Physics Handbook PDF

ALLEN Physics Handbook  C HAP TE R...

ALLEN Physics Handbook  C HAP TE R Physics HandBook ALLEN Units, Dimension, Measurements and Practical Physics Fundamental or base quantities Systems of Units MKS CGS FPS MKSQ MKSA The quantities which do not depend upon other (i) Length Length Length Length Length quantities for their complete definition are known (m) (cm) (ft) (m) (m) as fundamental or base quantities. (ii) Mass Mass Mass Mass Mass e.g. : length, mass, time, etc. (kg) (g) (pound) (kg) (kg) (iii) Time Time Time Time Time (s) (s) (s) (s) (s) Derived quantities (iv) – – – Charge Current The quantities which can be expressed in terms (Q) (A) of the fundamental quantities are known as derived Fundamental Quantities in quantities. S.I. System and their units N e.g. Speed (=distance/time), volume, S.N. Physical Qty. Name of Unit Symbol acceleration, force, pressure, etc. 1 Mass kilogram kg 2 Length meter m 0 Units of physical quantities 3 Time second s -2 4 Temperature kelvin K The chosen reference standard of measurement 5 Luminous intensity candela Cd E in multiples of which, a physical quantity is 6 Electric current ampere A 19 expressed is called the unit of that quantity. 7 Amount of substance mole mol e.g. Physical Quantity = Numerical Value × Unit 20 SI Base Quantities and Units LL S I U n its B a s e Q u a n tity N am e Sym bo l D efin itio n L e n gth m e te r m T h e m eter is the le ng th o f th e pa th tra vele d b y lig h t in on va cuum durin g a tim e in te rva l o f 1/(2 99 , 7 92 , 4 58 ) o f a seco nd (1 9 8 3 ) M ass kilogra kg T h e kilogra m is e qual to th e m ass o f the in tern atio n al i m p roto typ e o f th e k ilogra m (a platin um -iridium a lloy ss cy lin d e r) ke p t a t In tern atio n al B u rea u o f W eig h ts an d M ea su re s, a t S e vres, ne ar P a ris, Fra nc e. (1 8 8 9) T im e se co n d s T h e se co n d is th e dura tio n o f 9 , 1 9 2 , 6 31 , 77 0 pe rio ds Se o f the rad iatio n c orre spo nd in g to th e tran sition b etw e e n th e tw o h yp erfin e leve ls o f th e g ro u n d sta te of the A ce sium -1 3 3 a to m (1 9 6 7 ) E lectric C urren t am pe re A T h e a m p e re is th at c o n stan t cu rre nt w h ic h , if m a in ta in e d node06\B0AI-B0\Kota\JEE(Advanced)\Leader\Phy\Sheet\Hand book (E+L)\Eng\01_Unit & Dimension.p65 in tw o straig h t pa ralle l c o n du cto rs o f infin ite le ng th , of n egligible circular cro ss-section, an d p laced 1 m etre ap art in vacuum , w o uld prod uce be tw e en these con du ctors a fo rce e qual to 2 x 1 0 -7 N e w to n p e r m e tre o f le ng th. (1 9 4 8 ) T h erm o d yn am ic kelvin K The k elvin, is th e fra ctio n 1 /2 7 3.1 6 of the T e m p era ture th erm ody n am ic tem p erature o f the triple p o in t of w a ter. (1 9 6 7 ) A m ou nt o f m o le m ol T h e m ole is th e a m o unt o f substa nc e o f a sy stem , w h ic h S ubsta n ce co n ta in s a s m an y elem e nta ry e ntitie s as the re a re a to m s in 0.01 2 k ilo gram o f carb o n -1 2. (1 9 7 1 ) L um inous ca nde la Cd T h e can dela is th e lum in o us in te nsity, in a give n dire ctio n, Inte nsity o f a so u rce th at em its m o n oc h ro m a tic rad iatio n of fre quen cy 5 4 0 x 1 0 1 2 h e rtz an d th at ha s a rad ia nt inten sity in th at directio n of 1/ 6 8 3 w a tt pe r stera d ian (1 9 7 9 ). E 1 Physics HandBook CH APTER ALLEN Supplementary Units Limitations of dimensional analysis Radian (rad) - for measurement of plane angle In Mechanics the formula for a physical quantity Steradian (sr) - for measurement of solid angle depending on more than three other physical quantities cannot be derived. It can only be checked. Dimensional Formula This method can be used only if the dependency is Relation which express physical quantities in terms of of multiplication type. The formulae containing appropriate powers of fundamental units. exponential, trigonometrical and logarithmic functions can't be derived using this method. Use of dimensional analysis Formulae containing more than one term which are added or subtracted like s = ut +½ at2 also To check the dimensional correctness of a given can't be derived. physical relation To derive relationship between different physical The relation derived from this method gives no quantities information about the dimensionless constants. To convert units of a physical quantity from one If dimensions are given, physical quantity may not system to another N be unique as many physical quantities have the same a b c dimensions. æM ö æL ö æT ö n1u1= n2u2 Þ n2=n1 ç 1 ÷ ç 1 ÷ ç 1 ÷ è M2 ø è L 2 ø è T2 ø It gives no information whether a physical quantity 0 is a scalar or a vector. where u = MaLbTc -2 SI PREFIXES E 19 The magnitudes of physical quantities vary over a wide range. The CGPM recommended standard prefixes for magnitude too large or too small to be expressed more compactly for certain powers of 10. 20 Power of Power of Prefix Symbol Prefix Symbol 10 10 LL PREFIXES 1018 exa E 10-1 deci d USED FOR 10 15 peta P 10-2 centi c on DIFFERENT 1012 tera T 10-3 milli m POWERS 10 9 giga G 10-6 micro m OF 10 i 10 6 mega M 10-9 nano n ss 10 3 kilo k 10-12 pico p 10 2 hecto h 10-15 femto f 1 - 18 10 deca da atto a Se 10 A P hysical quan tity Unit P hys ical q uantity U nit node06\B0AI-B0\Kota\JEE(Advanced)\Leader\Phy\Sheet\Hand book (E+L)\Eng\01_Unit & Dimension.p65 A ngular acceleration rad s- 2 F requen cy h ertz Mo ment o f in ertia kg – m 2 Resistan ce kg m 2 A- 2 s-3 Self in ductan ce h en ry Surface ten sio n n ew ton/m UNITS -1 -1 OF Magnetic flux w eber Universal gas con stant joule K mo l IMPORTANT Po le str en gth A –m Dipole mo ment co ulo mb–meter PHYSICAL V iscosity po is e Stefan co n stant watt m - 2 K -4 QUANTITIES Reactance oh m Perm ittivity o f free space (e0) co ulo mb 2 /N– m 2 P erm eability o f free space Specific heat J/kg°C weber/A-m (m0 ) Strength of m agnetic n ewton A -1 m - 1 P lan ck's co nstan t jo ule –se c field A stronom ical dis tance Parsec En tro py J/K 2 E  C HAP TE R Physics HandBook ALLEN DIMENSIONS OF IMPORTANT PHYSICAL QUANTITIES Physical quantity Dimensions Physical quantity Dimensions Momentum M L T– 1 1 1 Capacitance M–1 L–2 T4 A 2 Calorie M1 L 2 T –2 Modulus of rigidity M1 L– 1 T –2 Latent heat capacity M L T– Magnetic permeability M L T – A– 0 2 2 1 1 2 2 Self inductance M L T – A– Pressure M L– T – 1 2 2 2 1 1 2 Coefficient of thermal conductivity M L T– K – Planck's constant M L T– 1 1 3 1 1 2 1 Power M L T– Solar constant M L T– 1 2 3 1 0 3 Impulse M1 L 1 T –1 Magnetic flux M1 L2 T –2 A– 1 Hole mobility in a semi conductor M– L T A Current density M L– T A 1 0 2 1 0 2 0 1 M L– T– M L– T – N Bulk modulus of elasticity 1 1 2 Young modulus 1 1 2 Potential energy M L T– Magnetic field intensity M L– T A 1 2 2 0 1 0 1 Gravitational constant M – L T– Magnetic Induction M T– A – 1 3 2 1 2 1 0 Light year M0 L1 T0 Electric Permittivity M– L– T A 1 3 4 2 -2 Thermal resistance M–1 L–2 T3 K Electric Field M1L 1T–3A- 1 E Coefficient of viscosity M1 L– 1 T–1 ML T – A– 19 Resistance 2 3 2 SETS OF QUANTITIES HAVING SAME DIMENSIONS 20 LL S.N. Quantities Dimensions on 1. Strain, refractive index, relative density, angle, solid angle, phase, distance gradient, relative permeability, relative permittivity, angle of contact, Reynolds [M 0 L 0 T 0] number, coefficient of friction, mechanical equivalent of heat, electric susceptibility, etc. i 2. Mass or inertial mass [M 1 L 0 T 0 ] ss 3. Mom entum and impulse. [M 1 L 1 T – 1] 4. Thrust, force, weight, tension, energy gradient. [M 1 L 1 T – 2] Se 5. Pressure, stress, Young's modulus, bulk modulus, shear modulus, modulus of rigidity, energy density. [M 1 L – 1 T – 2 ] A 6. Angular momentum and Planck's constant (h). [ M 1 L2 T –1] 7. Acceleration, g and gravitational field intensity. [ M 0 L1 T –2] node06\B0AI-B0\Kota\JEE(Advanced)\Leader\Phy\Sheet\Hand book (E+L)\Eng\01_Unit & Dimension.p65 8. Surface tension, free surface energy (energy per unit area), force gradient, spring constant. [ M 1 L0 T –2] 9. Latent heat capacity and gravitational potential. [ M 0 L2 T –2] 10. Thermal capacity, Boltzmann constant, entropy. [ ML 2 T – 2K – 1 ] 11. Work, torque, internal energy, potential energy, kinetic energy, mom ent of force, (q2 /C), (LI2 ), (qV), (V 2 C), (I 2 Rt), V2 t , (VIt), (PV), (RT), (mL), (mc DT) [M 1 L 2 T – 2] R 12. Frequency, angular frequency, angular velocity, velocity gradient, radioactivity R 1 , , 1 [M 0 L 0 T – 1] L RC LC 13. ælö 12 æmö 1 2 æL ö ç ÷ ,ç ÷ ,ç ÷ , (RC), ( LC ) , time [ M 0 L 0 T 1] ègø è k ø èR ø 14. (VI), (I2 R), (V 2/R), Power [ M L 2 T – 3] E 3 Physics HandBook CH APTER ALLEN Gravitational constant (G) 6.67 × 10 –11 N m 2 kg –2 KEY POINTS Speed of light in vacuum (c) 3 × 10 8 ms –1 Trigonometric functions SOME FUNDAMENTAL CONSTANTS Permeability of vacuum (m 0 ) 4p × 10 –7 H m –1 sinq, cosq, tanq etc and their Permittivity of vacuum (e 0 ) 8.85 × 10 –12 F m –1 arrangement s q are Planck constant (h) dimensionless. 6.63 × 10 –34 Js Atom ic mass unit (am u) 1.66 × 10 –27 kg Dimensions of differential Energy equivalent of 1 amu 931.5 MeV é dny ù éyù coefficients ê n ú = ê n ú 9.1 × 10 –31 kg º 0.511 ë dx û ë x û Electron rest mass (m e ) MeV Dimensions of integrals Avogadro constant (N A ) 6.02 × 10 23 mol –1 é ydx ù = [ yx ] ëê ò úû Faraday constant (F) 9.648 × 10 C mol 4 –1 Stefan–Boltzmann constant (s) 5.67× 10 –8 W m –2 K –4 We can't add or subtract two physical quantities of Wien constant (b) 2.89× 10 –3 mK different dimensions. N Rydberg constant (R ¥ ) 1.097× 10 m –1 7 Independent quantities may Triple point for water 273.16 K (0.01°C) be taken as fundamental quantities in a new system of 0 22.4 L = 22.4× 10 –3 m 3 Molar volum e of ideal gas (NTP) units. mol –1 -2 E PRACTICAL PHYSICS 19 Rules for Counting Significant Figures For example : 3.0 × 800.0 = 2.4 × 103 20 For a number greater than 1 The sum or difference can be no more precise than All non-zero digits are significant. the least precise number involved in the mathematical LL All zeros between two non-zero digits are operation. Precision has to do with the number of significant. Location of decimal does not matter. positions to the RIGHT of the decimal. The more on If the numbe is without decimal part, then the position to the right of the decimal, the more precise terminal or trailing zeros are not significant. the number. So a sum or difference can have no Trailing zeros in the decimal part are significant. more indicated positions to the right of the decimal i ss For a Number Less than 1 as the number involved in the operation with the Any zero to the right of a non-zero digit is significant. LEAST indicated positions to the right of its decimal. All zeros between decimal point and first non-zero For example : 160.45 + 6.732 = 167.18 (after Se digit are not significant. rounding off) A Significant Figures Another example : 45.621 + 4.3 – 6.41 = 43.5 All accurately known digits in measurement plus (after rounding off) node06\B0AI-B0\Kota\JEE(Advanced)\Leader\Phy\Sheet\Hand book (E+L)\Eng\01_Unit & Dimension.p65 the first uncertain digit together form significant Rules for rounding off digits : figure. 1. If the digit to the right of the last reported digit is Ex. 0.108 ® 3SF, 40.000 ® 5SF, less than 5 round it and all digits to its right off. 1.23 × 10-19 ® 3SF, 0.0018 ® 2SF 2. If the digit to the right of the last reported digit is Significant Digits greater than 5 round it and all digits to its right off The product or quotient will be reported as having and increased the last reported digit by one. as many significant digits as the number involved 3. If the digit to the right of the last reported digit is a in the operation with the least number of significant 5 followed by either no other digits or all zeros, digits. round it and all digits to its right off and if the last For example : 0.000170 × 100.40 = 0.017068 reported digit is odd round up to the next even Another example : 2.000 × 104 / 6.0 × 10–3 = digit. If the last reported digit is even then leave it 0.33 × 107 as is. 4 E  C HAP TE R Physics HandBook ALLEN For example if we wish to round off the following Error in Product and Division number to 3 significant digits : 18.3682 A physical quantity X depend upon Y & Z as X = Ya Zb The answer is : 18.4. Another example : Round then maximum possible fractional error in X. off 4.565 to three significant digits. DX DY DZ = a + b The answer would be 4.56. X Y Z Rounding off Error in Power of a Quantity 6.87® 6.9, 6.84 ® 6.8, 6.85 ® 6.8, am Dx é æ Da ö æ Db ö ù 6.75 ® 6.8, 6.65 ® 6.6, 6.95 ® 7.0 x= n then = ± êm ç ÷ + n ç ÷ ú è ø è b øû b x ë a Order of magnitude The quotient rule is not applicable if the numerator Power of 10 required to represent a quantity and denominator are dependent on each other. 49 = 4.9 × 101 » 101 Þ order of magnitude =1 XY 51 = 5.1 × 101 » 102 Þ order of magnitude = 2 e.g if R =. We cannot apply quotient rule X+Y 0.051 =5.1 × 10-2 » 10-1order of magnitude = -1 EN to find the error in R. Instead we write the equation Errors Whenever an experiment is performed, two kinds 1 1 1 as follows = +. Differentiating both 0 of errors can appear in the measured quantity. R X Y (1) random and (2) systematic errors. -2 1. Random errors appear randomly because of dR dX dY the sides, we get - 2 =- 2 -. operator, fluctuations in external conditions and R X Y2 19 variability of measuring instruments. The effect of random error can be some what reduced by taking r x y Thus 2 = 2 + the average of measured values. Random errors Y2 20 R X have no fixed sign or size. LL Least count 2. Systematic errors occur due to error in the The smallest value of a physical quantity which can procedure, or miscalibration of the instrument etc. n be measured accurately with an instrument is called Such errors have same size and sign for all the the least count of the measuring instrument. io measurements. Such errors can be determined. Vernier Callipers A measurement with relatively small random error ss Least count = 1MSD – 1 VSD is said to have high precision. A measurement with (MSD ® main scale division, VSD ® Vernier scale division) small random error and small systematic error is Se said to have high accuracy. Least Count Error :– If the instrument has known A least count, the absolute error is taken to be equal node06\B0AI-B0\Kota\JEE(Advanced)\Leader\Phy\Sheet\Hand book (E+L)\Eng\01_Unit & Dimension.p65 to the least count unless otherwise stated. 0 1 2 3 4 5 6 14 15 absolute error in a measurement Relative error = size of the measurement A. Systematic errors : They have a known sign. The systematic error is removed before beginning calculations. Bench error Ex. A vernier scale has 10 parts, which are equal to 9 and zero error are examples of systematic error. parts of main scale having each path equal to 1 Propagation of combination of errors 9 mm then least count = 1 mm – mm = 0.1 mm Error in Summation and Difference : 10 x = a + b then Dx = ± (Da+Db) [Q 9 MSD = 10 VSD] E 5 Physics HandBook CH APTER ALLEN Zero Error Main scale Main scale Main scale 0 1 0 1 0 1 0 5 10 0 5 10 0 5 10 Vernier scale Vernier scale Vernier scale without zero error with positive zero error with negative zero error (i) (ii) The zero error is always subtracted from the reading to get the corrected value. If the zero error is positive, its value is calculated as we take any normal reading. EN Negative zero error = – [Total no. of vsd – vsd coinciding] ×L.C. Screw Gauge 0 pitch Least count = total no. of divisions -2 on circular scale 19 Spindle Circular (Head) scale Ratchet 05 10 20 Linear (Pitch) Scale Thimble Sleeve LL Ex. The distance moved by spindle of a screw gauge for each turn of head is 1mm. The edge of the humble is n 1mm io provided with a angular scale carrying 100 equal divisions. The least count = = 0.01 mm 100 ss Zero Error Positive Zero Error If there is no object between the jaws (i.e. jaws are in (2 division error) i.e., +0.002 cm contact), the screwgauge should give zero reading. But Se due to extra material on jaws, even if there is no object, it A gives some excess reading. This excess reading is called Zero error. Circular scale node06\B0AI-B0\Kota\JEE(Advanced)\Leader\Phy\Sheet\Hand book (E+L)\Eng\01_Unit & Dimension.p65 Negative Zero Error (3 division error) i.e., –0.003 cm 15 10 Circular scale 5 Zero of the circular 0 scale is below the 0 zero of main scale Main scale 95 10 reference line zero of the circular 90 5 scale is above the 0 0 zero of main scale 95 Main scale 90 reference line 85 6 E  C HAP TE R Physics HandBook ALLEN Basic Mathematics used in Physics Quadratic Equation Binomial Theorem 2 - b ± b - 4ac n(n - 1) 2 n(n - 1)(n - 2) 3 Roots of ax2 + bx + c=0 are x = (1+x)n = 1 + nx + x + x +.... 2a 2 6 b Sum of roots x1 + x2 = – ; n ( n - 1) 2 n(n - 1)(n - 2) 3 a (1–x)n = 1 – nx + x - x +..... 2 6 c Product of roots x1x2 = If x m2 now for mass m1, m1 g – T = m1a then for body m1: (F – f1) = m1a for mass m2,T – m2 g = m2 a a a (m1 - m2 ) net pulling force Acceleration = a = g= F (m1 + m2 ) total mass to be pulled m1 f1 f1 m2 2m1 m2 2 ´ Pr oduct of masses Tension = T = g= g Fig. 1(a) : F.B.D. representation of action and reaction forces. (m1 + m2 ) Sum of two masses Reaction at the suspension of pulley : For body m2 : 4m1 m2 g m2 F R = 2T = f1=m2a Þ action of m1 on m2: f1 = (m1 + m2 ) m1 + m2 Case – II Pulley system a For mass m1 : m1 A single fixed pulley changes the direction of force T only and in general, assumed to be massless and T = m1 a EN frictionless. For mass m2 : m2g – T = m2 a SOME CASES OF PULLEY Acceleration: m2 a 0 Case – I : m2 g m1 m2 a= and T = g (m1 + m2 ) (m1 + m2 ) -2 T T 19 m2 a a m1 FRAME OF REFERENCE 20 LL Inertial frames of reference : A reference frame which is either at rest or in uniform motion along the straight line. A non–accelerating frame of reference is called an inertial frame of reference. n All the fundamental laws of physics have been formulated in respect of inertial frame of reference. io Non–inertial frame of reference : An accelerating frame of reference is called a non–inertial frame of reference. Newton's laws of motion are not directly applicable in such frames, before application we must add pseudo force. ss Pseudo force: Se The force on a body due to acceleration of non–inertial frame is called fictitious or apparent or pseudo force and r r r A is given by F = - ma0 , where a0 is acceleration of non–inertial frame with respect to an inertial frame and m is mass of the particle or body. The direction of pseudo force must be opposite to the direction of acceleration node06\B0AI-B0\Kota\JEE(Advanced)\Leader\Phy\Sheet\Hand book (E+L)\Eng\04_NLM & Friction.p65 of the non–inertial frame. When we draw the free body diagram of a mass, with respect to an inertial frame of reference we apply only the real forces (forces which are actually acting on the mass). But when the free body diagram is drawn from a non– r r inertial frame of reference a pseudo force (in addition to all real forces) has to be applied to make the equation F = ma to be valid in this frame also. r r r r r åF real + Fpseudo = ma (where ar is acceleration of object in non inertial reference frame) & Fpseudo = -ma a r (where a 0 is acceleration of non inertial reference frame). 24 E  CH APTER Physics HandBook ALLEN Man in a Lift Graph between applied force and force of (a) If the lift moving with constant velocity v upwards friction or downwards. In this case there is no accelerated motion hence no pseudo force experienced by observer f block applied force inside the lift. \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ So apparent weight W´=Mg=Actual weight. Friction force (f) Limiting friction (b) If the lift is accelerated upward with constant f=fL acceleration a. Then forces acting on the man w.r.t. observed inside the lift are Kinetic friction n (i) Weight W=Mg downward io ct fri (ii) Fictitious force F0=Ma downward. ic at St So apparent weight W´=W+F0=Mg+Ma=M(g+a) (c) If the lift is accelerated downward with acceleration 45° Applied force F a g. Then as in Case (c). Apparent weight W´ =M(g–a) is negative, i.e., the LL N man will be accelerated upward and will stay at the ceiling of the lift. n Re f an of su d N l io lta FRICTION nt Applied Friction is the force of two surfaces in contact, or f force ss the force of a medium acting on a moving object. (i.e. air on aircraft.) W Fr ictional fo rces ar ise due t o molecular Se fS m s N interactions. In some cases friction acts as a tan l = = = mS A N N supporting force and in some cases it acts as w Angle of repose : The maximum angle of an opposing force. inclined plane for which a block remains node06\B0AI-B0\Kota\JEE(Advanced)\Leader\Phy\Sheet\Hand book (E+L)\Eng\04_NLM & Friction.p65 w Cause of Friction: Friction arises on account of stationary on the plane. strong atomic or molecular forces of attraction between the two surfaces at the point of actual contact. N w Types of friction fs Friction q Mgcosq si n Mg q Mg Static friction Kinetic friction R (No relative motion (There is relative motion tanq R=m s between objects) between objects) E 25 Physics HandBook C HAP TE R ALLEN Dependent Motion of Connected Bodies Normal constraint : displacements, velocities & Method I : Method of constraint equations accelerations of both objects should be same along C.N. v2 dx i d2 x å x i = constant Þ å dt = 0 Þ å 2i = 0 dt v1 a2 a1 q r For n moving bodies we have x1, x2,...xn r No. of constraint equations = no. of strings e.g. a2 = a1 tan q & v2 = v1 tan q KEY POINTS Method II : Method of virtual work : Aeroplanes always fly at low altitudes because The sum of scalar products of tension forces applied according to Newton's III law of motion as aeroplane displaces air & at low altitude density of air is high. by connecting links of constant length and displacement of corresponding contact points equal to zero. Rockets move by pushing the exhaust gases out so r r r r r r they can fly at low & high altitude. EN åT × x = 0 Þ åT × v =0 Þ åT ×a =0 Pulling (figure I) is easier than pushing (figure II) on a rough horizontal surface because normal reaction is less in pulling than in pushing. 0 F -2 F m q q m T m m x1 x2 Here 2a2 = a1 19 T Fig. I Fig. II T While walking on ice, one should take small steps to avoid slipping. This is because smaller step 2T 20 increases the normal reaction and that ensure smaller 1 a1 2 a2 friction. LL n IMPORTANT NOTES io ss Se A node06\B0AI-B0\Kota\JEE(Advanced)\Leader\Phy\Sheet\Hand book (E+L)\Eng\04_NLM & Friction.p65 26 E  CH APTER Physics HandBook ALLEN CIRCULAR MOTION Definition of Circular Motion r r r In vector form velocity v =w´r When a particle moves in a plane such that its r r r r dv d r r dw r r dr distance from a fixed (or moving) point remains w Acceleration a = = (w ´ r ) = ´ r +w´ dt dt dt dt constant then its motion is called as circular motion r r r r r r = a ´ r + w ´ v = a t + aC with respect to that fixed point. That fixed point is called centre and the distance is called radius of circular path. dv w Tangential acceleration: a t = = ar dt