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Physics Mechanics Notes

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Published in: Physics
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Physics Notes- 1. Mechanics 2. Magnetism 3. Electromagnetic Induction

Pankaj J / Mumbai

11 years of teaching experience

Qualification: B.Tech/B.E. (KU - 2008)

Teaches: Biology, Chemistry, Mathematics, Physics, IIT JEE Mains, GMAT, GRE, IELTS, SAT

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  1. UNIT & DIMENSIONS AND MEASUREMENT STRAIGHT LINES PHYSICAL QUANTITIES The quantities which can be measured by an instrument and by means of which we can describe the laws of physics are called physical quantities. Till class X we have studied many physical quantities eg. length, velocity, acceleration, force, time, pressure, mass, density etc. Fundamental or Basic Quantities PHYSICAL QUANTITIES Suplimentary Quantities Derived Quantities Fundamental Units : These are the elementary quantities which covers the entire span of physics. Any other quantities can be derived from these. All the basic quantities are chosen such that they should be different, that means independent of each other. (i.e., distance, time and velocity cannot be chosen as basic quantities as V = d/t). An International Organization named CGPM : General Conference on weight and Measures, chose seven physical quantities as basic or fundamental. Length (L) Unit / metre (m) symbol Fundamental Units Time (T) Mass (M) Temperature Electric Luminous Amount of Current (A) Intensity (Cd) Substance (mol) (K) second (s) kilogram (kg) Kelvin (K) Ampere (A) candela (Cd) mole (mol) These are the elementary quantities (in our planet) that's why chosen as basic quantities. In fact any set of independent quantities can be chosen as basic quantities by which all other physical quantities can be derived. i.e., (p) Area velocity Density Can be chosen as basic quantities (R) (S) Sp. Heat Resistance Luminous Capacity (mol) mole quantities) (L) Length But (A) Area (on some other planet, these might also be used as basic (V) Velocity cannot be used as basic quantities as Area = (Length)2 so they are not independent. Supplementary quantities Besides seven fundamental quantities two supplementary quantities are also defined. They are 0 Plane angle — Unit = radian (rad) Solid angle — Unit = Steradian (sr) Derived quantities Physical quantities which can be expressed in terms of basic quantities (M,L,T....) are called derived quantities. displacement i.e., Momentum P = mV = (m) time ML Ml Ll T- 1 Here [ Ml Ll T— I ] is called dimensional formula of momentum , and we can say that momentum has 1 Dimension in M (mass) 1 Dimension in L (meter)
  2. and —1 Dimension in T (time) The representation of any quantity in terms of basic quantities (M,L,T....) is called dimensional formula and in the representation, the powers of the basic quantities are called dimensions. Physical quantity Frequency Force Pressure, stress Energy, work, quantity heat Power, radiant flux Quantity of electricity, Electric charge Electric potential, Potential difference, Electromotive force Capacitance Electric resistance Conductance magnetic flux Magnetic magnetic flux magnetic induction Inductance Luminous luminous Power Luminance Activity of a radio of field, density, flux, nuclide/radioactive source OTHER CLASSIFICATION : Name hertz newton pascal joule watt coulomb volt farad ohm mho weber henry lumen lux becquerel Symbol Hz pa w c c s Wb 1m Ix SI Unit Expression in terms of other units N/m2 or Nm -2 Nm J/s or Js WIA or WA I Vs or J/A Wb/m2 Wb/A 2 Im/m Expression in terms of SI base Units s kg m s-2 or kg m/s 2 kgm Is 2 or kg/s2 m kg m2 s 2 or kg rn2/s2 kg m2 s-3 or kg m2/s3 Kg rn2s3 A I or kg m2/s3 A A2s4 kg- -2 m 2 -3 kg m s rn-2 kg -1 2 -2 kg m s kg s- kg rn2 s-2 A 2 cd /sr m- cd sr-l s If a quality involves only length, mass and time (quantities in mechanics), then its unit can be written in MKS, CGS or FPS system. S system meter kg sec C S system FPS system sec cm gram sec foot pound For MKS system : In this system Length, mass and time are expressed in meter, kg and sec. respectively. It comes under Sl system. For CGS system : In this system ,Length, mass and time are expressed in cm, gram and sec. respectively. For FPS system : In this system, length, mass and time are measured in foot, pound and sec. respectively.
  3. Example 1: Solution : Find the unit of speed. distance Speed = time m — m/ s = ms s Prefixes to the Power of 10: The physical quantities whose magnitude is either too large or too small can be expressed more compactly by the use of certain Sl refixes. Factor of 10 10 102 -3 10 106 109 -12 10 103 106 109 1012 Example 2 : Solution : Prefix deci centi m illi micro nano ico kilo me a ia tera S mbol d c m n k Fill in the blank by suitable conversion of units 1 kg m2s 2 g cm s Ikg m2s-2 = 107g cm2s 2 DIMENSIONS The dimensions of a physical quantities are the powers to which the base quantities are raised to represent that quantity. (a) (b) (c) Application of dimensional analysis In conversion of units from one system to other. (ii) To check the dimensional correctness of a given physical relation. (iii) To establish the relation among various physical quantities. (iv) To find dimensions of physical constants or co-efficients. Limitations of dimensional analysis (i) by this method the value of dimensionless constant cannot be calculated. (ii) by this method the equation containing trigonometric, exponential and logarithmic terms cannot be analyzed. (iii) if a physical quantity in mechanics depends on more than three factors, then relation among them cannot be established. Dimensions of commonly used Physical Quantities S.No. 2. 3. 4. 5. 6. 7. Physical Quantity (Mechanics) Velocity = displacement/time Acceleration = velocity/time Force = mass x acceleration Work = force x displacement Energy Torque = force x perpendicular distance Power = work/time SI Units m/s 2 m/s 2 = Newton or N kg-m/s 2 2 = N-m = Joule or kg-m /s N-m J/s or watt Dimensional formula M OLT M OLT -2 -2 MLT ML2T -2 ML2T -2 ML2T -2 ML2T -3
  4. S.No. Physical Quantity (Mechanics) Momentum = mass x velocity 8. Impulse = force x time 9. Angle = arc/radius 10. AL AV Strain = — or 11. Stress = force/area 12. Pressure = force/area 13. Modulus of elasticity = stress/strain 14. Frequency = 1/ time period 15. Angular velocity = angle/time 16. 2 Moment of inertia = (mass) (distance) 17. Surface tension = force/length 18. Gravitational constant 19. Thermodynamic temperature 20. Heat 21. Specific heat 22. Latent heat 23. Universal gas constant 24. 25. Boltzmann's constant 26. Stefan's constant 27. Planck's constant Solar constant 28. Thermal conductivity 29. Thermal resistance 30. Enthalpy 31. Entropy 32. Example 3 : Check the accuracy of the relation v = Solution : SI Units Kg-m/s Kg-m/s or N-s radian or rad no units N/m2 N/m2 N/m2 per sec or hertz (Hz) rad/s kg-m2 N/m N-m2/kg 2 kelvin (K) joule Jkg J kg I J mol-1 1
  5. RHS = _ As LHS = RHS 1 L MLT 2 ML I Dimensionally the formula is correct. SIGNIFICANT FIGURES (a) (b) The rules for determining the number of significant figures (i) All the non-zero digits are significant. (ii) All the zeros between two non-zero digits are significant. If the number is less than 1 , the zeros on the right of decimal point but to the left of 1 st non- (iii) zero digit are not significant. (iv) All the zeros to the right of the last non-zero digit (trailing zeros) in a number without a decimal point are not significant, unless they come from experiment. (v) The trailing zeros in a number with a decimal point are significant. Significant figure in algebraic operation In multiplication or division, the number of significant digits in the final result should be equal to the number of significant digits in the quantity, which has the minimum number of significant digits. (ii) In addition or subtraction the final result should retain as many decimal places as are there in the number with the least decimal place. Example 4 : Add and subtract 428.5 and 17.23 with due regards to significant figures Solution : We have Sum 428.50 17.23 445.73 Difference 428.50 17.23 41 1.27 Rounding off the results of the above sum and difference to the first decimal, We have correct sum 445.7 and correct difference 41 1.3. ERROR ANALYSIS IN EXPERIMENTS (a) (b) (c) Errors in sum or difference Let X-A+B Maximum absolute error in X is, AX = +(AA + AB) i.e., the maximum absolute error in sum and difference of two quantities is equal to sum of the absolute errors in individual quantities. Errors in product Let, X = AB AX AA AB Maximum possible value of x Maximum fractional error in product of two (or more) quantities is equal to sum of fractional errors in the individual quantities. Errors in division Let, X
  6. AX AA AB The maximum value of x or, the maximum value of fractional error in division of two quantities is equal to the sum of fractional errors in the individual quantities. (d) Example 5 : Solution : Errors in quantity raised to some power Let X AX AA AB Maximum value of x The sides of a rectangle are (10.5 ± 0.2) cm and (5.2 ± 0.1) cm. Calculate its perimeter with error limit. C = (10.5 ± 0.2) cm b = (5.2 ± 0.1) cm P = + b) = 2 (10.5 + 5.2) = 31.4 cm AP = ± 2 (AC + Ab) = ± 0.6 Hence perimeter = (31.4 ± 0.6) cm.

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