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Presentation on Units and dimensions

1. UNITS AND DIMENSIONS Chapter- 1

2. What is Physics ? Physics , the study of matter and energy, is an ancient and broad field of Science. The word 'physics' comes from the Greek 'knowledge of nature,' and in general, the field aims to analyze and understand the natural phenomena of the universe. It's often considered to be the most fundamental science. It provides a basis for all other sciences - without physics, you couldn't have biology, chemistry, or anything else! Mathematics is the language of Physics. Without knowledge of Mathematics it would be much more difficult to discover, understand, and explain laws of nature. Units - All physical quantities are measured w.r.t . standard magnitude of the same physical quantity and these standards are called UNITS. eg . second, meter, kilogram, etc. So the four basic properties of units are:— They must be well defined. 2. They should be easily available and reproducible. 3. They should be invariable e.g. step as a unit of length is not invariable. 4. They should be accepted to all. e.g. if some body has to study 4 hrs, the numeric part 4 says that it is 4 times of the unit of time. The second part says that the unit chosen for time is hour.

3. SET OF FUNDAMENTAL QUANTITIES A set of physical quantities which are completely independent of each o ther . Physical quantities is called Set of Fundamental Quantities. Physical Quantity Units(SI) Units(CGS) Notations Mass kg(kilogram) g M Length m (meter) cm L Time s (second) s T Temperature K ( kelvin ) °C Theta Current A (ampere) A I or A Luminous intensity cd (candela) — cd Amount of substance mol — mol

4. Physical Quantity Definition (SI Unit) Length (m) The distance travelled by light in vacuum in 1/299,792,458 second is called 1 metre. Mass (kg) The mass of a cylinder made of platinum- iridium alloy kept at International Bureau of Weights and Measures is defined as 1 kilogram. Time(s) The second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium - 133 atom. Electric Current (A) If equal currents are maintained in the two parallel infinitely long wires of negligible cross-section, so that the force between them is 2 × 10–7 Newton per metre of the wires, the current in any of the wires is called 1 Ampere.

5. Physical Quantity Definition (SI Unit) Thermodynamic The fraction 273.16 1 of the thermodynamic Temperature (K) temperature of triple point of water is called 1 Kelvin Luminous Intensity( cd ) 1 candela is the luminous intensity of a black body of surface area 1/600,000 m2 1 placed at the temperature of freezing platinum and at a pressure of 101,325 N/m2 , in the direction perpendicular to its surface. Amount of substance (mole) The mole is the amount of a substance that contains as many elementary entities as there are number of atoms in 0.012 kg of carbon-12

6. There are two supplementary units too: 1. Plane angle (radian) angle = arc / radius Theta= l / r 2. Solid Angle ( steradian ) System of Units : The common system of units are : FPS system : The units of length, mass and time are respectively foot, pound and second. CGS system : The units of length, mass and time are respectively centimeter, gram and second. MKS system : The units of length, mass and time are respectively metre , kilogram and second. The International system of units (SI units).

7. Derived Quantities The Physical quantities that depend upon other physical quantity for its measurement are known as derived quantities. The measurement of derived quantities directly depends upon other quantities. So in order to measure the derive quantity, one must measure the quantities that it depends upon. Except 7 fundamental quantities, all other quantities are derived quantities. Some examples of derived quantities are: Derived Units Volume l (or lit) 0.001 m 3  1000 cm 3 Force Newton (SI)N1 kg · m/s 2 Dyne (CGS) 1 g ·cm/s 2 Pressure Pascal (SI)Pa1 N/m 2 Energy, work  Joule (SI)J 1 N · m = 1 kg · m 2 /s 2 Erg (CGS)  1 dyne · cm = 1 g · cm 2 /s 2 Gram-calorie cal 4.184 J = 4.184 kg · m 2 /s 2 PowerWatt W1 J/s = 1 kg · m 2 /s 2

8. Dimensions Dimensions of a physical quantity are the powers to which the fundamental quantities must be raised to represent the given physical quantity. In mechanics all physical quantities can be expressed in terms of mass (M), length (L) and time (T). Example : Force = mass x acceleration =       Or,  So, the dimensions of force are 1 in mass, 1 in length and – 2 in time Dimensionless quantity In the equation     then the quantity is called dimensionless. Examples : Strain, specific gravity, angle. They are ratio of two similar quantities. A dimensionless quantity has same numeric value in all system of units

9. Uses of Dimension Dimensional consistency of of any equation with physical sense must be identical. Otherwise, an equality in one system would be broken upon conversion to another system. This fact is used to obtain derived units from fundamental units. Example In the LMT class, the dimension of mass is M, the dimension of acceleration is LT−2, the Dimension of force can be obtained (derived) from Newton’s second law: f = ma [f] = [m] [a] = MLT−2 In other words, in the LMT class, the dimension of force is LMT−2. We can determine the unknown exponent “?” in the following equation by requiring the same units on both sides: E = mc? ML2T−2 = M(LT−1)? ? = 2 This is one technique of Dimensional Analysis, which can allow us to identify the controlling physical quantities in unfamiliar or complicated quantities.

10. Limitations in Dimensional Analysis Dimensional method can not be used to derive equation involving addition or subtraction. In some cases, it is difficult to guess the factors while deriving the relation connecting two or more physical quantities. Equations using trigonometric, exponential or logarithmic expression can not be deduced. If dimensions are given , physical quantity may not be unique as many physical quantities having same dimensions. For example dimensional formula of a physical quantity is ML2T-2, it may be work or energy or torque.

11. Limitations in Dimensional Analysis It cannot be used if the physical quantity is dependent on more than three unknown variables. This method cannot be used in an equation containing two or more variables with same dimensions. Equation of frequency of a tuning fork f=(d/L2)v can not be derived by theory of dimension but can be checked. By Jai Sharma of class 11 th science A