Physics Numericals. Subtopic wise numericals with solutions. Shortcuts to enable quick problem solving.

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1 Physics Numeicals SD. XII Sci. Salient Featues Subtopic wise numeicals with solutions. Shotcuts to enable quick poblem soling. Pactice poblems fo eey subtopic. Includes soled boad numeicals. Numeical based multiple choice questions fo effectie pepaation. Solutions/hints to pactice poblems and multiple choice questions aailable in downloadable PDF fomat at Pinted at: Repo Knowledgecast Ltd., Mumbai aget Publications Pt. Ltd. No pat of this book may be epoduced o tansmitted in any fom o by any means, C.D. ROM/Audio Video Cassettes o electonic, mechanical including photocopying; ecoding o by any infomation stoage and etieal system without pemission in witing fom the Publishe. P.O. No _JUP

2 Peface In the case of good books, the point is not how many you can get though, but athe how many can get though to you. SD XII Sci.: PHYSICS NUMERICALS is a complete and thoough guide to the numeical aspect of the HSC pepaation. he book is pepaed as pe the Mahaashta State Boad syllabus.subtopic wise segegation of Soled Numeicals in each chapte help the student to gain knowledge of the boad spectum of poblems in each subtopic Fomulae which fom a ital pat of poblem-soling ae poided in eey chapte. Solutions and calculations hae been boken down to the simplest fom possible (with log calculation poided wheee needed) so that the student can tackle each and eey poblem with ease. Poblems fo pactice ae poided to test the igilance and aletness of the students and build thei confidence. Boad Numeicals till the latest yea hae been poided to help the student get accustomed to the diffeent standads of boad numeicals. Numeical based multiple choice questions ae coeed sub-topic-wise to pepae the student on a competitie leel. Solution/hints to pactice poblems and multiple choice questions can be downloaded in PDF fomat fom ou website www. tagetpublications.og he jouney to ceate a complete book is stewn with tiumphs, failues and nea misses. If you think we e nealy missed something o want to applaud us fo ou tiumphs, we d loe to hea fom you. Please wite to us on : mail@tagetpublications.og A book affects etenity; one can nee tell whee its influence stops. Yous faithfully Authos Edition: Second Best of luck to all the aspiants! Disclaime his book is intended to be study mateial expessing iews and elaboating concepts fo ease of undestanding fo students and puely fo thei benefits. We make no epesentations as to accuacy, completeness, coectness, suitability, o alidity of any infomation though this study mateial. And, shall not be held liable o esponsible fo any eos, omissions, o diffeences in this infomation o any losses, injuies o damages aising fom its use. All infomation is poided on as it is basis depending upon the undestanding of the autho and his/he elaboation of such concepts along with adoption and inspiation fom aious othe texts in elation to the topics as mentioned in this study mateial. It is the eade s esponsibility to eify thei own facts. hough this study mateial we ae only explaining and elaboating aious concepts as may be necessay fo the students in the pesent famewok and context. he iews and opinions expessed in this study mateial ae puely as pe the undestanding of the authos and do not necessaily eflect the official policy o position of any othe agency, oganization, employe o company. Assumptions made in this analysis ae not eflectie of the position of any othe than the authos - and since we ae citically thinking human beings with pesonified opinions, these iews ae always subject to change, eision and ethinking at any time. Please do not hold us to them in pepetuity. Reades shall not misconceie this wok with any othe wok. his wok is puely inspied upon the couse wok as suggested and pescibed by the Mahaashta State Boad of Seconday and Highe Seconday Education, Pune. All the facts and figues so stated hae been puely adopted fom aious eseach points puely fo the pupose of epesentation and explaining the students and eades at lage as a pat of fai dealing. By poducing anything and eeything in this book the autho does not intend to claim copyights on any such mateial but has been puely adopted and used fo the pupose of epesentation and fo bette undestanding of the students with a pue intention to educate the public at lage fo a bette India.

3 01 Cicula Motion Chapte 01: Cicula Motion Fomulae Section 1: Angula Displacement, Relation Between Linea Velocity and Angula Velocity 1. Angula elocity: i. = whee, = linea elocity = adius of the cicle along which paticle pefoms cicula motion. ii. = t whee, = angula displacement of the paticle in cicula motion duing time inteal t. iii. = n whee, n = fequency of eolution of paticle in cicula motion. i. = whee, = peiod of eolution of paticle pefoming cicula motion.. Angula displacement: = t 3. ime peiod: i. = ii. 4. Fequency of eolution: n = 1 = 5. Linea elocity: i. = ii. = n Section : Angula Acceleation 1. Angula acceleation: i. = t whee, = change in the angula elocity of a paticle in cicula motion duing a time inteal t. ii. n = t whee, n = change in fequency of the paticle in cicula motion duing a time inteal t. Section 3: Centipetal and angential Acceleation 1. Centipetal (o adial) acceleation: a = = =. angential acceleation: a = 3. Resultant o total acceleation: a = at a ata cos whee, = angle made by a with a t a = a a when = Fo U.C.M.: a = a = = a t = 0 t Section 4: Centipetal and Centifugal Foces 1. Centipetal foce: i. F c = ii. F c = iii. F c = m i. F c = 4 mn. 4 m F c = whee, m = mass of paticle pefoming cicula motion Section 5: Motion of a Vehicle along a Cued Unbanked Road 1. he necessay centipetal foce: F c = mg = whee, m = mass of ehicle = elocity of the ehicle = adius of the cue oad 1

4 Std. XII Sci.: Physics Numeicals = coefficient of fiction between the tyes of the ehicle and the suface of the oad.. he maximum elocity with which a ehicle can take a tun safely without skidding: = g 3. he maximum angula elocity with which a ehicle can take a tun safely without skidding: g = Section 6: Banking of Roads Fo motion of ehicles along a banked cue oad: 1. he pope elocity o optimum elocity: = g tan whee, = angle of banking. he maximum elocity without skidding: s tan max = g 1 s tan whee, s = coefficient of fiction between the tyes of the ehicle and suface of the oad 3. Angle of banking: = tan 1 o tan = g g 4. Height of inclined oad: h = d sin whee, d = distance between the two font o ea wheels. 5. he maximum elocity with which a ehicle can go on a banked cued oad without toppling: d = g H whee, H = height of cente of gaity (C.G.) of the ehicle fom the oad. Section 7: Conical Pendulum 1. Linea speed of bob: = g tan. Angula elocity: = g l cos 3. Peiodic time: = l cos = = g gtan = h g whee, l = length of conical pendulum h = the height of the fixed suppot fom the cente of the cicle o axial height of the cone = semietical angle of the cone. 4. ension in the sting: mg = cos Section 8: Vetical Cicula Motion 1. Velocity at any point in etical cicula motion: i. P = L g (1 cos) ii. L = 5g iii. H = g i. M = 3g whee, P = elocity of the paticle at any point P along the cicle. L = Minimum elocity at the lowest point on the cicle so that it can safely tael along the etical cicle (looping the loop). H = Minimum elocity of the paticle at the highest point on the cicle so that the sting will not be slackened. M = Minimum elocity of the paticle at a mid-way point so that it can tael along the cicle. = adius of the etical cicle. = angle between the position ectos at the gien position of paticle and that of the lowest point on the etical cicle.

5 Chapte 01: Cicula Motion. Relation between elocities at diffeent points in etical cicula motion: i. L = H + 4g ii. = + g M H 3. ension at: i. Any point P, P = + mg cos ii. Lowest point L, L = L + mg = H + 5mg iii. Highest point H, H = H mg = i. Midway point M, M = M = L 5 mg L mg 4. otal enegy at any point: E = 1 + mg (1 cos ) = 5 mg Section 9: Kinematical Equations Analogy between tanslatoy motion and cicula motion No. anslatoy Motion Cicula Motion 1. Linea elocity = d dt Angula elocity = d dt. Linea acceleation d d a= dt dt Angula acceleation d d = dt dt 3. Linea momentum p = m Angula momentum L = I 4. Linea impulse = F (t) = p Angula impulse = (t) = L 5. Foce F = m a oque = I 6. Wok W = F. Wok W =. 7. Kinetic enegy of tanslation E t = 1 Kinetic enegy of otation E = 1 I 8. Equations of linea motion Equations of otational motion i. = u + at i. = 1 + t ii. s = ut + 1 at ii. = 1 t + 1 t iii. u = as iii. 1 = i. s n = u + a (n 1) i. n = 1 + (n 1) 3

6 Std. XII Sci.: Physics Numeicals Shotcuts 1. Angula acceleation (): If numbe of otations o eolutions is gien, then = (n n 1 ) ad/s t. If linea elocity is gien, apply a = 4 and if angula elocity is gien, apply a = to find a. 3. If numbe of eolutions in a paticula time is gien, apply a = 4 n. 4. If some mass is placed on a otating body, then angula motion changes due to fictional foce between the mass and the body. Coefficient of fiction can be calculated by applying the fomula, = mg o = = g = 4 n g g 5. Beaking tension is the maximum centipetal foce which is gien by the elation, F = = m 6. Speed of a ehicle on a banked oad o cicula tun depends upon the cuatue of the oad. tan = g 7. o aoid skidding, g whee is the coefficient of fiction between the tyes and oad. d = h whee d is the distance between two wheels. 8. In case of conical pendulum if l be the length and be the adius of the hoizontal cicle then height of the igid point of suspension is calculated by the fomula, h = l 9. Use the expessions, L = 5g and = g fo the following cases: i. Bucket full of wate whiled in a etical cicle. ii. Moto cyclist iding in a etical cicle in hollow sphee. 10. Fo looping the loop of adius, the minimum height fom which the body should be eleased is gien by, h = i. Fo a ehicle moing oe a conex bidge which is in the shape of an ac of a cicle, N = mg ii. Fo the moto cyclist at the uppe most point in the globe of death in a cicus, N = mg whee, N = nomal eaction acting on the ehicle o the motocycle. Soled Examples Section 1: Angula Displacement, Relation Between Linea Velocity and Angula Velocity Example 1.1 A wheel of adius mete is making 60 eolutions pe minute. Calculate the angula elocity of any point on the im. Gien: n = 60.p.m. = 60/60 = 1 ps, = m o find: Angula elocity () Fomula: = n Calculation: Fom fomula, = 1 = ad/s Ans: he angula elocity of any point on the im of the wheel is ad/s. Example 1. Find the angula displacement of a paticle moing on a cicle with angula elocity (/3) ad/s in 15 s. Gien: t = 15 s, = (/3) ad/s o find: Angula displacement () Fomula: = t Calculation: Fom fomula, = 15 3 = 10 π ad Ans: he angula displacement of the paticle is 10 ad.

7 Chapte 01: Cicula Motion Example 1.3 What is the angula speed of the second hand of a clock? If the second hand is 10 cm long, then find the linea speed of its tip. Gien: = 10 cm = 10 1 m, = 60 s o find: i. Angula speed () ii. Linea speed () Fomulae: i. = ii. = Calculation: Fom fomula (i), = 3. = 6014 = ad/s Fom fomula (ii), = = = m/s. Ans: i. he angula speed of the second hand of a clock is ad/s. ii. he linea speed of the tip of the second hand is m/s. Example 1.4 If a body otates in a hoizontal cicle of adius 15 cm with an angula elocity of 0.8 ad/s, then what is its linea elocity? Gien: = 15 cm = 0.15 m, = 0.8 ad/s o find: Linea elocity () Fomula: = Calculation: Fom fomula, = = 0.1 m/s. Ans: he linea elocity of the otating body is 0.1 m/s. Example 1.5 he linea elocity of a body is 0. m/s. If it otates in a hoizontal cicle haing adius 0.5 m, what is its angula elocity? Gien: = 0.5 m, = 0. m/s o find: Angula elocity () Fomula: = Calculation: Fom fomula, = = = 0.4 ad/s Ans: he angula elocity of the otating body is 0.4 ad/s. Example 1.6 A paticle is eoling in a cicle of adius 10 cm with linea speed of 0 m/s. Find (i) its peiod of eolution and (ii) fequency. Gien: = 10 cm = m, = 0 m/s o find: i. Peiod of eolution () ii. Fomulae: i. = Fequency (n) ii. n = 1 Calculation: Fom fomula (i), = 0 = = s Fom fomula (ii), n = 1 = n = s 1 Ans: i. he peiod of eoling paticle is s ii. he fequency of the paticle is n = s 1. Example 1.7 What is the angula elocity of the minute hand of a clock? If the minute hand is 5 cm long, what is the linea elocity of its tip? [Oct 04] Gien: R = 5 cm = 5 10 m, = 1h = 3600 s o find: i. Angula elocity () ii. Linea elocity of the tip () Fomulae: i. = ii. = Calculation: Fom fomula (i), = 3.14 = 3600 = ad/s Fom fomula (ii), = = m/s Ans: Fo the minute hand of the clock, i. he angula elocity is ad/s. ii. he linea elocity of the tip is m/s. 5

Written as per the revised syllabus prescribed by the Maharashtra State Board of Secondary and Higher Secondary Education, Pune.

Written as per the revised syllabus prescribed by the Maharashtra State Board of Secondary and Higher Secondary Education, Pune. Witten as pe e evised syllabus pescibed by e Mahaashta State oad of Seconday and Highe Seconday Education, Pune. Pecise Physics I SD. XII Sci. Salient Featues Concise coveage of syllabus in Question nswe