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Unformatted text preview: LUJ/us along the mat along
along the lie? force
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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 pointmass 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 ”9 313
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
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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 ests: 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. Tetherball sea4: 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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 Winter '07
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