Intended as supplementary material for undergraduate physics students, this wide-ranging collection of problems in applied mathematics and physics features complete solutions. The problems were specially chosen for the inventiveness and resourcefulness their solutions demand, and they offer students the opportunity to apply their general knowledge to specific areas.
Numerous problems, many of them illustrated with figures, cover a diverse array of fields: kinematics; the dynamics of motion in a straight line; statics; work, power, and energy; the dynamics of motion in a circle; and the universal theory of gravitation. Additional topics include oscillation, waves, and sound; the mechanics of liquids and gases; heat and capillary phenomena; electricity; and optics.
About the Author
M. P. Shaskol'skaya and I. A. El'tsin were on the faculty of Lomonosov State University in Moscow.
Read an Excerpt
Selected Problems in Physics with Answers
By M. P. Shaskol'skaya, I. A. El'tsin, W. J. F. Reynolds
Dover Publications, Inc.Copyright © 2013 Dover Publications, Inc.
All rights reserved.
1. Two passengers with stop-watches decide to measure the speed of a train: one by the click of the wheels passing over the junctions between rails (knowing that the length of each rail is 10 m), the other by the number of telegraph-poles passing the window (knowing that the distance between them is 50 m). The first passenger starts his stop-watch on the first click and stops it on the 156th. He finds that 3 min have passed. The second passenger starts his stop-watch when the first telegraph-pole appears in the window and stops it when the 32nd pole appears. He too finds 3 min have passed. The first passenger calculates that the train's speed is 31.2 km/hr, and the second that it is 32 km/hr. Which of them has made a mistake and how? What is the real speed of the train?
2. The journey from port A to port B lasts exactly 12 days. Every midday a steamer sets out from A for B and another from B for A. How many steamers does each boat meet in the open sea?
3. What exposure should be given to a photograph of a car moving at a speed of 36 km/hr (Fig. 1), so that the image on the negative should not be blurred—that for this the image should not move more than 0·1 mm? The length of the car is 3 m, and the resulting image is 1·5 cm long.
4. A car travels a distance from A to B at a speed of 40 km/hr (v1) and returns at a speed of 30 km/hr (v2). What is its average speed for the whole journey?
5. A boy is throwing balls into the air, throwing one whenever the previous one is at its highest point. How high do the balls rise if he throws twice a second?
6. Two stones fall down a shaft, the second one beginning its fall 1 sec after the first. Find the second stone's motion in relation to that of the first. Ignore air-resistance.
7. Two planes are flying at the same speed of 200 m/sec in opposite directions. A machine-gun mounted in one plane fires at the other at right angles to their line of flight (Fig. 2). How far apart will the bullet-holes made in the side of the second plane be, if the machine-gun fires 900 rounds per minute? What role does air-resistance play in this?
8. A billiards-ball is at point A on a billiards-table whose dimensions are given in Fig. 3. At what angle should the ball be struck so that it should rebound from two cushions and go into pocket B? Assume that in striking the cushion, the ball's direction of motion changes according to the law of reflection of light from a mirror, i.e. the angle of reflection equals the angle of incidence.
9. You are given three billiards-tables of different lengths and the same width. Balls are struck simultaneously from the edge of one of the long sides of each table (Fig. 4) with velocities which are equal in direction and magnitude. Is it possible that these balls should not return to the side from which they started at exactly the same moment?
10. A bucket is left out in the rain. Will the speed at which the bucket is filled with water be altered if a wind starts to blow?
11. A tube is mounted on a trolley which moves uniformly in a horizontal plane (Fig. 5). At what angle to the horizontal should the tube be inclined so that a drop of rain, falling perpendicularly, should reach the bottom of the tube without touching its sides? The raindrop's rate of fall, v1 is 60 m/sec (which does not alter, thanks to the effect of air-resistance). The speed of the trolley, v2 is 20 m/sec.
12. To find the speed of a river's current, a boatman decides to carry out the following experiment. He lowers a wooden bucket into the water and himself sets off downstream, rowing. After 40 min he reaches a point A, 1 km from his starting-point and turns back. He picks up the bucket, turns round again and, rowing downstream once more, reaches A for the second time 24 min later. What is the speed of the current, assuming that the speeds of both current and boat are constant, and also that no time is wasted on turning round? How long does the oarsman spend on rowing upstream to meet the bucket? What is the boat's speed relative to the water?
13. Why is it that when a car is moving forward on a cinema screen, the wheels often appear to be turning backwards?
14. If a disk with one or more holes pierced in it is placed in front of a beam of light which lights up drops of water falling one after the other and the disk is then rotated, the beam will light up the drops intermittently. The number of flashes will depend on the speed of rotation of the disk and on the number of holes in it. This method of illumination is called stroboscopic; it permits periodic phenomena to be observed which are taking place at such a speed that it is impossible to observe them with ordinary lighting. If the number of revolutions of the stroboscope is so chosen that in the time between two flashes the drops have time to move a distance equal to the distance between successive drops, then the drops will appear to be stationary. Find the number of revolutions of the disk necessary for this, if the disk has two holes, if the distance between the drops, s = 2 cm, and the height from which the drops fall is h = 22·5 cm.
15. A disk with holes pierced in it at distances of 1 cm along the circumferences of concentric circles (Fig. 6), is lit from behind by a lamp. The disk rotates with a speed of 30 rev/min. At what distance from the centre of the disk shall we see a continuous circle of light? The human eye does not distinguish between alternating periods of light and dark, if their frequency is greater than 16 to the second.
16. A hoop of radius R rolls without slipping along a horizontal plane with constant speed v. What is the acceleration of different points on the hoop's circumference?
17. A man holds one end of a plank, while the other end rests on a drum (Fig. 7). The plank is horizontal. Then the man moves the plank forward, making the drum roll without slipping along a horizontal plane; no slipping takes place either between plank and drum. How far must the man move before reaching the drum, if the plank is of length l?
18. A hoop is thrown on to a rough horizontal plane with a linear speed v. At the same time the hoop is given a rotatory movement in a direction such that the hoop will roll in the direction of the throw (Fig. 8). What angular velocity, ω?, will make the hoop roll along the plane without slipping, if the hoop's radius is R?
19. When a wheel is in motion, the upper spokes often seem to merge, while the lower ones are distinct. Why is this?
20. At what speed should an aeroplane fly horizontally on the latitude of Leningrad (60°) so that the pilot should be able to see the sun always in the south? The radius of the earth is 6300 km.
21. Two men decide to fight a duel with revolvers in unusual circumstances: they are to fire while standing on a roundabout of radius R, which is turning with an angular velocity of ω. The first duellist stands at the centre O of the roundabout, the second at its edge. How should they each aim so as to hit his opponent? Which is in the more favourable position? Assume that the first duellist's bullet is fired from O at a velocity v.CHAPTER 2
The Dynamics of Motion in a Straight Line
22. A bomb is dropped from an aeroplane flying horizontally at a constant speed. Where will the aeroplane be when the bomb hits the ground?
23. The barrel of a gun and the centre of a target, hung from a thread, are in a horizontal straight line (Fig. 9). Will the bullet hit the target if the thread breaks and the target begins to fall freely at the moment the bullet leaves the muzzle? Assume that there is no air-resistance.
24. Which raindrops fall faster, big ones or little ones? Why?
25. Two spheres of the same radius and the same material fall through the air from the same height; one sphere is solid, the other is hollow. Which will fall faster?
26. A tube in the shape of a rhombus with rounded corners is placed in a vertical plane as shown in Fig. 10. A ball is allowed to roll inside the tube along sides AB and BC, and then allowed to roll along sides AD and DC. In which case will it roll faster? The length of the rhombus's side is A.
27. A load of mass m begins to slip without friction down the inclined face of a wedge lying on a horizontal plane surface; there is no friction either between wedge and plane. The mass of the wedge is M, the angle of inclination of the wedge's top surface with the horizontal is α. Find the acceleration of the load and of the wedge relative to the plane, the force exerted by the load on the wedge and by the wedge on the plane.
28. In Fig. 11 is shown a thin ring of radius R. Equal forces are acting at points A, B, C, D, which are the vertices of an inscribed square, in the directions shown in the diagram. Two equal forces also act at points A and B, along the line of the diagonals of the square. The forces acting along the sides of the square are each of 1 kg, and those acting along the diagonals each equal [square root of 2]kg. Find the resultant of all the forces and its point of application. How will the ring move under the action of the forces given?
29. A scale-pan is attached to a spiral spring, whose extension is subject to Hooke's law; in the scale-pan is a weight (Fig. 12). With what force should the scale-pan be pulled downwards so that when it is released there should be a moment at which the weight ceases to exert pressure on the scale-pan.
30. Two laminas of mass m1 and m2 are joined by a spring (Fig. 13). With what force should the upper lamina be pressed downwards so that when the force is removed the upper lamina should spring back and raise the lower lamina a little too? The coefficient of elasticity of the spring is k. Assume that Hooke's law is applicable throughout. Ignore the mass of the spring.
31. A cyclist moves with uniform velocity down an inclined plane. What is the size and direction of the plane's reaction?
32. A plank inclined at an angle of a to the horizontal lies on two supports A and B (Fig. 14), over which it can slip without friction under the action of its own weight Mg. With what acceleration and in what direction should a man of mass m move along the plank so that it should not slip?
33. A fly is sitting at the bottom of a test-tube. The test-tube falls freely, maintaining its vertical position (Fig. 15). How will the duration of the test-tube's fall be affected if the fly, during the test- tube's fall, flies up from the bottom of the test-tube to the top?
34. A bird is enclosed in a box standing on one pan of a pair of scales. While the bird is sitting on the bottom of the box, the scales are balanced by weights in the other scale-pan. What will happen to the scales if the bird takes off and hovers inside the box?
35. A balloon descends with constant velocity v. What amount of ballast must be jettisoned from the balloon so that it should rise with the same velocity v? The air-resistance is proportional to the velocity. The weight and carrying capacity of the balloon are known.
36. A bullet travels vertically upwards, reaches its highest point and falls back vertically downwards. At what points of its trajectory does the bullet's acceleration have its maximum and its minimum value? Take into account air- resistance, which increases in proportion to the increase of the bullet's velocity.
37. A spring is put into a large tube and occupies the tube's full length when not subject to outside forces. A sphere is placed on top of the spring and compresses it to approximately half its previous length (Fig. 16). Then the tube begins to fall in an inclined position. What will happen to the sphere?
38. A balance is mounted on a stationary trolley, with a weight suspended from one end, while the other end is linked to the floor of the trolley by a spring (Fig. 17). If the trolley be accelerated in a horizontal direction by a constant force, the weight will be inclined at an angle in the direction opposite to the line of acceleration. Will this alter the tension of the spring?
39. A piston is fixed in the cylindrical part of a vessel containing compressed air. The volume of the cylindrical part is small by comparison with that of the whole vessel (Fig. 18). If the piston be released from the forces which hold it in place, it will be pushed downwards out of the vessel (there is no friction between the sides of the cylinder and the piston). How will the time taken for the piston to move down the cylindrical part be affected if: (1) a small sphere be placed on top of the piston? (2) the weight of the piston be increased by an amount equal to the weight of the small sphere?
40. Two boys A and B attach a dynamometer by a ring to a nail driven into a wall and fasten a cord to the dynamometer's hook; they then take turns to pull the cord to see which of them is stronger. When A pulls, the dynamometer registers 42 kg, and when B pulls it registers 35 kg. What will it register if the boys take it down from the nail and take hold, one of the cord and the other of the hook, and then pull in opposite directions (Fig. 19)? (In neither of the cases given do their feet slip on the ground.)
41. In the film 'Brave People' the hero of the film jumps from a train moving along a level track on to the buffer-mounting and uncouples the last two carriages. In what cases is this possible?
42. Two weights of mass m1 and m2 are joined by a non-elastic cord passing over a fixed pulley (Fig. 20). Find the acceleration of the loads, the tension in the cord and the force exerted on the pulley's axle. Ignore the mass of the pulley.
43. Through the middle of a rod of length 2l = 2m, passes a horizontal axle O, about which the rod can rotate. To the ends of the rod are fixed loads M1 and M2 of mass 1 and 7 kg respectively (Fig. 21). The rod is brought into a horizontal position and then smoothly released. What force will it exert on the axle at the instant after release? Neglect the mass of the rod and the friction in the axle.
44. A chain is lying on an absolutely smooth table, half of it hanging over the edge of the table (Fig. 22a). How will the time it takes to slip off the table be affected if two weights of equal mass be attached, one to each end (Fig. 22b)?
45. A rope passes over a weightless pulley A, with a load M1 attached to one end and to the other a weightless pulley B, carrying loads M2 and M3 on the ends of its rope. The whole system is hung, by pulley A, from a spring balance (Fig. 23). Find the acceleration a of load M1 and the reading T of the spring balance, taking M2 ≠ M3, M1 > M2 + M3.
46. A homogeneous chain of length l and mass m hangs partly from a table and is held in equilibrium by the force of friction. Find the coefficient of static friction if it be known that the greatest length of the chain that can be hanging from the table without the whole chain slipping is l1.
47. If a locomotive cannot move a heavy train from rest, the driver acts as follows: he puts the locomotive into reverse and then, having pushed the train back a little, switches into forward gear. Explain why this procedure allows the train to be moved forward.
48. According to Newton's law only an outside force impressed by another body can alter the state of motion of a given body. Then what outside force brings a car or any other self-moving vehicle to a stop under braking?
49. A long-handled broom lies horizontally on the forefingers of a pair of hands held wide apart (Fig. 24). What will happen if the left hand remains still and we move the right hand towards page the left, keeping it constantly at the same level? What will happen if the right hand remains still and we move the left hand towards it? What will happen if we move both hands towards one another at the same time?
Excerpted from Selected Problems in Physics with Answers by M. P. Shaskol'skaya, I. A. El'tsin, W. J. F. Reynolds. Copyright © 2013 Dover Publications, Inc.. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.
Table of Contents
Translation Editor's Foreword
II. The Dynamics of Motion in a Straight Line
IV. Work; Power; Energy: The Law of Conservation of Momentum; The Law of Conservation of Energy
V. The Dynamics of Motion in a Circle
VI. The Universal Theory of Gravitation
VII. Oscillation: Waves: Sound
VIII. The Mechanics of Liquids and Gases
IX. Heat and Capillary Phenomena