Physicalquantities
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1) The document discusses physical quantities and units of measurement. It defines physical quantities as quantities that can be measured such as length, mass, time, etc. 2) Standard SI units are provided for different base physical quantities. Prefixes are also introduced to denote multiples and submultiples of units. 3) Derived quantities are defined as combinations of base quantities, and their derived units are obtained from the relationships between quantities. 4) The concept of dimensions is introduced to represent the relationship between quantities and base units. Dimensional analysis is used to check the consistency of physical equations.
![Example :
v = u + 2as
v = L T-1
= L T-1
u = L T-1
= L T-1
2as = L T-2
= L2
T-2
L
[v]
=
[u]
≠
[2as]
The dimension is not consistent
The
equation is
incorrect.
Case 2](https://image.slidesharecdn.com/physicalquantities-130512093505-phpapp02/85/Physicalquantities-24-320.jpg)
![Solution :
Unit for E = N m-2
= ( ) m-2kg m s-2
= kg m-1
s-2
[E] = M L-1
T-2
a) t =
Aρ
E
gl3
b) t =
ρ
EAl c) t =
AE
ρ
l
g
[t] = T
Aρ
E
gl3 =
M L-1
T-2
½
(L T-2
) L3M L-3
[ρ] =
Mass
volume
=
L-3
L-1
T-2
L2
T-1
= T
+ xx
The equation is
dimensionally consistent](https://image.slidesharecdn.com/physicalquantities-130512093505-phpapp02/85/Physicalquantities-31-320.jpg)
![Solution :
a) t =
Aρ
E
gl3
b) t =
ρ
EAl c) t =
AE
ρ
l
g
[t] = T
ρ
EAl = L
½
[ρ] =
Mass
volume
M L-3
M L-1
T-2
= L
½
L-2
T-2
= L
L-1
T-1
= T
Theequationis
dimensionallyconsistent](https://image.slidesharecdn.com/physicalquantities-130512093505-phpapp02/85/Physicalquantities-32-320.jpg)
![Solution :
a) t =
Aρ
E
gl3
b) t =
ρ
EAl c) t =
AE
ρ
l
g
[t] = T
AE
ρ
l
g
=
½M L-1
T-2
M L-3
L
L T-2
= L2
T-2 1
T-1
= L2
T-1
The dimension
is not consistent
Hence,
the equation
is incorrect](https://image.slidesharecdn.com/physicalquantities-130512093505-phpapp02/85/Physicalquantities-33-320.jpg)
![Solution :
a) T =
h
mpc2
h = Tmpc2
h = Tmpc2[ ] [ ]
= T M (L T–1
)2
= M L2
T–1
Unit = kg m2
s-1](https://image.slidesharecdn.com/physicalquantities-130512093505-phpapp02/85/Physicalquantities-47-320.jpg)
Physicalquantities
- 1. experiment Careful observations precise Accurate measurement Quantities that can be measured
- 2. Quantities that can be measured Examples: Length Mass Time Weight Electric current Force Velocity energy 50 kg Numericalvalue unit Describing a physical quantity:
- 3. 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 ……
- 4. unit Is the standard use to compare different magnitudes of the same physical quantity
- 5. 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 standard
- 7. International Prototype Metre standard bar made of platinum-iridium. This was the standard until 1960 The metre is the length equal to 1650763.73 wavelengths in vacuum of the radiation corresponding to the transition between the levels 2p10 and 5d5 of the krypton-86 atom In 1960
- 8. The metre is the length of the path travelled by light in vacuum during a time interval of 1⁄299792458 of a second. In 1975 …………………
- 9. Prefix Factor Symbol pico 10-12 p nano 10-9 n micro 10-6 µ Milli 10-3 m centi 10-2 c deci 10-1 d kilo 103 k Mega 106 M Giga 109 G Tera 1012 T
- 10. 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
- 11. 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
- 12. 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 Ω Or V A-1 Capacitance F or C V-1 Heat capacity J K-1 Specific heat capacity J kg-1 K-1
- 13. Dimensions of Physical quantities Is the relation between the physical quantities and the base physical quantities Example 1 : Dimension of area = Area Length x Breadth= = L x L = L2 Unit of area = m2
- 14. Example 2 : Dimension of velocity = Velocity Displacement Time= = L T = L T-1 Unit of velocity = m s-1
- 15. Example 3 : Dimension of acceleration = Acceleration Change of velocity Time= = L T-1 T = L T-2 Unit of acceleration = m s-2
- 16. Example 4 : Dimension of force = Force Mass x Acceleration= = M x L T-2 = M L T-2 Unit of force = kg m s-2 or N
- 17. Example 5 : Dimension of energy = Energy Force x Displacement= = M L T-2 x L = M L2 T-2 Unit of energy = kg m2 s-2 or J
- 18. Example 5 : Dimension of energy = 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 OR
- 19. Example 6 : Dimension of electric charge = Charge Current x Time= = A x T = A T Unit of area = A s
- 20. Example 7 : Dimension of frequency = Frequency 1 . Period= = 1 T = T-1 Unit of frequency = s-1
- 21. Example 8 : Dimension of strain = Strain Extension Original length= = L L = 1 Unit of frequency = no unit dimensionless
- 22. Uses of Dimensions Dimensional homogeneity of a physical equation 1st Physical quantity 2nd Physical quantity+ _= Both have same dimensions
- 23. Example : v2 = u2 + 2as v2 = L T-1 2 = L2 T-2 u2 = L T-1 2 = L2 T-2 2as = L T-2 = L2 T-2 L Every term has the same dimension The equation is dimensionally consistent Case 1
- 24. Example : v = u + 2as v = L T-1 = L T-1 u = L T-1 = L T-1 2as = L T-2 = L2 T-2 L [v] = [u] ≠ [2as] The dimension is not consistent The equation is incorrect. Case 2
- 25. A physical equation whose dimensions are consistent need not necessary be correct :
- 26. Example : v2 = u2 + as v2 = L T-1 2 = L2 T-2 u2 = L T-1 2 = L2 T-2 as = L T-2 = L2 T-2 L Every term has the same dimension The equation is dimensionally consistent Case 3 The constant of proportionality is wrong The equation is incorrect.
- 27. Example : v2 = u2 + 2as + v2 = L T-1 2 = L2 T-2 u2 = L T-1 2 = L2 T-2 2as = L T-2 = L2 T-2 L Every term has the same dimension The equation is dimensionally consistent Case 4 Has extra term The equation is incorrect. s2 t2 s2 t2 = L2 T2 = L2 T-2
- 28. The truth of a physical equation can be confirmed experimentally :
- 29. Example : The period of vibration t of a tuning fork depends on the density ρ, 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) t = Aρ E gl3 b) t = ρ EAl c) t = AE ρ l g 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
- 30. Use the data above to confirm the choice of the right equation. Hence, determine the value of the constant A. For steel: density ρ = 8500 kg m-3 , E = 2.0 x 1011 Nm-2
- 31. Solution : Unit for E = N m-2 = ( ) m-2kg m s-2 = kg m-1 s-2 [E] = M L-1 T-2 a) t = Aρ E gl3 b) t = ρ EAl c) t = AE ρ l g [t] = T Aρ E gl3 = M L-1 T-2 ½ (L T-2 ) L3M L-3 [ρ] = Mass volume = L-3 L-1 T-2 L2 T-1 = T + xx The equation is dimensionally consistent
- 32. Solution : a) t = Aρ E gl3 b) t = ρ EAl c) t = AE ρ l g [t] = T ρ EAl = L ½ [ρ] = Mass volume M L-3 M L-1 T-2 = L ½ L-2 T-2 = L L-1 T-1 = T Theequationis dimensionallyconsistent
- 33. Solution : a) t = Aρ E gl3 b) t = ρ EAl c) t = AE ρ l g [t] = T AE ρ l g = ½M L-1 T-2 M L-3 L L T-2 = L2 T-2 1 T-1 = L2 T-1 The dimension is not consistent Hence, the equation is incorrect
- 34. Solution : a) t = Aρ E gl3 b) t = ρ EAl c) t = AE ρ l g 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 Plot a grapht against l 3 2 t against l
- 35. Frequency /Hz 256 288 320 384 480 Length l /cm 12.0 10.6 9.6 8.0 6.4 period, t, s 1 256 3.91x10-3 period = 1 frequency 1 288 3.47x10-3 1 320 3.13x10-3 1 384 2.60x10-3 1 480 2.08x10-3
- 36. l 3 2 ,s 3 2, cm ( l )3 ( 12)3 41.6 ( 10.6)3 34.5 ( 9.6 )3 29.7 ( 8.0 )3 22.6 ( 6.4 )3 16.2
- 37. t against l 3 2 Graph of 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 tx10-3 /s 5 10 15 20 25 30 35 40 45
- 38. Compare the graph with the equation a) a) t = Aρ E gl3 y = mxGeneral equation c ≠ 0 c = 0 Therefore the equation is incorrect
- 39. t against l Graph of 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 tx10-3 /s 2 4 6 8 10 12 14 16 18 20 l /cm
- 40. b) t = Al ρ E y = mxGeneral equation c = 0 c = 0 Therefore the equation is correct Compare the graph with the equation b)
- 41. Use the data above to confirm the choice of the right equation. Hence, determine the value of the constant A. For steel: density ρ = 8500 kg m-3 , E = 2.0 x 1011 Nm-2 b) t = Al ρ E y = mx m = A ρ E m = 3.91x10-3 s 12 cm = 3.91x10-3 s 12 m 100 = 0.03258333 0.03258333 = A ρ E
- 42. Use the data above to confirm the choice of the right equation. Hence, determine the value of the constant A. For steel: density ρ = 8500 kg m-3 , E = 2.0 x 1011 Nm-2 b) t = Al ρ E y = mx m = A ρ E 0.03258333 = A ρ E 0.03258333 = A 8500 . 2.0 x 1011 A = 1.58 x 102
- 43. Example : The dependence of the heat capacity C of a solid on the temperature T is given by the equation : C = αT + βT3 What are the units of α and β in terms of the base units?
- 44. Solution : C = αT + βT3 All the terms have same unit Unit for C = J K-1 = kg m s-1 K-1 2 = kg m2 s-2 K-1 Unit for αT == kg m2 s-2 K-1 Unit for α = kg m2 s-2 K-1 unit for T = kg m2 s-2 K-1 K = kg m2 s-2 K-2
- 45. Solution : C = αT + βT3 All the terms have same unit Unit for βT3 == kg m2 s-2 K-1 Unit for β = kg m2 s-2 K-1 unit for T3 kg m2 s-2 K-1 K3 = = kg m2 s-2 K-4
- 46. Example : A recent theory suggests that time may be quantized. The quantum or elementary amount of time is given by the equation : T = h mpc2 where h is the Planck constant, mp = mass of proton and c = speed of light. a)What is the dimension for Planck constant? b)Write the SI unit of Planck constant.
- 47. Solution : a) T = h mpc2 h = Tmpc2 h = Tmpc2[ ] [ ] = T M (L T–1 )2 = M L2 T–1 Unit = kg m2 s-1