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Dynamics - LUJ/us along the mat along along the lie force...

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Unformatted text preview: LUJ/us along the mat along along the lie? force tai accei- this ’02 /r here are :menturn .0. of length ai circle, hat is the must he 'orces on ere is no irections (3.52) (3.53) E, which Exercise ig con~ ;. 3.11). at 3.2. 3.3. 3.4. 3.5. 3.6. 3.7. 3.8. 3.6 Problems Double Atwood’s machine Me A doubie Atwood‘s machine is shown in Fig. 3.12, with masses mt. mg, and m3. Find the accelerations of the masses. Infinite Atwood’s machine 1H”: Consider the infinite Atwood’s machine shown in Fig. 3.13. A string passes over each pulley, with one end attached to a mass and the other end attached to another pulley. All the masses are equal to m, and all the pulleys and strings are massiess. The masses are held fixed and then simnltaneously released. What is the acceleration of the top mass? (You may define this infinite system as follows. Consider it to be made of N pulleys, with a nonzero rnass replacing what would have been the (N + i)th pulley. Then take the limit as N wt» 00.) Line of pulleys a- N +2 equal masses hang from a system of pulleys, as shown in Fig. 3.14. What are the accelerations of all the masses? Ring of pulieys *7». Consider the system of pulleys shown in Fig. 3.15. The string (which is a loop with no ends) hangs over N fixed puileys that circle around the underside of a ring. N masses, m1, m2, , mN, are attached to N pulleys that hang on the string. What are the accelerations of all the masses? Sliding down a plane aa- (a) A block starts at rest and slides down a frictionless plane inclined at an angle 9. What should 6 he so that the block travels a given horizontal distance in the minimum amount of time? (in) Same question, but now let there be a coefficient of kinetic friction ,u. between the block and the plane. Sliding sideways on a plane am: A hlock is placed on a plane inclined at an angle 6. The coefficient of friction between the block and the plane is it 2 tan 6. The block is given a kick so that it initiaiiy moves with speed V horizontaliy along the plane (that is, in the direction perpendicular to the direction pointing straight down the plane). What is the speed of the block after a very Eong time? Moving plane am: A block of mass m is held motionless on a frictionless plane of mass M and angie of inclination 6 (see Fig. 3.16). The plane rests on a fric— tionless horizontal surface. The block is released. What is the horizontal acceleration of the plane? 71 m2 1113 Fig. 3.12 Fig. 3.13 Fig. 3.14 Fig. 3.16 72 Using F=ma Section 3.3: Solving differential equations i; 3.9. Exponential force a A particle of mass m is subject toa force Fm Le magi“ . The initial position and speed are zero. Find x(t).‘ ‘ \p 3.10. mkx force m A particle of mass m is subject to a force F (x) = ~loc, with k > G. The initial position is x0, and the initial speed is zero. Find x(r). -< {23.11. Failing chain in: A chain with length 8 is held stretched out on a frictionless horizontal table, with a length yo hanging down through a hole in the table. The chain is released. As a function of time, find the length that hangs down through the hole (don’t bother with t after the chain loses contact with the table). Also, find the speed of the chain right when it loses contact with the table.19 J 3. i2. Throwing a beach ball mix A beach ball is thrown upward with initial speed 110. Assume that the drag force from the air is Fd : ~mom. What is the speed of the ball, 'Uf, right before it hits the ground? (An implicit equation is sufficient.) Does the ball spend more time or less time in the air than it would if it were thrown in vacuum? 313. Balancing a pencil ass: Consider a pencil that stands upright on its tip and then falls over. Let’s idealize the pencil as a mass m sitting at the end of a massless rod of length 8.29 (a) Assume that the pencil makes an initial (smali) angle 60 with the vertical, and that its initial angular speed is we. The angle will eventually become large, but while it is small (so that sin 6 8 9), what is 6 as a fiinction of time? (b) You might thinic that it should be possible (theoretically, at least) to make the pencil balance for an arbitrarily longtime, by mak— ing the initial 90 and we sufficiently small. However, it turns out that due to Heisenberg’s uncertainty principle (which puts a con straint on how well we can know the position and momentum of ‘9 Assume that the hole is actually a short frictionless tube bent into a gradual right angle, so that the chain’s horizontat momentum doesn’t cause it to overshoot the hole. For a description of what happens in a similar prohiem when this constraint is removed, see Cation (£989). 20 It actually involves only a trivial modification to do the problem correctly using the moment of inertia and the torque. But the point-mass version is quite sufficient for the present purposes, 3.7 Exercises dotted circle. You are required to do this in such a way that the string remains in contact with the pole at ail times. (You will have to move tbounces ; inclined your hand around the pole, of course.) What is the speed of the mass as Drizontal. a function of time? There is a special value of the time; what is it and .orizontal why is it special? . .1: 3.23. Aforce F9 2 mm to. j _ . Consider a particle that feels an angular force only, of the form F9 3 Lot 63 be 1 5 mid. Show that f m m, where A and B are coastants of :th of the - _j integration, determined by the initial conditions. (There’s nothing all ' that physical about this force. It simply makes the F 2 ma equations solvable.) (3.54) _ 3.24. Free particle was Consider a free particle in a plane. With Cartesian coordinates, it is easy to use F :2 ma to show that the particie moves in a straight line. The task of this problem is to demonstrate this result in a much more cum- bersome way, using polar coordinates and Eq. (3.51). More precisely, show that 0059 m ro/r for a free particie, where my is the radius at closest approach to the origin, and 8 is measured with respect to this radius. g, in a cir- rvelocity ' i 3.7 Exercises ”9- 3-13 3 similar ' _ Section 3.2: Free—body diagrams 3.25. A peculiar Atwood’s machine at lies in _ (a) The Atwood’s machine in Fig. 3.18 consists of n masses, m, m / 2, own and - i: m /4, . . . , m/2n_i. Ail the pulleys and strings are massless. Put a M sieration ' '- mass m /2"‘1 at the free end of the bottom string. What are the {studied . accelerationsof all the masses? m1 m2 at alter it _ (13) Remove the mass ”2/2”“2 (which was arbitrarily small, for very H large n) that was attached in part (a). What are the accelerations Fig. 3:39 of ali the masses, new that you’ve removed this infinitesimal rface, is _ . piece? around a .. 3.26. Keeping the mass still a 7) You In the Atwood’s machine in Fig. 3.19, what should M be, in terms of red so in ' . m; and 1412, so that it doesn’t move? m. Your ; j m tong the - 3.27. Atwood s 1 9: _ _ .. Consider the Atwood’s machine in Fig. 3.20. It consists oft’hree pulleys, 2m m 0_ Find ' ' a short piece of string connecting one mass to the bottom pulley, and a continuous long piece of string that wraps twice around the bottom Fig. 320 75 ties back vay from direction the bend .the rope 'or it — all one (in a the crack tribution, intinuoes t1 separa- e) length than the iiy in the ireceriing ranges in tins, you noks onc— io above. e spacing aningless, ; a dimen- ieompare ie masses. atter what ae second ails or" an ch(l989). he second ords, it is 1 scenario gration in (with the e inelastic most by Wes 5.9 Problems 5.9 Problems Section 5.1." Conservation of energy in one dimension 5.1. Minimum iength a . The shortest configuration of string joining three given points is the one shown in the first setup in Fig. 5.19, where ali three angies ai‘e 120°. M Explain how you could experimentaily prove this fact by cutting three holes in a table and making use of three equal masses attached to the ends of strings, the other ends of which are connected as shown in the second setup in Fig. 5.?9. ' 5.2. Reading to zero a A particle moves toward x m 0 under the influence of a potential V(x) = —A Ext", where A > G and n > 0. The particle has barely enough energy to reach x = 0. For what values of n wiil it reach x = 0 in a finite time? 5.3. Leaving the sphere a A email mass rests on top of a fixed frictionless sphere. The mass is given a tiny kick and siiries downward. At what point does it lose contact with the sphere? 5.4. Pulling the pucks m r {3 J4; . (a) A massless string of length 28 connects two hockey packs that lie on frictionless ice. A constant horizontal force F is applied to the midpoint of the string, perpendicular to it (see iiig. 5.20). By calculating the work done in the transverse direction. find how much kinetic energy is lost when the pucks collide, assuming they stick together. (b) The answer you obtained above should be very ciean and nice. Find the sEick solution that makes it transparent why the answer is so nice. 5.5. Constant y 12* 4'3 “l A bead, under the influence of gravity, slides down a frictionless wire ‘ whose height is given by the function y(x). Assume that at position (any) 2 (0, O), the wire is vertical and the bead passes this point with a given speed so downward. What shoald the shape of the wire be (that is, what is y as a function ofx) so that the vertical speed remains no at ali times? Assume that the carve heads toward positive x. 7‘4 If the three points form a triangle that has an angle greater than 120°, then the string simply passes through the point where that angle is. We won’t worry about this case. 120° 120° Fig. 5.19 Fig. 5.20 120° 173 178 Fig. 5.26 (top view) hand Conservation of energy and momentum approximation where am is much iarger than m2, and assume that the belts bounce elasticaily. Also assume, for the sake of having a nice ciean problem, that the bails are initially separated by a small distance, and that the balls bounce instantaneously. (b) Now consider 21 balls; Bl, . . . ,Bm having masses m1, m2, . . . , m” (with in; >> m2 >> >>‘ in”), standing in a verticat stack (see Fig. 5.26). The bottom ofB; is a height h above the ground, and the bottom of B" is a height}: + 8 above the ground. The hails are dropped. In terms of a, to what height does the top ball bounce? Note: Make assumptions and approximations simiiar to the ones in part (a). Ifh = E meter, what is the minimum number ofbails needed for the tOp one to bounce to a height of at least 1 kilometer? To reach escape velocity? Assume that the balls stiil bounce elasticalty (which is a bit absurd here), and ignore wind resistance, etc, and assume that E is negligibie. 5.24. Maximal deflection their A mass M collides with a stationary mass m. IfM < m, then it is possi— ble for M to bounce directly backward. However, ifM > m, then there is a maximal angle of deflection of M. Show that this maximal angle equals sin‘1(m/M). Hint: It is possible to do this problem by working in the lab frame, but you can save yourself a lot of time by consider- ing what happens in the ‘CM frame, and then shifting back to the lab frame. Section 5.8: Inherently inelastic processes Note: In the problems involving chains in this section (with the exception of Problem 5.29), we ’11 assume that the chains are ofthe type described in thefirst scenario near the end of Section 5. 8. I ' 5.25. Colliding masses a A mass M, initiaily moving at speed V, collides and sticks to a mass m, - .' ' initiaily at rest. ASSunie M >> m, and work in this approximation. What _ are the final energies of the two masses, and how much energy is lost to : _ ' heat, in: '- (a) The iab frame? (b) The frame in which M is initially at rest? 5.26. Pulling a chain as 31% A chain with length L and mass density 0* light; lies straight on a friction- “: _ less horizontal surface. You grab one end and pull it back along itself, in a parallel manner {see Fig. 5.27). Assume that you puil it at constant 5.9 Probiems 1‘79 sume that speed 2). What force must you apply? What is the totai work that yon 3f having do, by the time the chain is straightened out? How much energy is lost ited by a to heat, if any? 3:: _ . 9 mg 5.27. Puiling a chain again as- ‘ tack (see -'= ' “£613 A chain with mass density a kg/m lies in a heap on the fioor. You mid, and . ' J} grab an end and pull horizontaliy with constant force F. What is the :balls are - position of the end of the chain, as a function of time, while it is unrav- bounce? _ I oiling? Assume that the chain is greased, so that it has no friction with the ones ' itself. ' cc de d for 5.28. Falling chain *1: hand To reach A chain with length L and mass density a" kg/m is heid in the position iastically shown in Fig. 5.28, with one end attached to a support. Assume that etc, and oniy a negligible length of the chain starts out beiow the support. The chain is reieased. F ind the force that the support applies to the chain, as a function of time. ’ is pOSSi~ 5.29. Falling chain (energy conserving) Mm hen there Consider the setup in the previous problem, but now let the chain be of I. cal angle the type in the second scenario described in Section 5.8. Show that the _ working total time it takes the chain to straighten out is approximately 85% of F'g' 5'28 considcru - the time it would take if the ieft part were in freefall (as it was in the i; o the lab . previous problem); you wiil need to solve something numericaliy. Also, show that the tension at the left end of the infinitesimal bend equals the tension at the right end at ail times.27 ¥ 5.30. Failing from a table was: ( Keep?” (a) A chain with length L lies in a straight iine on a frictionless table, lescrtbed except for a very email piece at one end which hangs down through a hole in the table. This piece is released, and the chain siides down through the hoie. What is the speed of the chain at the instant it imass m) loses contact with the table? (See Footnote 3.19.) on. What ; (b) Answer the same qeestion, but now let the chain lie in a heap on a is test to ' table, except for a very small piece atone end which hangs dowa through the hole. Assume that the chain is greased, so that it has no friction with itself. Which of these two scenarios has the larger finai speed? . friction— 27 The “ends” of the bend actually aren’t well defined, because the chain is at least a little bit curves? _ everywhere. But since we’re assamiug that the horizontal span of the chain is very smail, we can ng itseif, ' define the height of the bend to he, say, £00 times this horizontal span, and this height is still constant - negiigible compared with the total height of the chain. imuin. Make - yo. Explain 3 considered that the new ength of the no, as shown uolination is ad the plane ' the spring. ing (relative ad to it) that 2‘ it? 11 you found the plane to 1g the plane. the, so that ing is at its m is placed in position. is rotated to a wall, and )0th kinetic ion) of the orth before at and held it It has one ole, and the em length, 5.46. 5.47. 5.48. 5.49. 663 5.10 Exercises with the mass touching the inside surface of the circle at the bottom. (Whatever negligible length of spring remains is essentially horizontal.) The spring is then released, and the mass gets pushed initially to the right and then up along the circle; the setup at a random later time is shown in Fig. 5.33. Let t be the equilibrium length of the spring. What is the minimum value of E for which the mass remains in contact with the circle at all times? Spring and hoop *7? A fixed hoop of radius R stands vertically. A spring with spring constant k and relaxed length of zero is attached to the top of the hoop. (a) A block of mass m is attached to the unstretched spring and dropped from the top of the hoop. If the resulting motion of the mass is a linear vertical oscillation between the top and bottom points on the hoop, What is k? (b) The block is now removed from the spring, and the spring is stretched and connected to a bead, also of mass m, at the bottom of the hoop, as shown in Fig. 5.34. The head is constrained to move along the hoop. It is given a rightward kick and acquires an initial speed on. Assuming that it moves frictionlessiy, how does its speed depend on its position along the hoop? Constant 5: an A head, under the influence of gravity, slides along a frictionless wire whose height is given by the fimction y(x}. Assume that at position (x, y) = (0, 0), the wire is horizontal and the bead passes this point with a given Speed ”00 to the right. What should the shape of the wire be (that is, what is y as a ninetion of x) so that the horizontal speed remains no at all times? One solution is simply y = 0. Find the other.28 Over the pipe as A frictionless cylindrical pipe with radius r is positioned with its axis parallei to the ground, at height It. What is the minimum speed at which a ball must be thrown (from ground level) in order to make it over the pipe? Consider two cases: (a) the ball is allowed to touch the pipe, and (b) the ball is not allowed to touch the pipe. Pendulum projectile m A pendulum is held with its string horizontal and is then released. The mass swings down, and then on its way back up, the string is out when 23 Solve this exercise in the Spirit of Problem 5.5, that is, by solving a differential equation. Once you get the answer, you’ll see that you could have just written it down without any calculations, based on your knowledge of a certain kind ci‘ physical motion. Fig. 5.33 Fig. 5.34 183 184 Fig. 5.38 Conservation of energy and momentum 5.51. 5.52. 5.54. it makes an angle 6 with the vertical; see Fig. 5.35. What shouid 9 he so that the mass travels the largest horizontal distance by the time it returns to the height it had when the string was out? . Centered projectile motion '1”: A mass is attached to one end of a massless string, the other end of which is attached to a fixed support. The mass swings around in a vertical circie as shown in Fig. 5.36. Assuming that the mass has the minimum speed necessary at the top of the circle to keep the string from going siack, at what Eocation should you cut the string so that the resulting projectile motion of the mass has its maximum height iocated directiy above the center of the circle? Beads on a hoop 'in'r Two beads of mass m are initially at rest at the top of a frictionless hoop of mass M and radius R, which stands vertically on the ground. The heads are given tiny kicks, and they slide down the hoop, one to the right and one to the left, as shown in Fig. 5.37. What is the largest vaiue of m /M for which the hoop never rises up off the ground? Stationary bowl est-s: A hemispherical bowl of mass M rests on a table. The inside surface of the bowl is frictionless, while the coefficient of frictionxhetween the bottom of the bowl and the table is [L = 1. A particie of mass m is released from rest at the top of the bowl and siides down into it, as shown in Fig. 5.38. What is the largest value of m/M for which the bowl never slides on the table? Him: The angle you’re concerned with is not 45°. . Leaving the hemisphere saws A point particle of mass m sits at rest on top of a frictionless hemisphere of mass M, which rests on a frictionless table. The particle is given a tiny kick and slides down the (recoiling) hemisphere. At what angle 6 (measured from the top of the hemisphere) does the pafiicle lose contact with the hemisphere? [n answering this question for m ¢ M, it is suf~ ficient for you to produce an equation that 6 must satisfy {it’s a cubic). However, fost‘the special case of m = M , the equation can be solved without too much difficuity; find the angle in this case. Tether-ball sea-4: A smali ball is attached to a massless string of length L, the other end of which is attached to a very thin pole. The ball is thrown so that it initially travels in a horizontal circle, with the string making an angle 69 with the vettical. As time goes on, the string wraps itseli...
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