dyne /cm^2 (10^4)$ $ = 0.046 \times10^{5-4} \ dyne/cm^2$ $ = 0.046 \times 10^1 \ dyne/cm^2$ $ = 0.46 \ dyne/cm^2$ 2. $1 \ gm / cm^3 = 1000 \ kg/m^3$ $1 \ gm/cm^3 = 1 \times 10^{-3} (10^{-2})^3 \ kg/m^3$ $ = 1 \times 10^{-3} (10^{-6}) \ kg/m^3$ $ = 1 \times 10^{-3+6}) \ kg/m^3$ $ = 1 \times 10^{3} \ kg/m^3$ $ = 1000 \ kg/m^3$