S. I. UNITS, STATICS

Transcription

1 UNIT S. I. UNITS, STTICS S.I. UNITS Introduction uantities which can be measured either directly or indirectly are known as physical quantities. Every physical quantity is measured in terms of some standard which is called the unit of that quantity. When we say that a table is 6 m long, we take metre as the unit of length and 6 as the magnitude of the physical quantity. Length, mass, electric current, etc. which do not depend on other quantities are known as basic or fundamental quantities. uantities which are defined in terms of fundamental quantities are called derived quantities or derived units. The fundamental units of the CGS system are inconveniently small and hence not suitable in the fields of physical sciences and engineering. To overcome this difficulty, the MKS system was introduced. It is a coherent system based on the fundamental quantities length, mass and time. The units of these quantities in the MKS system are metre, kilogram and second respectively System International (S I Units) MKS System of units was found more convenient as compared to the other two (CGS and FS system). However, in engineering applications large and different conversion factors were needed to make the units correspond to those used in practice. Besides, physical parameters, associated with other branches of physics, like optics, heat and thermodynamics, etc. could not be conveniently expressed in terms of these fundamental units. In order to overcome these difficulties, it was decided in an International Conference on Weights and Measures in 1960 to adopt seven fundamental units and two supplementary

2 Engineering hysics units. This system is commonly referred to as S I (System International of Units). S I is a coherent system based on seven fundamental quantities and two supplementary quantities. The details are given in Table 1.1 The following definitions in the International Conference on Weights and Measures were adopted. Metre: The metre is the length equal to wave length in vacuum of the radiation corresponding to the specified transition between the levels p 10 and 5d 5 of the Krypton atom. Table 1.1 Fundamental and Supplementary Units in S I System Basic hysical uantities Base Unit Symbol Dimension Length metre m L Mass kilogram kg M Time second s T Intensity of Electric current ampere Thermodynamic temperature kelvin K θ Luminous intensity candella cd I uantity of a substance mole mol n Supplementary S I Units 1. lane angle radian rad. Solid angle steradian sr Kilogram: Kilogram is the unit of mass which is equal to the mass of the international prototype of the kilogram, placed at the International Bureau of Weights and Measures in Sevres (near aris) in France. Time: Second is the duration of periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the atom caesium. mpere: The ampere is that constant current, which if maintained in two straight parallel conductors of infinite length of negligible circular cross-section, and placed one metre apart in vacuum, would produce between these conductors a force equal to 10 7 per metre of length.

3 S I Units, Statics 3 1 Kelvin: Kelvin the unit of thermodynamic temperature is th fraction of thermodynamic temperature of the triple point of water. Celsius Kelvin Candela. (Luminous intensity): This is a fundamental physical quantity used in photometry. Mole (mol): mole of a substance contains vogadro s number of particles of that substance. i.e., particles. part from these, the two supplementary units are defined in the following way: adian: This is the unit of plane angle and is given by 360 1radian= = 57 14' 44 '. π Steradin: This is the unit of a solid angle which is defined as the ratio of the intercepted of the spherical surface to the square of the radius. Thus the solid angle made by a sphere at its centre is 4πr r = 4π sr Conventional Standard uantity (Unit) In order to measure any physical quantity, some definite and convenient quantity of the same kind is taken as the standard, in terms of which the quantity as a whole is expressed. The conventional standard quantity is termed as unit. For example, when we say that mass of a body is 100 kg, it implies that a kg (i.e., a kilogram) is used as the standard (or Unit) for measurement and the mass of the body is 100 times the unit. If we simply say kg not using 100 with it, then it has no meaning. Similarly if we simply say 100 not using kg with it, then also it has no meaning. Thus in order to express a physical quantity or parameter, its numerical value must be followed by the unit in which it is measured. standard adopted as a unit should possess the following features: i. It should be universally acceptable and be of reasonable size, ii. Its magnitude should be definite and definable, iii. Its magnitude should not change with time, temperature and pressure and vi. It should be reproducible.

4 4 Engineering hysics Some onderous oints While using S I Units the following two points have to be kept in mind. i. The letter s is not added to a symbol to indicate its plurality. For example, 10 kg is not written as 10 kgs. The letter s is the symbol for second and may create confusion if used for plural. ii. Units with names of the scientists should not start with a capital letter as is done while writing the name of a person i.e., write newton (or N) not Newton; watt (W) not Watt; joule (or J) and not Joule dvantages of S I Units The main advantages of S I Units are now summarized: i. It is comprehence It means that its seven base units cover all disciplines of science and technology. ii. The system is coherent The unit of any derived physical quantity can be obtained as a product or quotient of two or more fundamental units. iii. The SI units are internationally used refixes and Units When the magnitude of a physical quantity is very large or very small, prefixes are used to express them more conveniently. The following table gives commonly used prefixes. Table 1.: refixes Used with Symbols Multiple refix Symbol Fraction refix Symbol 10 1 tetra T 10 1 deci d 10 centi c 10 9 giga G 10 3 milli m 10 6 mega M 10 6 micro µ 10 3 kilo k 10 9 nano n 10 hecto h 10 1 pico p femto f 10 1 deca da atto a Some Derived S I Units Very important derived S I Units of commonly used physical parameters are given in Table 1.3

5 S I Units, Statics 5 Table 1.3: Some Features of Derived S I Units uantity Unit Short form Symbol rea square metre m Volume cubic metre m 3 Frequency herts s 1 Hz Density (mass density) kilogram per cubic metre kg m 3 Velocity metre per socond m s 1 ngular velocity radian per socond rad s 1 cceleration metre per socond square ms Force newton kg ms N ressure pascal N/m a Viscosity pascal second Ns/m Work, Energy and joule N m quantity of heat ower watt J/s Electric charge coulomb s otential, potential difference, volt W 1 V electro motive force Electric field newton/coulomb N/C esistance ohm V/ Ω Capacitance farad s/v F Manetic flux weber V s Wb Inductance henry V s/ H Magnetic flux density tesla Wb/m T Magnetic field strength newtom/ampere metre kg/s Luminous flux lumen cd sr lm Luminance candela/m cd/m Illumination lux lm/m Wave number per metre m 1 Entropy joule/kelvin J/K Specific heat joule per kilogram per kelvin J/kg K Thermal conductivity watt metre per kelvin W/m K adiant intensity watt per sterdian W/sr ctivity (or a radioactive per second s 1 source) ermeability newton per ampere square N/ Dimensions of hysical uantities It has been pointed out earlier that a physical quantity can be expressed in terms of a combination of fundamental units of mass (M), length (L), time (T), temperature (θ), current () and light intensity (I). Thus velocity and acceleration of a body can be written as:

6 6 Engineering hysics velocity = displacement L = = LT = M LT time T velocity LT acceleration = = = LT = M LT time T The equation velocity M LT 1 is called a dimensional equation, whereas the expression M LT 1 is called dimensional formula of velocity. This velocity is said to have zero dimension is mass, 1 dimension length and 1 dimension is time Dimension of Some Common hysical arameters hysical uantity elation with other uantities Dimesional Formula Volume length breadth height L 3 Density mass/volume ML 3 Force mass acceleration MLT Stress force/area ML 1 T ower work/time ML T 3 Surface tension force/length MT ngle lengths of arc/radius No dimension Coefficient of thermal heat distance MLT 3 θ 1 conductivity area temp time esistance volt/ampere ML T 3 1. STTICS 1..1 Introduction hysical quantities, like mass, time, temperature energy, charge which are specified by their magnitudes only are called scalars. Scalars can be added, subtracted and multiplied algebraically. hysical quantities like displacement, that are specified by a magnitude and a direction are called vectors. nother commonly known vector is the velocity of a body, which is described by the speed of the body in a definite direction. Force, pressure, momentum, torque, electric and magnetic fields are some other examples of vector quantities. 1.. Vectors Vector quantities possess both magnitude and direction. They are added and subtracted according to special laws such as parallelogram law of addition e.g., force, velocity, current density, intensity of electric field, etc. It should be noted that all physical quantities having both magnitude and direction are not necessarily vectors. ll vectors obey the laws of vector algebra. For example, the electric current and time have both magnitude and direction; but they are scalars, because they do not obey the laws of vector algebra.

7 S I Units, Statics 7 epresentation of a vector: vector quantity is represented by a straight line with an arrow head; the length of the line representing its magnitude and the arrow head indicating its direction r 1..3 ddition of Two Vectors (i) When vectors are acting in the same direction: The magnitude of the resultant vector is equal to the sum of the magnitudes of the two vectors and the direction is the same as that of the two given vectors. ur ur ur ur ur ur = + B; = + B (ii) When two vectors are acting in opposite directions: The magnitude of resultant vector is the difference between the two vectors and the direction is that of the bigger vector r r r r r r = + ( B); = B (iii) When two vectors are inclined at an angle: The sum of two vectors can be determined by (a) the law of triangle of vectors or (b) the law of parallelogram of vectors (a) Triangle Law of Vectors If two sides of a triangle represent two given vectors in the magnitude and direction and in the same order, then the third side of the triangle in the reverse order represents the vector sum of the vectors. = + B O If O represents vector r r and B represents vector in the anticlockwise order, then r O B in the clockwise order represents the vector sum of r and r r = r + r (b) arallelogram Law of Vectors (Law of arallelogram of Forces) If two adjecent sides of a parallelogram represent two vectors in magnitude and direction, then the diagonal of the parallelogram starting from the point of intersection of the two vectors represent the vector sum of the two vectors. θ

8 8 Engineering hysics = + B C O If O represents r and OB represents O r, then the vector sum r is represented by the diagonal O r C of the parallelogram OCB. (c) nalytical Method of Vector ddition Draw BC perpendicular to O produced. Let α be the angle which the resultant makes with the direction r. In the triabgle OCB, we have O α = + OB = OC + CB = (O+C) + CB = O + O C + C + CB = O + O C + B θ = + + cos θ [since C = B cos θ] = + + cosθ 1..4 Equilibrium of a Body Under the ction of Three Concurrent Forces Consider two forces and acting on a body at a point O. Their resultant can be obtained by the parallelogram method or by the triangle method which we have studied earlier. To balance the resultant, we would require an equal and opposite force. Thus the forces and can be balanced by a force equal in magnitude to the resultant but opposite in direction. The force is called equilibrant. The equilibrant of a set of forces acting on a body is that force which along with the set of forces keeps the body in equilibrium. In Fig. 1.1 the equilibrant of the forces and acting at the point O is. Its magnitude is equal to that of the resultant in the opposite direction. C B

9 S I Units, Statics 9 C C B B Fig. 1.1 Equilibrant of forces Let us now apply the triangle law of vectors to find the resultant of and. In the figure, the forces and are represented by the sides B and BC of triangle BC. Then the resultant is obviously given by C, both in magnitude and direction. To keep the body in equilibrium, the third force to be applied should be equal to that represented by C but in the reverse direction. Hence in a vector diagram, the position of the arrow head will tell us whether a given vector is resultant or equilibrant. C in figure is the resultant of and but C is the equilibrant of the same set. Hence the condition of equilibrium of a body under the action of three concurrent forces can be summarized in the form of a law known as law of triangle of force. Law of Triangle of Forces If three forces acting at a point be represented in magnitude and direction by the sides of a triangle taken in order, they will be in equilibrium. Converse of the law of triangle of forces is also true. If three forces acting at a point be in equilibrium they can be represented in magnitude and direction by the sides of a triangle which is drawn so as to have the sides respectively parallel to the directions of the forces. Lami s Theorem If three forces acting on a particle keep it in equilibrium, each force is proportional to the sine of the angle between the other two., and three forces acting at a point keeping it in equilibrium, Fig. 1.. If α, β and γ are the angles apposite to each of them respectively, = = sin α sinβ sin γ

10 10 Engineering hysics β γ α C γ x C α B z β y Fig. 1. This law is a direct consequence of the triangle law. Since the forces are in equilibrium, they can be represented by the sides of the triangle BC taken in order. general property of any triangle is that each side is proportional to the sine of the angle opposite to it. Thus in the triangle BC drawn with the sides parallel to the forces,, and, B BC C = = sin x sin y sin z Here x, y and z are the angles of the triangle BC. But by the triangle law of forces, the sides of the triangle are proportional to the respective force. From the Fig. 1. sin x = sin (180 α) = sin α sin y = sin (180 β) = sin β sin z = sin (180 γ) = sin γ Hence sin α = = = constant sin β sin γ Thus, if 3 forces acting on a particle are in equilibrium, each force is proportional to the sine of the angle between the other two Experimental Verification of the Laws The simple arrangement used to verify the above discussed laws is generally called parallelogram law apparatus as shown in Fig. 1.3.

11 S I Units, Statics 11 X Y Z Fig. 1.3 Two smooth aluminium pulleys are fixed on a vertical drawing board as shown in this figure. long string is passed over the pulleys. nother string is knotted at a point O as indicated. sheet of paper is pinned to the board behind the strings. Weights, and placed at the ends of the strings and adjusted such that the knot O rests in a convenient position against the paper. The position of the point O and the directions of the strings OX, OY and OZ are marked on the paper. The paper is taken out and the experiment is repeated for different values, and. Verification of the arallelogram Law Taking a suitable scale, lengths O, OB and OC are selected along the directions OX, OY and OZ. The parallelogram ODB is completed and the diagonal OD is drawn. D x y B O 1 O p C r q 1 Z C 1 Fig. 1.4

12 1 Engineering hysics Since OC represents the equilibrant of and it should be equal and opposite to the resultant of and. In all cases, it will be seen that OC = OD and angle COD = 180. This verifies the parallelogram law. Verification of Lami s Theorem To verify Lami s theorem, the angles α, β and γ, opposite to the forces are measured from the traces of the string position. The ratios, and are calculated. In all sin α sinβ sin γ cases = = which verifies the theorem. The experiment is repeated for sin α sin β sin γ different values of, and. Forces ngles α β γ sinα sinβ sinγ Verification of Triangle Law The law of triangle of forces is verified indirectly by verifying the converse of the law. If we have 3 forces in equilibrium, they can be represented by the sides of any triangle which is drawn so as to have its sides parallel to the direction of the forces. Using a pair of set squares draw lines, O 1 1, 1 C 1 and C 1 O 1, parallel to the lines of action of the three forces, and respectively to form the triangle O 1 1 C 1,, and O 1 1 C 1 1 CO are 1 1 calculated. It will be seen that these ratios are the same. Hence the sides of the triangle represent the forces in magnitude. Since the sides are parallel to the lines of action of the forces, they represent magnitudes as well as directions. The experiment is repeated for different values of, and.

13 S I Units, Statics 13 Forces Sides of triangle O 1 1 C 1 1 OC 1 1 O C 1 O 1 C 1 SHOT UESTIONS 1 Mark uestions 1.1 Express 5 C in kelvin scale. 1. What are the two supplementary SI units? 1.3 Give the dimensions for work done. 1.4 The wave length of sodium light is nanometre. Express it in metre. 1.5 What is the unit of electric charge? 1.6 What is the dimension of the physical quantity stress? 1.7 The power of the given electric lamp is 100 Watts. Is it the correct way of writing this physical quantity? 1.8 Strain in elasticity is a physical quantity without any dimension. Is it right or wrong? 1.9 Give two scalar quantities Force acting on unit area is called Give the value of vogadro s number. 1.1 epresent a vector Write zero degree kelvin in celsius scale. Marks uestions 1.1 Explain radian the unit of plane angle. Express one radian in degree. *1. What are the two parameters required to represent a physical quantity? 1.3 Get the unit and dimensions of acceleration. *1.4 MLT is the dimensions of a physical quantity. Name the physical quantity. From that get the dimension of pressure.

14 14 Engineering hysics 1.5 How will you add when two vectors act in the same direction? *1.6 Define Lami s theorem. 1.7 Explain the apparatus required to verify parallelogram law of forces *1.8 Get the unit of thermal conductivity. 1.9 State the converse of triangle law of forces Express 60 in radian measure. *1.11 π Convert radian into degree. 6 nswers to Starred uestions 1. Let us assume the distance between two points is 3 metre. If the magnitude 3 is absent, then the representation of the physical quantity is meaningless. Similarly if metre is absent, then also the true meaning is unattainable. So both the magnitude and the unit are required t represent a physical quantity. 1.4 The general expression for force is F = ma The dimension are m M & a = L 1 T T F= MLT nswer Force for unit area is pressure i.e. = F MLT = L Thus the dimension of pressure is ML T --1 nswer 1.6 If three forces acting on a particle keep it in equilibrium, each force is proportional to the sine of the angle between the other two. 1.8 For solving this one requires the general definition which is explained below: In the steady state the quantity of heat conducted through a rod of uniform cross-section is depending on the area of cross-section, time of conduction and temperature gradient. i.e. µ µt θ θ d µ 1

15 S I Units, Statics 15 ( θ1 θ) i.e. = λ t d θ1 λ = ( θ1 θ) t d When we apply the units, the unit of λ coefficient of thermal conduction is joule metre m sec K = joule sec m K = W m 1 K 1 This is the unit of λ π radian corresponds to π 360 π radian corresponds to = π EVIEW UESTIONS 10 Marks uestions 1.1 (a) Describe the conventional standards used to represent a physical quantity. (b) How metre and second in S I units are defined? (c) What are the main advantages of S I units? 1. (a) Explain the dimensions of physical quantities with suitable examples. (b) Obtain the unit and dimensions of resistance. (c) Magnitude and unit are essential to represent any physical quantity. Explain. 1.3 (a) Distinguish scalar and vector quantities with suitable examples. (b) Explain addition and subtraction vectors acting in the same direction and also in the opposite directions. (c) Explain parallelogram law of forces. 1.4 (a) State Lami s theorem. (b) Explain how Lami s theorem is verified. (c) If the resultant of two forces 6N and 8N is 1N, find the angle between the two forces. 1.5 (a) State parallelogram law of forces.

16 16 Engineering hysics (b) Describe with a neat diagram an experimental arrangement to verify this law. (c) Discuss how the law of parallelogram of forces is verified. OBLEMS ND SOLUTIONS 1.1 Derive the dimensional formula for the kinetic energy of a body. 1 mv is the expression for kinetic energy. The dimensional formula for velocity is LT 1 ; hence that for K-E is ML T. 1. The value of g in S.I. unit is 9.8 m/s. Find its value in ft/s in FS system The dimensions of g are LT. If the units and the numerical magnitudes of a physical quantity in two systems of unit are U 1, n i and U n respectively, then n 1 U 1 = n U thus n 1 (L 1 T 1 ) = n (L T ) L T n 1 = n L T 1 1 = 9.8 1metre 1sec 1ft 1sec 3.81 ft = 9.8 1ft n 1 = 3.15 ft/s n = nswer ft/s 1.3 The wave length associated with the green line of mercury spectrum is 546 nm. Obtain the dimension of lanck constant. Calculate also the frequency of the line. ch E = hν= λ The dimension of energy F distance i.e E MLT L and ν T 1.

17 S I Units, Statics 17 Thus, the dimension of h is ν = ML T 1 T 8 c 3 10 = λ = ML T ν= Hz nswer 1.4 Explain what is meant by dimensions? Write down the dimensional formulae for the following physical quantities: (a) ) momentum (b) ) surface tension (c) ) angular velocity It has been pointed out in many places that a physical quantity can be expressed in terms of a combination of fundamental units of mass (M), length (L), time (T), temperature (θ), current () and light intensity (I). Thus, velocity and acceleration of a body can be written as Velocity = Displacement Length = Time Time = LT cceleration = -1 Velocity LT = Time T = LT nswer The equatin for veleocity namely LT 1 is called a dimensional equation, whereas the expresssion M LT 1 is called dimensional formula. Thus, velocity is said to have zero dimension in mass, 1 in length and in time. (a) Momentum : mass velocity : MLT force MLT (b) Surface tension : : : MT length L angle angle 1 (c) ngular velocity : : : T time time 1 nswer 1.5 Deduce the dimensions of universal constant of gravitation. From Newton s law of universal graviation, we know that force F, between two bodies of masses m 1 and m, kept at a distance d apart is given by F = G = G F d mm mm 1 d 1

18 18 Engineering hysics where G is a constant called gravitational constant G = MLT M L G: M LT 1 3 nswer 1.6 Get the unit of electrical resistance and its dimension. lso that of electric charge Ohm s law states V = I = V :volt/ampere I nswer Dimension of = V I But V = El with F = q E; or E = F q E = V = Thus, = F MLT MLT q I t T MLT L T MLT L T = M L T 3 i.e. the dimension of resistance 3 ML T Dimension of Electric Charge, q = It = T nswer

19 S I Units, Statics famous relation in physics relates moving mass m to rest mass m o in terms of its speed v and the speed of light c.. boy writes wrongly as m= Guess where to put c? m o -v. The dimensions of physical quantities appearing on both sides must be possible. This is possible only when you put v c instead of v. The correct equation is Get the dimension of gas constant = V MLT L T ML T θ 1 3 = L θ m = m o 1 ( v / c nswer 1 MLT nswer θ 1.9 Convert one newton into dynes Let n 1 newton be equal to n dynes n 1 (N 1 )=n (N ) n 1 (M 1 L 1 T 1 )=n (N ) (M L T ) n = = 1 1 M L T n M L T 1 1 M 1 L 1 T 1 n1 M L T = = newton= 10 dyne nswer

20 0 Engineering hysics 1.10 Two forces 15N and 10N act at a point at an angle 30. Find the resultant. = 15 N = 10N = 30 D C θ α θ O D or = + + cos θ = cos 30 = 4. newton nswer 1.11 The resultant of two unequal forces acting at an angle 150 is perpendicular to the smaller of the forces. If the larger force is 3 newton, find out the smaller force and the resultant. D C Let and represent the smaller force and the resultant respectively. is perpendicular to. From triangle CD D sin 60 = = CD 3 or = 3 sin 60 =.6 newton lso, + = 3 3N B

21 S I Units, Statics 1 or = 9 = 9.6 = 9.6 = 1.5 N nswer 1.1 body is acted on by two forces 150 N towards north and another of 00 N towards west. Find the resultant in magnitude and direction. N 150N α 00N = = 6500 = 50 newton nswer tan α = = α= 36.9 nswer 1.13 Find graphically the resultant of two forces 10 N and 5 N acting at a point inclined at an angle 60. Verify by calculation B 60 O ( ) O corresponds to 10 units O (.7) corresponds to 10.7 = 13.5 O = 13.5 nswer