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Physics 1-3A (Sem-l & 2) 4. Suppose P be a point in the space. 5. Now from Fig. 1.1.1, x = x' + vt ...(1.1.1) y = y' 1 •••(1.1.2) z = Z No relative motion ...(1.1.3) t = t' J •••(1.1.4) 6. Eq. (1.1.1) to eq. (1.1.4) are position and time transformation equations in sand s' frame. 7. Differentiating eq. (1.1.1) w.r.t. t on both sides, dx dx' vdt' - = -+-- ...(1.1.5) dt dt dt dx dx'. vdt' - = -+-- ( .: t =t' dt = dt') dt dt' dt' => u = u' + v ...(1.1.6) 8, Differentiating eq. (1. 1.2) ~.r.t. t, dy dy' ...(1.1.7) dt = & dy dy' (.: t=t') dt = dt' => u = u' ...(1.1.8) 9. Similarly, tl = u,Y ...(1.1.9) 10. Now differentiating ~q. (l~1.6) w.r.t. t, we get, dux du: dv -- = --+ dt dt dt du du: ( .: v = constant) d/ = dt dux _ du: (.: t =t') dt - dt' => a =a' ...(1.1.10) 11. Similarly on differeritiati~geq. (1.1.8) and eq. (1.1.9), we get a y =a'y . ...(1.1.11) az = a'z ...(1.1.12) 12. Eq. (1.1.10), eq. (1.1.11) and eq. (1.1.12) shows that the acceleration is invariant in both frames. 1:~. So a frame of reference moving with constant velocity is an inertial frame. Que 1:2:: '1 Derive the Galilean transfonnation equations and show that its acceleration components are invariant. Answer I 1. Suppose we are in an inertial frame ofreference s and the coordinates of some event that occurs at the time t are x, y, z as shown in Fig. 1.2.1. I~A(Sem·l&2) Relativistic Mechanics 2. An observer located in a different inertial frame s' which is moving with -+ respect to s at the constant velocity v, will find that the same event. occurs at time t' and has the position coordinates x', y' and z'. y y! s s'_ .. v o x 0' x' z z' ... 3. Assume that v is in positive x direction. 4. When origins ofs and s' coincide, measurements in the x direction made in s is greater than those ofs' by v -+ t (distance). 5. Hence, x'=x-vt (1.2.1) y' = y (1.2.2) z'= z (1.2.3) t'= t (1.2.4) These set of equations are known as Galilean transformations. 6. Differentiating eq. (1.2.1) with respect to t, we get dx' = dx _ v dt} no relative motion dt dt dt dx' dx --= --v (.: t = t' dt' = dt) dt' dt dy' dy 7. Similarly dt' = dt dz' dz and dt' = dt 8. Since, dx'/dt' = ur', the x-component ofthe velocity measured in s', and dx/dt = ur' etc., then, u'=u r r -v ...(1.2.5) u' = u y y ...(1.2.6) 9. u/ = Uz ...( 1.2.7) Eq. (1.2.5), eq. (1.2.6) and eq. (1.2.7) can be written collectively in the vector form as -+ u' = u - v ...(1.2.8) 10. To obtain the acceleration transformation, we differentiate the eq. (1.2.5). eq. (1.2.6) and eq. (1.2.7) with respect to time such that du; = .!!:....(u -v)~ dUr aktureference.com dt' dt r dt
__ l'hysics 1-5 A (Sem-l & 2) duy' = duy and dU; = duz Similarly, dt' dt dt' dt dux' ,duy' ,du; , Since, --= a '--=a '--=a dt' x 'dt' y' dt' z du du du -_X_=a . --y-=a ; __z_=a dt x' dt y dt z Then we get a'= a ...(1.2.9) .• x a y '= a y ...(1.2.10) a'= a ...(1.2.11) z z or writing these equations collectively, ;;, = U, The measured components,ofacceleration ofa particle are independent of the uniform relative velocity ofthe reference frames. ,i Tn other words, acceleration remains invariant when passing from one inertial frame to another that is in uniform relative translational motion. '~ue 1.3. IShow that the distance between points isinvariant under ";,,iilean transformations. Answer·'· :1 Let (xl'.Y1' Zl) and (x2' Y 2, Z2) be the coordinate oftwo points P and Q in l'est frame 5. Then the distance between them will be d = J(x2 ·- XI )2 + (Y2 - YI)2 + (Z2 - ZI)2 .Now the distance between them measured in moving frame 5' is d' = J(x 2 '- Xl ')2 + (Y2 '- YI ')2 + (Z2 '- Zl ')2 U sing Galilean transformation x 2' = x 2 - v/,Y2' =Y2 -v/ and Z2' = Z2 -v,t Xl' = Xl -v/,YI ' =YI -vi and Zl' = Zl -v,t 1, Hence ,,i' J[(x~ - V x t)-(~xtW-+[(Y2 -vyt)-(YI _vy t)]2 + [(Z2 -vzt)-(Zl -vzt)]2 '" J(x~ - XI )2 + (Y2 .,.- YI)2 + (Z2 - ZI)2 d'= d Que 1.4;,/.,1 Discuss the objective and outcome ofMichelson-Morley experiment. An~~~~~!i~ A. Objective of Michelson-Morley Experiment: The main objective of conducting the Michelson-Morley experiment was to confirm the existence ofstationary ether. According to Morley ifthere exist some imaginary medium like 'ether' in the earth atmosphere, there should be some tilue difference between relative motion ofbody with respect to earth and against the motion of earth. I~A(Sem-l & 2) Relativistic Mechanics 3. Due to this time difference there exist some path difference and if such path difference occurs, Huygens concept is correct and if it does not occur then Huygens concept is wrong. B. Michelson-Morley Experiment: 1. In Michelson-Morley experiment there is a semi-silvered glass-plate P aild two plane mirrorsMI andM2 which are mutually perpendicular and equidistant from plate P. 2. There is a monochromatic light source in front of glass plate P, 3. The whole arrangement is fixed on a wooden stand and that wooden stand is dipped in a mercury pond. So it becomes easy t.o rotate, 4. Let vbe the speed ofimaginary medium (ether) w.r.t. earth and (' is the velocity oflight, so time taken to move the light ray from plate P to M, and reflected back, C T = + = d[ .2 l I c+v c-v (C2_V~)J 2dc 2d T1 = 2 c [ 1-;:J=F ~; J 5. Expanding binomially, TI = 2d[I+~J ,.. (1.4.1) 2 C c [Neglecting higher power term] 6. Time taken to move a light ray from plate P to M, and to reflect back, 2d 2d 2d [ v~ J I T2 = .Jc 2 _ v 2 = = --;;- 1 - c 2 "--2 C M 1 M'l I I' I ':./r~':. ':.1 I , 1 I , I ' , , , , M 2 M'2 , I I , ;' .... I' I III ! ~'" ,;', -II S > > 2 N ,. -....-- I I ,./ ,./ P' I I .... ;' I I I- d" "I I..! L T _.iht~~~~i~*l#Ji1~~nt. aktureference.com
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