UNIT # 01 (PART I) BASIC MATHEMATICS USED IN PHYSICS, UNIT & DIMENSIONS AND VECTORS. 8. Resultant = R P Q, R P 2Q

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1 J-Phsics UNI # 0 (PAR I) ASIC MAHMAICS USD IN PHYSICS, UNI & DIMNSIONS AND VCORS XRCIS I. nclosed area : A r so da dr r Here r 8 cm, dr da 5 cm/s () (8) (5) 80 cm /s. Slope d d 6 9 if angen is parallel o he ais hen d d ( ) (+) 0 or 0 or. p n F dp d n ()n+ () +n F 0 + n 0 n e e 4. e side of cube be hen d cm/s V dv d cm /s 5. Check A e forces be A and and < A hen A Resulan 4 5 N 9. Required uni ecor A i ˆ 6ˆj kˆ A 6 i 6ˆj k ˆ 7. For zero resulan, sum of an wo forces remaining force. R P Q, R P Q R P 0 P Q P 0 P Q.P 0 R P +Q + P.Q P + Q P Q RQ 4. a c RP and b c RQ bu RP RQ a b c RP RQ a b c 5. cos r W r r 0 N Saring poin 0 / S 0 unis node6\ : \Daa\04\Koa\J-Adanced\SMP\Ph\Soluion\Uni & \0 asic Mahs.p65 R A A cos R 8 and A sin A 8 + A 64 (A ) (A+) 64 A 4 A 0N & 6N c c c 0. c ± ˆ ˆ 4 4 i j kˆ 4 ˆ ˆi ˆj ˆi ˆ j 6i ˆ 8ˆ j Use R A + + Acos or see opions 0. Displacemen m. Required angle

2 J-Phsics. A A A sin A cos an 60 A A A cos 60 A A A A 4. A. Acos Projecion of A A. on A cos A.ˆ 5. P Q R Q R P Q R + P RP cos cos Now P Q R 0 P R Q P + R + PRcos Q cos 6. Resulan cos + ( + ) + ( )cos cos g cc kg kg m m u 7. n u n u n u n M M (n ) 000g 00cm 0g0cm 0. n 0 N 0 Uni of force in new ssem So uni of force in new ssem 0.N O R As [F] M so uni of force (0g) (0 cm( 0.s) (0 kg) (0 m) 00(s) 0. (kg) (m) (s) 0. N 8. mus be dimensionless 9. ensionforce bu surface ensionforce / lengh 4. F M, A A 4. [a], [C] [] [b] [] 8. Projecion on plane 0 9. Veloci of one ball ˆ ˆ i j Veoci of second ball ˆ ˆ i j Angle beween heir pah :. cos 5. e e cos sin. In a clockwise ssem kˆ ˆj ˆi ˆi ˆj kˆ 4. r 0 4 î (6 8) ĵ ( ) + ˆk (4) i ˆ ˆj 4kˆ XRCIS II. A an insan + 5 u m 4m 5m ms - Differeniaing w.r.. ime Here d, d u u 8 + d m/s. + 4 d d bu d d 0 So d d node6\ : \Daa\04\Koa\J-Adanced\SMP\Ph\Soluion\Uni & \0 asic Mahs.p65

3 J-Phsics. I MR R R 8 R cal 4. J cal4. kgm s 4. kg m s di R dr MR dr dr 8 M 5 4 / R (5R 4 ) () () () 4 kg m s 6. Angle beween a and b, cos a.b ab 4 4. nowen G kg kg km 6 0 kg m newon engh a, ime a raio of uni of lengh 9 7. n u n u n u M n u M and raio of uni of ime / M M 8 7. Q 0 R P Q R sin 0 sin 90 sin 50 P / Q R / P 9. [k] [] [ ] [M ] [ ] M P Q R k (consan) Force Area Modulus of elasici P : Q: R k : k : k : : 0.. P + b c / / a V (R+b) V c wae eloci 8. a b c a b c a.b b.c c.a a b c 0, b c a 0 a b b c c a 0 c a b 0 & a b c a b c node6\ : \Daa\04\Koa\J-Adanced\SMP\Ph\Soluion\Uni & \0 asic Mahs.p65 P (R +b) V C a V AV m V n m c and n. A m A m [] M M [ ] laen hea 4. C, G M, h M M hc G 9. a b c a b c (a b b c c a) + + +(cos + cos + cos ) cos + cos + cos 0. â bˆ cos cos

4 J-Phsics â bˆ sin. a a, cos a z a z a cos 5 a 5 z 5 Now a a a 50 4a a 5 50 a 5 a 5 a 5. C A C A + + Acos If C < A + hen cos < 0. herefore > Area of riangle a b b c c a 6. F F F F4 4ˆi 5ˆj 5k ˆ 5ˆi 8ˆj 6k ˆ i ˆ 4ˆj 7k ˆ + i ˆ ˆj kˆ moion will be in z plane 8. r F ˆi ˆj kˆ 7 5 4ˆj kˆ 4i ˆ 8ˆj 6kˆ ˆ ˆ k i j ˆi ˆ j d 0 ( k)d 0 k. Here 45 so inclinaion of AC wih ais is 45. So uni ecor along AC ˆi ˆj cos 45 ˆi sin 45 ˆ j a b. 7a 5b 0. 7a 5b + 6 a b a 4b 7a b 0 and 7a + 8b 0 a b (i)...(ii) adding (i) and (ii) b + 46 a b 0 a b b So 7a 5b + 8b 0 a b abcos b cos cos (/) For riangle AC : A C CA 0 Now A C CA A C CA CA 0 CA CA 9. r a cos i ˆ a sin j ˆ eloci dr a sin i ˆ a cos j ˆ d r Acceleraion a cos ˆi a sin j ˆ r rue /False. If XRCIS III A A 0 hen A. wo ecors are alwas coplanar. 0. A A cos 8, A A sin 8 8 an 60, 0 8. Displacemen dr di ˆ dj ˆ bu + k 5 so d + kd 0 k k ˆ ˆ dr di dj ˆi ˆ j d Work done is zero if F.dr 0 4 Fill in he blanks W F.d 0ˆi ˆj 8kˆ..0i ˆ ˆj 7kˆ 6i ˆ 5ˆj kˆ 0i ˆ ˆj 8kˆ 4ˆi 7ˆj 0kˆ J. Required ecor baˆ 7 4 i ˆ 4ˆj 4 i ˆ 4ˆj 5 5i ˆ 0ˆ j 5 node6\ : \Daa\04\Koa\J-Adanced\SMP\Ph\Soluion\Uni & \0 asic Mahs.p65

5 J-Phsics. a b ˆi ˆj ˆi ˆj kˆ kˆ 0 5. e unknown displacemen be s W N S hen i 5 cos 7 i sin 7 j s 6i s j ˆ 0 0 ˆ ˆ ˆ ˆ a b ˆi ˆj k i ˆ 6. Area ˆ kˆ ˆj 7. According o quesion 8ˆ Ai ˆ Aj ˆ 8ˆ A ˆj ˆi 8 A 5 A 8. According o quesion u u 0 and u u u 0 & u u 4 u cos ab PV M R PV 4. V [R][PV] (M )( ) M [nerg] Comprehension O V m/s m/s. For shores disance d 0 50 sec W. min Comprehension. a, b a + b 0.. dr d r dr ai ˆ bj ˆ a 0, ai ˆ ˆ bj 9. k / k / s/m Comprehension 4 node6\ : \Daa\04\Koa\J-Adanced\SMP\Ph\Soluion\Uni & \0 asic Mahs.p65 0. a b c P d a b c M M M a + b + c 0, a b + c 0, a c a 5 6, b, c Comprehension. [b] [V] a. P [a] [PV V ]. [PV] [R], [Pb] [PV] [R] a PV P R and V V ab PV V PV R V V 5. e uni of lengh, ime and mass be, and M respeciel. According o quesion 9.8 (7.) (448) M 00 M (7.) (448) M 0 M b soling aboe equaion m. soling aboe equaion s. soling aboe equaion M 544. M 544. kg

6 J-Phsics XRCIS IV(A). A 4, A 6 so A + 0 and A + 9 (i) 0 4 6m and 9 6 m (ii) lengh m (iii) an an an 6. (i) e he angle beween A and is, hen (ii) (iii) A cos A 90 Resulan erj A ˆ ˆ ˆ ˆ ˆ i j k i j 0 0 i ˆ ˆj kˆ ˆi ˆj e j e j i j k i j k i j Projecion of resulan on ais Required ecor j A e j j i j k. (i) Componen of A along F HG i j k I KJ A. i ˆ ˆj. ˆj kˆ ˆj kˆ A. ˆ ĵ k Componen of (ii) (iii) A A A ˆ i ˆ ˆj ˆj kˆ 5 Area of he parallelogram A 6 i j k 0 0 i 6 j k b g 7 unis Uni ecor perpendicular o boh A & A n i 6 j k 6 i j k A Componen along he ecor i j (A cos ) (A.) (i 4j).(i j) (i j) ( ) 4 (i j) 7 (i j) Componen along he ecor i j (A cos ) (A.) (i 4j).(i j)(i j) ( ) ( 4) (i j) (i j) 6. e wo forces are A and hen A + P, A Q A P Q Resulan k A A cos, P Q P Q P Q P Q P Q cos P Q (P Q ) cos P Q ( cos ) ( cos ) P cos Q sin 7. A A A cos (0) (6) (0)(6) cos sin 60 6 / an 0 6 cos an 7 8. Resulan force in erical direcion 50 N N 00 N 50cos cos 0 node6\ : \Daa\04\Koa\J-Adanced\SMP\Ph\Soluion\Uni & \0 asic Mahs.p65

7 J-Phsics N Resulan force in horizonal direcion 00sin0 50 sin N so resulan pull (5) (5 ) 7.9N 9. a, b b a. (i) displacemen () (4) (5) 50m Disance r p 6 i 4 ˆj (i) ˆ 7 88 k 4 ˆ 6 ˆi 7 ˆj i 4j 5. F 5Pj, F 4Pi, F 0P (6i 8j)P 5 (( i i) (j 4j)) F4 5P Pi 9Pj 5 F F F F F 5Pj 4Pi 6Pi 8Pj 4 Pi 9P j Pi 4Pj F P ( ) (4) 0P 4 5 an 4 an ( ) (ii) (7) (5) 74m. e c is c ˆ ˆ i c j hen according o quesion c c 5 7. Displacemen (0) (40) 50km 40 an 5 N o 0 w N 40 km 0km s c + c 5 and a.c 0 c + 4 c 0...(i)...(ii) 9. Speed m / s 4. from equaion (i) and (ii) c ± 4, c dr (6 6) i + ( ) j m/s d a (6i 4j) m/s K.. 0. From graph m J 5J (a 0) 50 m/s d d d d node6\ : \Daa\04\Koa\J-Adanced\SMP\Ph\Soluion\Uni & \0 asic Mahs.p65 (i) F ma 6(6i 4j) (6i 44 j)n (ii) r F ( 6)i ( 4 )j 6i 44j [( 44 ) + (44 6 ) + 44 ] k ( ) k (iii) p m 6[(6 6)i ( )j] [6( )i 7 j] 7 a d a m/s d b0. (i) ˆi a b e kˆ 0 (ii) a b e b0 (iii) a a b e kˆ 0 0. Dimension of M Dimension of M b

8 J-Phsics Dimension of 0 Dimension of 0 M 0. (i) c (ii) Q m [ ] Dimension of c 0 ( ) 0 [ M ] [ M ][ M K ] [ K ] 0 0 Dimension of [ M ] [ M ] [ M K ] (iii) R PV M [M K mol ] n molk. Surface ension (S) XRCIS IV() work done nerg Area Area A S ( ). Dimension of joule M Value of joule in sar ssem (0 0 ) (0 8 ) (0 ) 0 0 sar joule 4. e i & OP i j i + d j 4. Dimensions of a M [a] M [] and [ ] [M ] 0 5. m [] k [d] [g] [M 0 0 ] [ ] k [M ] [ ], k + 0, K 0,, and K 6 so OP (i ˆ dj) ˆ i ˆ dkˆ (d is consan) which is independen of posiion. 5. Vecor PP ˆ ˆ 5 i 5k and P P 4ˆi kˆ P(,,0) 6. R a g b [] [ ] a [ ] b a + b, a b 0 a, b R g 7. [b] [] [ ] dimensions of [a] dimensions of RV a RV [M ] M k mol [k] [ ] [M 5 mol ] 8. Y () (a) (F) z [M ] [ ] [ ] [M ] z z, + + z, z z,, 4 Y F a 4 [M] z [] ++z z [] 8 M P(,, ) P(4,,5) e angle beween hese ecors be hen cos ˆ ˆ ˆ ˆ As PM PP sin 5 i 5k 4i k so PM 5 7 m 5 herefore 6. an 7 m.5 s m/s an node6\ : \Daa\04\Koa\J-Adanced\SMP\Ph\Soluion\Uni & \0 asic Mahs.p65

9 J-Phsics 7. Area of riangle (i j+k) A A ˆj k A 5m 4k ˆ ˆ j i ( i j + k ) ( i + j +k ) A i j c 4 k ˆ. ˆ ˆ A, Aˆi ˆj 4c 4 kˆ ˆi ˆj hus, A,, C i ˆ 6 ˆj 4k ˆ A 4, ˆi 96 ˆj 8 4 k ˆ ˆi 94ˆj 4k ˆ. a 5 cos i ˆ sin j ˆ 8. law of reflecion i r 4 4 i ˆ A ˆj ; A 0, 4 ˆ i 4 ˆ j ; C i ˆ ˆj 4 0, C d 5 cos i ˆ sin j ˆ herefore d 0 d 5 cos 5 sin (5sin ) 0 d 5 sin +5 5 cos 5 cos Similarl, / k M A k d A d 0 sin 4 M A d A 8 4 / (cos ) cos d cos 0 M oal M +M 4 k A sin + sin 5 sin ˆi cos ˆj hus, node6\ : \Daa\04\Koa\J-Adanced\SMP\Ph\Soluion\Uni & \0 asic Mahs.p65 0. m k an. dm k sec d dm m k sec k an d dm m d d sin cos sin % error is minimum when sin has maimum alue hence dr.i ˆ.8j ˆ.8 ˆ k A 4s,.ˆi 7.ˆj 8.8kˆ or 45 ˆ ˆ ˆ ˆ ˆ ˆ P F. 60 i 5 j 40k. i 7. j 8.8k 044W s 5 cos i sin ˆj and ˆ ˆ r i ˆj kˆ dr (i) ˆi j ˆ kˆ d (iii) a ˆj () a (iii) speed (i) a ˆi j ˆ kˆ ˆ ˆi j ˆ k a ˆ ˆ ˆ a j ˆ As a + a a N so a N a a ˆi j ˆ kˆ ; a

10 J-Phsics XRCIS V(A). he dimensions of orque and work are[m ]. As we know ha formula of eloci is [ ] Fi ll i n he blanks : XRCIS V() M /. h h M. [X] [ capaciance] [M Q ] and [Z] [Magneic inducion] [M Q ] herefore [Y] X M Q Z M Q M Q 4 4. Planck's consan (in erms of uni) Momenum (p) 4. Newon's formula (h) J-s [M ][] [M ] kg-ms [M][][ ][M ] dimensions of force dimensions of area dimensions of eloci gradien. lecrical conducii J I / A I / A I F / q F / I FA M a V A M A P a PV M M Si ngle Choi ce M 5. A A M his is onl possible if he alue of boh ecors A and A is zero. his occurs when he angle beween A and is. 7. Momen of ineria and momen of a force do no hae same dimensions. 8. Dimensions of inducance, i.e. henr are [M /Q ] 0. M F q MC C qq. F 4 0 r [M ] 0 [ 0 ] [M 4 A ] A [M 4 I ] [M 6 I ] [M ] V 7. 0 [M 4 I ] [] [M I ] [ ] 8. Dimension formula of olzman consan k [M ] [ ] M M [M ]; [M ] [M ] 9. (i) Dipole momen Charge engh [I] [ ] Dipole momen [I ] [ ] [ I ] q (ii) lecric flu (iii) lecric field F q 0 [I ] 4 [M I ] [M I ] [M ] lecric field [M I ] [I ] node6\ : \Daa\04\Koa\J-Adanced\SMP\Ph\Soluion\Uni & \0 asic Mahs.p65

11 J-Phsics MCQ'S q q. F 4 r 4. 0 M A 0 0 M A 4 0 ] M A + 4 F i i 4 r [M ] 0 A 0 0 MA di ol sec e (Hener) Ampere R ime consan; [] ohm sec I weber Ampere (Hener) Joule I (Ampere) 7. Mach he Column GM em ( A ) F R ( ) GM e M s F Work Mere s Coulomb Vol Mere Z(Hener) M Mere (Kg) (Mere) (S) R Kineic energ 8. Mach is I wih is II and selec he correc answer using he codes gien below he liss : is I A ns. ( C ) [II-J 0] is II P. olzmann consan.[m ] Q. Coefficien of iscosi.[m ] R. Planck consan.[m K ] S. hermal conducii 4.[M K ] Codes : P Q R S (A) 4 () 4 (C) 4 (D) 4 (P) olzmann consan nerg M emperaure K (Q) Coefficien of iscosi () : M K F A V, M M (R) Plank consan (h) : (S) M h; [h] hermal conducii Q K A M M [K] K M K R M (Mere) (S) node6\ : \Daa\04\Koa\J-Adanced\SMP\Ph\Soluion\Uni & \0 asic Mahs.p65 ( C ) ( D ) QV nerg F q QV nerg (farad)(ol) M m kg q q (r, s) GMe Work done R Mass (Veloci) (r, s)

Kinematics in two dimensions

Kinematics in two dimensions Lecure 5 Phsics I 9.18.13 Kinemaics in wo dimensions Course websie: hp://facul.uml.edu/andri_danlo/teaching/phsicsi Lecure Capure: hp://echo36.uml.edu/danlo13/phsics1fall.hml 95.141, Fall 13, Lecure 5