300 CREATIVE PHYSICS PROBLEMS PDF

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300 CREATIVE PHYSICS PROBLEMS with Solutions cot&=h'IGc ------ . -- Q(V ) ~t------------------- s sin a. , , LASZLO HOLIeS ANTHEM PR ESS LONDON· NEWYOR.K · DELH I 300 CREATIVE PHYSICS PROBLEMS with Solutions Laszlo Holies ANTHEM PR ESS LONDON· NEWYOR.K · DELH I Anthem Press An imprint of Wim...

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300 CREATIVE PHYSICS PROBLEMS

with Solutions cot&=h'IGc

Q(V ) ~t-------------------

s sin a.

,

,

LASZLO HOLIeS

------ . --

ANTHEM PR ESS LONDON· NEWYOR.K · DELH I

300 CREATIVE PHYSICS PROBLEMS with Solutions

Laszlo Holies

ANTHEM PR ESS LONDON· NEWYOR.K · DELH I

Anthem Press An imprint of Wimbledon Publishing Company

www.anthempress.com This edition first published in UK and USA 20 I 0 by ANTHEM PRESS 75-76 B lackfriars Road , London SE I 8HA, UK or PO Box 9779 , London SWI9 7ZG, UK and 244 Madison Ave. #116, New York, NY 10016, USA Copyright English translation

©

©

Laszl6 Holics 20 I 0

A. Gr6f, A. Salamon, A. Tasnadi , T. Tasnadi , Cs. T6th

Sponsored by Graphisoft Foundation The moral right of the authors has been asserted. All rights reserved. Without limiting the rights under copyright reserved above, no part of thi s publication may be reproduced, stored or introduced into a retrieval system, or transmitted , in any form or by any means (electronic , mechanical , photocopying, recording or otherwise), without the pri or written permission of both the copyright owner and the above publisher of this book .

British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library.

Library of Congress Cataloging in Publication Data A catalog record for this book has been requested. ISBN-13 : 978 I 84331 869 9 (H bk) ISBN-IO: 1843318695 (Hbk)

TABLE OF CONTENTS

How to Use This Book. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

Physical Constants alld Other Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vi

Part I. PROBLEMS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

I . Mechanics Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

I. I

Kinematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2

Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.3

Statics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

1.4

Fluids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

2. T hermodynam ics Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

2. I

Thermal expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

2.2

Ideal gas processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

2.3

Fi rst law of thermodynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

3. E lectrodynamics Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

3. I

Electrostatics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

3.2

Direct current. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

4. Magnetism Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

4. I

Magnetic field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

4.2

Induction (motional eml) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

4.3

Induction (transformer emt) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

4.4

A lternating current. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

5. Optics Proble ms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

III

Part II. SOLUT IONS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

6. Mechanics So luti ons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

6.1

Kinem ati cs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

6.2

Dynami cs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

6.:1

Stat ics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

:12:1

6.4

Fluid s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3:14

7. Thermodynamics So luti ons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

:142

7. 1

Th ermal expa nsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

342

7.2

Ideal gas processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

345

7.:1

Fi rst law of therm odynami cs .. ..... . ......... .. ....... . ... . .

:197

8. Electrodynami cs So luti ons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

429

8. 1

Elec trostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

429

8.2

Direct cu rrent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45 1

9. Mag neti sm So lut ion s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

470

9. 1

IV

Magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

470

9.2

In duc ti on (moti onal em l) ... . .. . .... .... ........ .. . ........ .

477

9.:1

Inducti on (transformer emf) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49:1

9.4

Alte rn ating current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

508

10. Opti cs Sol uti ons.. . . . . . . ... .... . .... . . ... ..... ... . .. . . .. ... . . . .. .

520

How to Use This Book

The bes t way of understandin g the laws of phys ics and learnin g how to so lve ph ys ics problem s is th ro ugh prac ti ce. Thi s book features almost three hundred probl ems and soluti ons worked out in detail. In Pa rt I, Problell1 s are arranged themati call y, starling in Charter I with probl e ms about mechanics, the branch of phys ics co ncerned with the behav iour of phys ica l bodies when subj ec ted to forces or disp lacements, and the subseque nt e nec t of the bodies on their enviro nme nt. Chapter 2 offers problems in thermodynamics, the study o f e nergy conversion between heat and mec hanical work , while the electrodynamics prob lems in Chapter :1 deal with the phenomena assoc iated with mov in g electrica l charges and their interac ti on with el ectric and mag neti c fields. Chapter 4 ' s rroble ms on magnetism seek to understand how materi als respond on the mi croscopi c le vel to an appli ed magneti c field . Lastl y, the optics probl ems in Chapt er 5 address the branch of ph ys ics th at studies the behav iour and phys ica l pro perti es of li ght. While the pro bl ems are arranged by topi c, the pro bl e ms within a sin gle topi c are ofte n arran ged by in creas in g le ve l of diOicull y. Indeed, many of these phys ics problem s are diOicult - ye t we e nco urage stude nts to try and solve the prob lems on their ow n, and to onl y co nsult the SO IUliO/ ls sec ti on in order to compare their ow n alle mpts with the correct results. We e ncourage creativ ity in problem- so lvin g, and these phys ics probl ems are intended as a means of deve lopin g the stude nt' s knowl edge of phys ics by appl yin g them to co nc rete prohlems.

v

Physical Constants and Other Data

Gravitational constant Speed of light (in vacuum) Elementary charge Electron mass Proton mass Neutron mass Charge-to-mass ratio of electron Unified atomic mass constant Boltzmann constant Plank constant A vogadro constant Gas constant Permittivity of free space Permeability of free space Coulomb constant Compton wavelength of electron

G c e

6.673 X 10 - 11 Nm 2 kg- 2 2. 998 x 10 8 ms- 1 1.602 x 10 - 19 C 9. 109 X 10 - 3 1 kg (5 11 .0 keV) 1.673 x 10 - 27 kg (938 .3 MeV) 1.675 x 10 - 27 kg (939.6 MeV) 1.759 x 1011 Ckg - 1 1.661 X 10- 27 kg 1.381 X 10- 23 JK - 1 6.626 X 10 - 34 Js 6.022 X 10 23 mo] - l 8.315 Jmol- 1 K- 1 8.854 x 10- 12 CV-1m- 1 2 411 x 10 - 7 Vs C - 1 m - 1 8.987 x 10 9 VmC - 1 2.426 x 10 - 12 m

Mean radius of the Earth R Sun-Earth distance (Astronomical Unit, AU) Mean density of the Earth p Acceleration due to gravity 9 Mass of the Earth Mass of the Sun I light year

6371 km 1.49 x 10 8 km 5520 kgm - 3 9.807 ms- 2 5.978 x 10 2 4 kg 1.989 X 10 30 kg 9.461 X 10 15 m

Surface tension of water Heat of vaporisation of water Tensile strength of steel

0.073 Nm - 1 2256 kJkg - 1 = 40.6 kJmol 500-2000 MPa

vi

'Y

L

1

Part I PROBLEMS

Chapter 1 Mechanics Problems

1.1 Kinematics Problem 1. A tra in is mo vin g at a speed rails. The tra in whi stles for a time o f T . whi stl e? The speed o f sound is c = 330 m /s; train does not reach the ra ilway ma n until the

o f v to ward s the railway man nex t to the H o w lo ng d oes the railwayman hear the v = 108 km / hour = 30 m is, T = 3 s; the end o f the whi stl e.

Problem 2. The speed o f a moto rboat in still wate r is fo ur times the speed o f a ri ver. Normally , the motorboat takes o ne minute to c ross the ri ver to the port straig ht ac ross on the other bank. One time, du e to a moto r probl e m, it was not able to run at full power, and it took four minutes to cross the ri ver al o ng the same path. By wh at fac tor was the speed of the boat in stili water reduced ? (Assume that the speed of the water is uniform throughout the wh o le width o f the river.) Problem 3. Consider a trough o f a se micircular cross secti on, and an inclined pl ane in it that lead s fro m a po int A to point B ly ing lo we r th an A . Prove that wherever point C is chosen o n the arc AB , an object will always get from A to B faster a lo ng the slopes AC B than alo ng the origin al s lope A B . The c ha nge o f direc ti o n at C does not involve a ch ange in speed . The e ffects of fricti o n are negli g ibl e. Problem 4. The acce le rati o n of a n objec t is unifo rml y Inc reas ing, and it is ao = 2 == 2 m /s at to = 0 sa nd al = 3 m /s 2 at t l = 1 s . The speed o f the obj ect at to = 0 s

is Vo = 1 m /s . a) De termine the speed of the o bject at t2 = 10 s. b) De termin e the v -t fun cti o n o f the mo ti o n, a nd the n plot it in the v - t coordin ate syste m. c) Estimate the di stance covered by the object in the first and last seco nd o f the time interval 0 < t < 10 s .

3

JOO C ,'ca t il'(' P l,ys ics PJ'O b lcJlJ.5 lI' i t l, So l ut io ns

Problem S. A n objec t moves on a circul ar path such th at it s distance covered is given hy the run cti on: 8 = 0 .5 / 2 III + 21m. The rat io o r the mag nitud\.:s o r it s accd erati ons at ti m\.:s I I = 2 s a nd 12 = 5 s is l : 2. Find the radi us or the circle. Prohlem 6. T he rad iu s or the tire o r a car is I?, The va lvt.: ca[l is at di stance /' rrom the ax is o r the wht.:e l. Tht.: car start s rrom rt.:st with out skiddin g, at constan t acce lerati on . Is it [loss il1k. in so me way . th at tht.: valve cap has no acce lerati on 1 turn roll ow in g th t.: hott om [lositi ol1 , a) in the

8

1

b) in th t.: - turn precedin g the bott om [lositi on'? Problem 7. A di sc or di amder 20 cII1 is rollin g at a speed or cl 111 /5 on the groun d, wi thout sli ppi ng. How long does it take until the speed o r point A first beco mes equ al to the present valu e o r the speeu o r po int J3 '! Prohlem X, A di sc o r radiu s R = 1 III roll s unirorml y, without skiduin g on horiw ntal ground , The speed of it s centre is u = 0. 5 m /s . Ld A stand ror the topm os t point at t = 0 a nd [3 ror the mid-point or the correspo ndin g radiu s. a) At what time will the speed o r point A f-i rst eq ua l the spt.:ed or po int [3 ') b) Fo ll ow in g on rro m part a) above, wht.: n the speed o r point A first equ als the speeu of point J] , what is thi s speed? c) Fo ll ow in g o n rrom part a) above , find the di stance travell ed by the centre o r the di sk up to the tim e when the speed o r point A first elJu als the speed o r point B. Prohlem 9. A cart moves on a mudd y road . The radius or its whee ls is R = 0.6 Ill . A small bit o r mud ddac hes rro m the rim at a he ight " = ~ R rrom the ground. a) Find the speed o r the cart ir the bit o r mud rall s back o n the whee l at the same height. b) Finu the kn gth or the arc on the rim th at co nn ec ts the points or detac hin g anu railin g back, c) Find tht.: ui sta nce covt.:red by tht.: car in th t.: mea ntime. Prohlem 10. A ball oon is ri sing vt.: rti ca ll y rrom the groun d in suc h a way th at with hi gh acc uracy it s acce krati on is a lin earl y dec rt.:as in g runctio n o r it s altitude above the ground k vd . At the mo mt.:nt o r re lease th t.: ve loc it y o r the ball oo n is wro, and its accelerati on is ([" .

4

1.2 Dy na.m ics

1. lU ce/JCl ll ies P ro b lem s

a) Determi ne the speed of the ba ll oo n at the height H , where its accelerati on becomes zero. b) What is the speed of the ba ll oo n at half of the altitude H ? c) How long cloes it take the ba ll oo n to reac h the altitu de H ? Prohlem 11. A massive ba ll is fall in g dow n from an initi al height of h = 20 m . With a gun held horizo nt all y, cl = 50 m far from the tra jec tory of the fa llin g ball , at the height of h' = 10 Ill , we are go in g to shoot at the fall ing ball . The bullet leaves the gun at a speed of l' = 100 111 / 5. At what time aft er the start of the fall should the gun be fired in order to hit the fall ing ba ll with th e bu ll et'? (The air res istance is neg li gibl e.) Prohlem 12. Two objects, one sli din g dow n from rest on a smooth (fricti onless) slope, the other be in g th row n fro m the po int 0, start their moti on at the same instant. Both get to the point P at the same ti me ancl at the same speed. Determine the initi al angle of the th row .

o

p

Prohlem 13. A projectil e is projec ted on the leve l ground at an angle of 30° with an initi al speed o f 400 Ill / S . At one point durin g its tra jec tory the project il e exp lodes into two pi eces. The two pieces reac h th e grou nd at the sa me moment ; one of them hit s the ground at exact ly where it was projec ted with a speed of 250 m /s . At what height did the ex pl os ion occur'? (Air drag and the mass of the ex pl osive materi al is negli gibl e, the 2 accel erati on due to grav ity can be considered as 10 Ill / 5 . ) Problem 14. T he bull et of a poac her fl yin g at a speed of v = 680 m/ s passes the gamekeeper at a d istance cl = 1 111. What was the di stance of the bull et fro m the ga mekeeper when he bega n to sense its shri eki ng sound ? The speed of propagati on of sound is c =34 0 Ill /s .

1.2 Dynamics Problem IS. A fr icti onless track co nsists of a hori zo nt al pa rt of un know n length, wh ich connec ts to a verti ca l semi circle of rad ius I' as shown. An objec t, whi ch is given an initia l ve locity v , is to move along the track in such a way that after leav in g th e sem icircle at the top it is to fa ll back to its initi al pos it ion. What shoul d the minimum length of the hor izo nta l part be '?

,,

r - -

-- - - ---

v " --........-

5

300 Cr eative Physics Problems with Solution s

Problem 16. A pointlike object of mass m starts from point J( in the figure . It slides along the full length of the smooth track of radius R, and then moves freely and travels to point C. 0 d C a) Determine the vertical initial velocity of the pointlike object. b) What is the minimum possible distance O C = d, necessary for the object to slide along Vo the entire length of the trac k? c) Find the normal forces exerted by the track at points A and B. 2 R=lm , h=2m, d=3m, m =0.5kg , use g= 10m/s ) B

,, :R , A t----'--'R_~_ - - - - - - - - - - - - - - - - - - - - - - - - - -.() I

,, :h , K

It m

(Let

Problem 17. A small object starts with a speed of Va = 20 m/s at the lowest point of a circular track of radius R = 8. 16 m. The small object moves along the track. Ho w big a part of the circular track can be removed, if you want to carry out the same trick ? (Neglect friction, 9 = 9 .8 m /s2 .) Problem 18. A small o bject of mass m = 0 .5 kg that hangs on a string of length L = 5.6 m is given a horizontal velocity of Va = 14 m /s . The string can withstand a maximum tension of 40 N without breaking . Where is the stone when the string breaks? 2

Useg=10m/ s . Problem 19. An object slips down the frictionless surface of a cylinder of radius R. a) Find the position in which the acceleration of the object is two thirds of the gravitational acceleration G. b) Find the direction of the obj ect's acceleration in that position.

Problem 20. Two horizontal tracks are connected through two circular slopes the radii of which are equal and R = 5 m. The tracks and the slopes are in a vertical plane and they join without a break or sharp corner. The height difference between the horizontal tracks is h = 2 m. An object moves from the track at the top onto the bottom one without friction. What is the maximum initial speed of the object when it starts, in order for it to touch the path at all times during its motion ? R

1"" - - - - - - - - - - - - - - -

Problem 21. A small object is moving on a

,

special slope consisting of a concave and a convex circular arc, both of which have a right angle at the centre and radius R = 0.5 m, and they join smoothly , with horizontal common tangent, as it is shown in the figure. Determine the distance covered by the object o n the slope, provided that it started from rest and it detaches from the slope at the altitude

6

~R . 4

(The friction is negligibly small.)

~R 4 R

1. lVl ech anics Problem s

1. 2 Dy na mics

Problem 22. A pe nd ulum , whose cord ma kes a n an g le 45 ° w ith th e vertical is released. Where w ill th e bob reac h its minimum ac ce le rati o n? Problem 23. ...