    Physical Quantities Quantities that can be measured Careful observations experiment   precise  Accurate measurement

Physical Quantities Quantities that can be measured Examples: Length Mass Time Weight Electric current Force Velocity energy Describing a physical quantity: 50 kg unit Numerical value

Different units can be used to describe the same quantity Example : The height of a person can be expressed in feet In inches In metres ……

unit Is the standard use to compare different magnitudes of the same physical quantity

Thermodynamic temperature standard SI Units Quantity SI unit Symbol Length metre m Mass kilogram kg Time second s Electric current Ampere A Thermodynamic temperature Kelvin K Amount of substance mole mol Light intensity Candela cd Base quantity

International Prototype Metre standard bar made of platinum-iridium International Prototype Metre standard bar made of platinum-iridium. This was the standard until 1960 The metre is the length equal to 1 650 763.73 wavelengths in vacuum of the radiation corresponding to the transition between the levels 2p10 and 5d5 of the krypton-86 atom In 1960

The metre is the length of the path travelled by light in vacuum during a time interval of 1⁄299 792 458 of a second. In 1975 …………………

prefixes Prefix Factor Symbol pico 10-12 p nano 10-9 n micro 10-6  Milli 10-3 m centi 10-2 c deci 10-1 d kilo 103 k Mega 106 M Giga 109 G Tera 1012 T prefixes

Is a combination of different base quantities Derived quantities Is a combination of different base quantities Derived unit = the unit for derived quantity = is obtained by using the relation between the derived quantities and base quantities

Derived quantity Derived unit Area m2 Volume m3 Frequency Hz Density kg m-3 Velocity m s-1 Acceleration m s-2 Force N or kg m s-2 Pressure Pa

Specific heat capacity Derived quantity Derived unit Energy or Work J or Nm Power W or J s-1 Electric charge C or As Electric potential V or J C-1 Electric intensity V m-1 Electric resistance  Or V A-1 Capacitance F or C V-1 Heat capacity J K-1 Specific heat capacity J kg-1 K-1

Dimensions of Physical quantities Is the relation between the physical quantities and the base physical quantities Example 1 : Area Dimension of area = Length x Breadth = L x L = = L2 Unit of area = m2

Example 2 : Velocity Dimension of velocity = Displacement Time = L T = = L T-1 Unit of velocity = m s-1

Example 3 : Acceleration Dimension of acceleration = Change of velocity Time = L T-1 T = = L T-2 Unit of acceleration = m s-2

Example 4 : Force Dimension of force = Mass x Acceleration = M x L T-2 = = M L T-2 Unit of force = kg m s-2 or N

Example 5 : Energy Dimension of energy = Force x Displacement = M L T-2 x L = = M L2 T-2 Unit of energy = kg m2 s-2 or J

OR Example 5 : Energy Dimension of energy = Mass x Velocity x Velocity = M x L T-1 x L T-1 = = M L2 T-2 Unit of energy = kg m2 s-2 or J

Example 6 : Charge Dimension of electric charge = Current x Time = A x T = = A T Unit of area = A s

Example 7 : Frequency Dimension of frequency = 1 . Period = 1 T = = T-1 Unit of frequency = s-1

Example 8 : Strain Dimension of strain = Extension Original length = L = = 1 dimensionless Unit of frequency = no unit

_ = + Uses of Dimensions Dimensional homogeneity of a physical equation Both have same dimensions _ 1st Physical quantity 2nd Physical quantity = +

Every term has the same dimension Case 1 Example : v2 = u2 + 2as Every term has the same dimension L T-1 2 L2 T-2 v2 = = L T-1 2 L2 T-2 u2 = = L T-2 L L2 T-2 2as = = The equation is dimensionally consistent

The equation is incorrect. Case 2 Example : v = u + 2as [v] = [u]  [2as] L T-1 L T-1 v = = L T-1 L T-1 u = = L T-2 L L2 T-2 2as = = The equation is incorrect. The dimension is not consistent

A physical equation whose dimensions are consistent need not necessary be correct :

dimensionally consistent The constant of proportionality is wrong Case 3 Example : v2 = u2 + as Every term has the same dimension L T-1 2 L2 T-2 v2 = = L T-1 2 L2 T-2 u2 = = L T-2 L L2 T-2 as = = The equation is incorrect. The equation is dimensionally consistent

dimensionally consistent Has extra term Case 4 Example : s2 t2 v2 = u2 + 2as + L T-1 2 L2 T-2 v2 = = Every term has the same dimension L T-1 2 L2 T-2 u2 = = L T-2 L L2 T-2 2as = = L2 T2 s2 t2 L2 T-2 = = The equation is incorrect. The equation is dimensionally consistent

The truth of a physical equation can be confirmed experimentally :

Example : The period of vibration t of a tuning fork depends on the density , Young modulus E and length l of the tuning fork. Which of the following equation may be used to relate t with the quantities mentioned? A E l g gl3  E AE  Al a) t = b) t = c) t = Where A is a dimensionless constant and g is the acceleration due to gravity. The table below shows the data obtained from various tuning forks made of steel and are geometrically identical. Frequency/Hz 256 288 320 384 480 Length l /cm 12.0 10.6 9.6 8.0 6.4

Use the data above to confirm the choice of the right equation Use the data above to confirm the choice of the right equation. Hence, determine the value of the constant A. For steel: density  = 8500 kg m-3 , E = 2.0 x 1011 Nm-2

dimensionally consistent Solution : A E The equation is dimensionally consistent l g gl3  E AE  b) t = Al a) t = c) t = Unit for E = N m-2 = ( ) m-2 kg m s-2 Mass volume [] = = kg m-1 s-2 [E] = M L-1 T-2 [t] = x x T + ½ M L-3 L-3 A E (L T-2) L3 = L2 gl3 T-1 = M L-1 T-2 L-1 T-2 = T

dimensionally consistent Solution : A E l g gl3  E AE  Al a) t = b) t = c) t = [t] = T ½ Mass volume  E M L-3 Al [] = = L M L-1 T-2 dimensionally consistent The equation is ½ L-2 = L T-2 L-1 = L T-1 = T

l l Solution : The dimension is not consistent [t] = T Hence, A E l g gl3  E AE  Al a) t = b) t = c) t = [t] = T Hence, the equation is incorrect ½ M L-1 T-2 l g L AE  = M L-3 L T-2 1 L2 T-2 = T-1 = L2 T-1

l l Solution : Which one is the right equation? t against l t against gl3  E AE  Al a) t = b) t = c) t = Which one is the right equation? Frequency/Hz 256 288 320 384 480 Length l /cm 12.0 10.6 9.6 8.0 6.4 t against l 3 2 t against l Plot a graph

1 period = frequency Frequency /Hz 256 288 320 384 480 Length l /cm 12.0 10.6 9.6 8.0 6.4 1 256 1 288 1 320 1 384 1 480 period, t, s 3.91x10-3 3.47x10-3 3.13x10-3 2.60x10-3 2.08x10-3 1 frequency period =

,s 3 2 3 2 ( 12)3 41.6 ( 10.6)3 34.5 ( 9.6 )3 29.7 ( 8.0 )3 22.6 ( 6.4 )3 16.2 l , cm ( l )3

l tx10-3/s 4.0 Graph of 3.5 t against 3.0 2.5 2.0 1.5 1.0 0.5 5 10 15 20 25 30 35 40 45

Compare the graph with the equation a) c  0 Therefore the equation gl3 a) t = Therefore the equation is incorrect General equation y = mx c = 0

l l tx10-3/s 4.0 Graph of 3.5 t against 3.0 2.5 2.0 1.5 1.0 0.5 /cm 2 4 6 8 10 12 14 16 18 20

Compare the graph with the equation b) c = 0 Therefore the equation  E Al b) t = Therefore the equation is correct General equation y = mx c = 0

Use the data above to confirm the choice of the right equation Use the data above to confirm the choice of the right equation. Hence, determine the value of the constant A. For steel: density  = 8500 kg m-3 , E = 2.0 x 1011 Nm-2  E Al b) t = m = y = mx 3.91x10-3 s 12 cm  E m = A 3.91x10-3 s 12 m 100 =  E 0.03258333 = A = 0.03258333

Use the data above to confirm the choice of the right equation Use the data above to confirm the choice of the right equation. Hence, determine the value of the constant A. For steel: density  = 8500 kg m-3 , E = 2.0 x 1011 Nm-2  E Al 8500 . 2.0 x 1011 b) t = 0.03258333 = A y = mx A = 1.58 x 102  E m = A  E 0.03258333 = A

Example : The dependence of the heat capacity C of a solid on the temperature T is given by the equation : C = T + T3 What are the units of  and  in terms of the base units?

All the terms have same unit Unit for C = J K-1 Solution : C = T + T3 All the terms have same unit Unit for C = J K-1 2 K-1 kg m s-1 = = kg m2 s-2 K-1 Unit for T = = kg m2 s-2 K-1 kg m2 s-2 K-1 unit for T kg m2 s-2 K-1 K Unit for  = = = kg m2 s-2 K-2

Solution : C = T + T3 All the terms have same unit Unit for T3 = = kg m2 s-2 K-1 kg m2 s-2 K-1 unit for T3 Unit for  = kg m2 s-2 K-1 K3 = = kg m2 s-2 K-4

Example : A recent theory suggests that time may be quantized. The quantum or elementary amount of time is given by the equation : h mpc2 T = where h is the Planck constant, mp = mass of proton and c = speed of light. What is the dimension for Planck constant? Write the SI unit of Planck constant.

Solution : h mpc2 a) T = h = Tmpc2 [ ] [ ] h = Tmpc2 T M (L T–1)2 = = M L2 T–1 Unit = kg m2 s-1