Physics for Scientists and Engineers with Modern Physics

Douglas C. Giancoli

Chapter 8

Conservation of Energy - all with Video Answers

Educators

+ 4 more educators

Chapter Questions

Problem 1

(1) A spring has a spring constant $k$ of 82.0 $\mathrm{N} / \mathrm{m} .$ How much
must this spring be compressed to store 35.0 $\mathrm{J}$ of potential
energy?

Derek Walkama

Derek Walkama

Numerade Educator

Problem 2

(I) A 6.0 -kg monkey swings from one branch to another 1.3 $\mathrm{m}$
higher. What is the change in gravitational potential energy?

Farhanul Hasan

Numerade Educator

Problem 3

(II) A spring with $k=63 \mathrm{N} / \mathrm{m}$ hangs vertically next to a
ruler. The end of the spring is next to the 15 -cm mark on the
ruler. If a. 2.5 $\mathrm{kg}$ mass is now attached to the end of the spring,
where will the end of the spring line up with the ruler marks?

Ravi Lall

Numerade Educator

Problem 4

(II) A 56.5 -kg hiker starts at an elevation of 1270 $\mathrm{m}$ and
climbs to the top of a $2660-\mathrm{m}$ peak. (a) What is the hiker's
change in potential energy? (b) What is the minimum work
required of the hiker? (c) Can the actual work done be
greater than this? Explain.

Farhanul Hasan

Numerade Educator

Problem 5

(II) A $1.60-\mathrm{m}$ tall person lifts a 1.95 -kg book off the ground so it is 2.20 $\mathrm{m}$ above the ground. What is the potential energy of the book relative to $(a)$ the ground, and $(b)$ the top of the person's head? (c) How is the work done by the
person related to the answers in parts $(a)$ and $(b) ?$

Ravi Lall

Numerade Educator

Problem 6

(II) A $1200-\mathrm{kg}$ car rolling on a horizontal surface has speed
$v=75 \mathrm{km} / \mathrm{h}$ when it strikes a horizontal coiled spring and
is brought to rest in a distance of 2.2 $\mathrm{m} .$ What is the spring
stiffness constant of the spring?

Farhanul Hasan

Numerade Educator

Problem 7

(1I) A particular spring obeys the force law $\vec{\mathbf{F}}=$
$\left(-k x+a x^{3}+b x^{4}\right) \hat{\mathbf{i}}$ (a) Is this force conservative?
Explain why or why not. (b) If it is conservative, determine
the form of the potential energy function.

Ravi Lall

Numerade Educator

Problem 8

(II) If $U=3 x^{2}+2 x y+4 y^{2} z,$ what is the force. $\vec{\mathbf{F}}$ ?

Farhanul Hasan

Numerade Educator

Problem 9

(II) A particle is constrained to move in one dimension
along the $x$ axis and is acted upon by a force given by
$\vec{\mathbf{F}}(x)=-\frac{k}{x^{3}} \hat{\mathbf{i}}$
where $k$ is a constant with units appropriate to the SI system.
Find the potential energy function $U(x),$ if $U$ is arbitrarily
defined to be zero at $x=2.0 \mathrm{m},$ so that $U(2.0 \mathrm{m})=0$

Ravi Lall

Numerade Educator

Problem 10

(II) A particle constrained to move in one dimension is
subject to a force $F(x)$ that varies with position $x$ as
$\vec{\mathbf{F}}(x)=A \sin (k x) \hat{\mathbf{i}}$
where $A$ and $k$ are constants. What is the potential energy
function $U(x),$ if we take $U=0$ at the point $x=0 ?$

Farhanul Hasan

Numerade Educator

Problem 11

(I) A novice skier, starting from rest, slides down a friction-
less $13.0^{\circ}$ incline whose vertical height is 125 $\mathrm{m} .$ How fast is
she going when she reaches the bottom?

Ravi Lall

Numerade Educator

Problem 12

(I) Jane, looking for Tarzan, is running at top speed $(5.0 \mathrm{m} / \mathrm{s})$ and grabs a vine hanging vertically from a tall tree in the jungle. How high can she swing upward? Does the length of the vine affect your answer?

Farhanul Hasan

Numerade Educator

Problem 13

(II) In the high jump, the kinetic energy of an athlete is
transformed into gravitational potential energy without the
aid of a pole. With what minimum speed must the athlete
leave the ground in order to lift his center of mass 2.10 $\mathrm{m}$
and cross the bar with a speed of 0.70 $\mathrm{m} / \mathrm{s} ?$

Ravi Lall

Numerade Educator

Problem 14

(II) A sled is initially given a shove up a frictionless $23.0^{\circ}$
incline. It reaches a maximum vertical height 1.12 $\mathrm{m}$ higher
than where it started. What was its initial speed?

Farhanul Hasan

Numerade Educator

Problem 15

(1I) A 55 -kg bungee jumper leaps from a bridge. She is tied
to a bungee cord that is 12 $\mathrm{m}$ long when unstretched, and
falls a total of 31 $\mathrm{m}$ . (a) Calculate the spring constant $k$ of
the bungee cord assuming Hooke's law applies. (b) Calcu-
late the maximum acceleration she experiences.

Ravi Lall

Numerade Educator

Problem 16

(II) $\mathrm{A} 72$ -kg trampoline artist jumps vertically upward from the top of a platform with a speed of 4.5 $\mathrm{m} / \mathrm{s}$ . (a) How fast is he going as he lands on
the trampoline, 2.0 $\mathrm{m}$ below (Fig. 31$) ?$ (b) If the trampoline
behaves like a spring of spring constant $5.8 \times 10^{4} \mathrm{N} / \mathrm{m},$ how
far does he depress it?

Farhanul Hasan

Numerade Educator

Problem 17

(II) The total energy $E$ of an object of mass $m$ that moves in
one dimension under the influence of only conservative
forces can be written as
$$E=\frac{1}{a} m v^{2}+U$$
Use conservation of energy, $d E / d t=0,$ to predict Newton's
second law.

Ravi Lall

Numerade Educator

Problem 18

(II) $\mathrm{A}$ 0.40-kg ball is thrown with a speed of 8.5 $\mathrm{m} / \mathrm{s}$ at an upward angle of $36^{\circ} .(a)$ What is its speed at its highest point,
and $(b)$ how high does it go? (Use conservation of energy.)

Farhanul Hasan

Numerade Educator

Problem 19

(II) A vertical spring (ignore its mass), whose spring constant is 875 $\mathrm{N} / \mathrm{m}$ , is attached to a table and is compressed down by 0.160 $\mathrm{m} .$ (a) What upward speed can it give to a 0.380 -kg ball when released? (b) How high above its original position (spring compressed) will the ball fly?

Ravi Lall

Numerade Educator

Problem 20

(II) A roller-coaster car shown in Fig. 32 is pulled up to point 1 where it is released from rest. Assuming no friction, Calculate the speed at points $2,3,$ and $4 .$

Farhanul Hasan

Numerade Educator

Problem 21

(II) When a mass $m$ sits at rest on a spring, the spring is compressed by a distance $d$ from its undeformed length (Fig. 33$a$ ). Suppose instead that the mass is released
from rest when it barely touches the undeformed spring (Fig. 33 $\mathrm{b} ) .$ Find the distance $D$ that the spring is compressed before it is able to stop the mass. Does $D=d ?$ If not, why not?

Ravi Lall

Numerade Educator

Problem 22

(II) Two masses are connected by a string as shown in
Fig. $34 .$ Mass $m_{\Lambda}=4.0 \mathrm{kg}$ rests on a frictionless inclined
plane, while $m_{\mathrm{B}}=5.0 \mathrm{kg}$ is initially held at a height of
$h=0.75 \mathrm{m}$ above the floor. $(a)$ If $m_{\mathrm{B}}$ is allowed to fall, what
will be the resulting acceleration of the masses? $(b)$ If the
masses were initially at rest, use the kinematic equations
to find their velocity just before $m_{B}$ hits the floor. (c) Use
conservation of energy to find the velocity of the masses just
before $m_{B}$ hits the floor. You should get the same answer as
in part $(b) .$

Farhanul Hasan

Numerade Educator

Problem 23

(II) A block of mass $m$ is attached to the end of a spring (spring stiffness constant $k ),$ Fig. $35 .$ The mass is given an initial displacement $x_{0}$ from equilibrium, and an initial speed $v_{0}$ . Ignoring friction and the mass of the spring, use energy
FIGURE 35 Problems $23,37,$ and 38

Ravi Lall

Numerade Educator

Problem 24

(1I) A cyclist intends to cycle up a $9.50^{\circ}$ hill whose vertical
height is 125 $\mathrm{m}$ . The pedals turn in a circle of diameter
36.0 $\mathrm{cm} .$ Assuming the mass of bicycle plus person is 75.0 $\mathrm{kg}$ ,
(a) calculate how much work must be done against gravity.
(b) If each complete revolution of the pedals moves the bike 5.10 $\mathrm{m}$ along its path, calculate the average force that must be exerted on the pedals tangent to their circular path. Neglect work done by friction and other losses.

Farhanul Hasan

Numerade Educator

Problem 25

(II) A pendulum 2.00 $\mathrm{m}$ long is released (from rest) at an
angle $\theta_{0}=30.0^{\circ}$ (Fig. 14). Determine the speed of the
$70.0-\mathrm{g}$ bob: $(a)$ at the lowest point $(\theta=0) ; \quad$ (b) at
$\theta=15.0^{\circ},(c)$ at $\theta=-15.0^{\circ}$ (i.e., on the opposite side).
d) Determine the tension in the cord at each of these
three points. $(e)$ If the bob is given an initial speed $v_{0}=1.20 \mathrm{m} / \mathrm{s}$ when released at $\theta=30.0^{\circ},$ recalculate the Speeds for parts $(a),(b),$ and $(c)$

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 26

II) What should be the spring constant $k$ of a spring designed
o bring a 1200 -kg car to rest from a speed of 95 $\mathrm{km} / \mathrm{h}$ so
hat the occupants undergo a maximum acceleration of 5.0 $\mathrm{g}$ ?

Farhanul Hasan

Numerade Educator

Problem 27

(III) An engineer is designing a spring to be placed at the bottom of an elevator shaft. If the elevator cable breaks when the elevator is at a height $h$ above the top of the spring, calculate the value that the spring constant $k$ should have so that passengers undergo an acceleration of no more than 5.0 $\mathrm{g}$ when brought to rest. Let $M$ be the total mass of the elevator and passengers.

Ravi Lall

Numerade Educator

Problem 28

(III) A skier of mass $m$ starts from rest at the top of a solid
sphere of radius $r$ and slides down its frictionless surface.
(a) At what angle $\theta$ (Fig. 36) will the skier leave the sphere? (b) If friction were present, would the skier fly off at a greater or lesser angle?
FIGURE 36 Problem 28

Farhanul Hasan

Numerade Educator

Problem 29

(I) Tiwo railroad cars, each of mass $56,000 \mathrm{kg}$ , are traveling
95 $\mathrm{km} / \mathrm{h}$ toward each other. They collide head-on and come
to rest. How much thermal energy is produced in this collision?

Ravi Lall

Numerade Educator

Problem 30

(1) A 16.0 -kg child descends a slide 2.20 $\mathrm{m}$ high and reaches
the bottom with a speed of 1.25 $\mathrm{m} / \mathrm{s} .$ How much thermal
energy due to friction was generated in this process?

Farhanul Hasan

Numerade Educator

Problem 31

$\begin{array}{llll}\text { (II) } & \text { Suppose } & \overrightarrow{\mathbf{A}}=1.0 \hat{\mathbf{i}}+1.0 \hat{\mathbf{j}}-2.0 \hat{\mathbf{k}} & \text { and } & \overrightarrow{\mathbf{B}}=\end{array}$
$-1.0 \hat{\mathbf{i}}+1.0 \mathbf{j}+2.0 \hat{\mathbf{k}},(a)$ what is the angle between these
two vectors? (b) Explain the significance of the sign in part (a).

Ravi Lall

Numerade Educator

Problem 32

(II) A 145 -g baseball is dropped from a tree 14.0 $\mathrm{m}$ above
the ground. $(a)$ With what speed would it hit the ground if
air resistance could be ignored? (b) If it actually hits the ground with a speed of $8.00 \mathrm{m} / \mathrm{s},$ what is the average force of air resistance exerted on it?

Farhanul Hasan

Numerade Educator

Problem 33

(II) A $96-\mathrm{kg}$ crate, starting from rest, is pulled across a floor
with a constant horizontal force of 350 $\mathrm{N}$ . For the first 15 $\mathrm{m}$
the floor is frictionless, and for the next 15 $\mathrm{m}$ the coefficient
of friction is $0.25 .$ What is the final speed of the crate?

Ravi Lall

Numerade Educator

Problem 34

(II) Suppose the roller-coaster car in Fig. 32 passes point 1
with a speed of 1.70 $\mathrm{m} / \mathrm{s}$ . If the average force of friction is
equal to 0.23 of its weight, with what speed will it reach
point 27 The distance traveled is 45.0 $\mathrm{m} .$

Farhanul Hasan

Numerade Educator

Problem 35

(II) A skier traveling 9.0 $\mathrm{m} / \mathrm{s}$ reaches the foot of a steady
upward $19^{\circ}$ incline and glides 12 $\mathrm{m}$ up along this slope before
coming to rest. What was the average coefficient of friction?

Ravi Lall

Numerade Educator

Problem 36

(II) Consider the track shown in Fig. $37 .$ The section $\mathrm{AB}$ is
one quadrant of a circle of radius 2.0 $\mathrm{m}$ and is frictionless.
$\mathrm{B}$ to $\mathrm{C}$ is a horizontal span 3.0 $\mathrm{m}$ long with a coefficient of kinetic friction $\mu_{\mathrm{k}}=0.25 .$ The section $\mathrm{CD}$ under the spring is frictionless. A block of mass 1.0 $\mathrm{kg}$ is released from rest at
A. After sliding on the track, it compresses the spring by
0.20 $\mathrm{m} .$ Determine: $(a)$ the velocity of the block at point $\mathrm{B}$ ;
(b) the thermal energy produced as the block slides from B to $\mathrm{C} ;(c)$ the velocity of the block at point $\mathrm{C} ;(d)$ the stiffness
constant $k$ for the spring.

Farhanul Hasan

Numerade Educator

Problem 37

(II) A 6.620 -kg wood block is firmly attached to a very light
horizontal spring $(k=180 \mathrm{N} / \mathrm{m})$ as shown in Fig. $35 .$ This
block-spring system, when compressed 5.0 $\mathrm{cm}$ and released,
stretches out 2.3 $\mathrm{cm}$ beyond the equilibrium position before stopping and turning back. What is the coefficient of kinetic friction between the block and the table?

Ravi Lall

Numerade Educator

Problem 38

(II) A 180 -g wood block is firmly attached to a very light horizontal spring, Fig. $35 .$ The block can slide along a table where the coefficient of friction is $0.30 .$ A force of 25 $\mathrm{N}$ compresses the spring 18 $\mathrm{cm} .$ If the spring is released from
this position, how far beyond its equilibrium position will it stretch on its first cycle?

Farhanul Hasan

Numerade Educator

Problem 39

You drop a ball from a height of $2.0 \mathrm{m},$ and it bounces back
to a height of 1.5 $\mathrm{m}$ (a) What fraction of its initial energy is lost
during the bounce? (b) What is the ball's speed just before and just after the bounce? (c) Where did the energy go?

Ravi Lall

Numerade Educator

Problem 40

(1I) A 56 -kg skier starts from rest at the top of a 1200 -m-
long trail which drops a total of 230 m from top to bottom.
At the bottom, the skier is moving 11.0 $\mathrm{m} / \mathrm{s} .$ How much.
energy was dissignated by friction?

Farhanul Hasan

Numerade Educator

Problem 41

(II) How much does your gravitational energy change when
you jump as high as you can (say, 1.0 $\mathrm{m} )$ ?

Ravi Lall

Numerade Educator

Problem 42

(III) A spring $(k=75 \mathrm{N} / \mathrm{m})$ has an equilibrium length of
1.00 $\mathrm{m} .$ The spring is compressed to a length of 0.50 $\mathrm{m}$ and a
mass of 2.0 $\mathrm{kg}$ is placed at its free end on a frictionless slope
which makes an angle of $41^{\circ}$ with respect to the horizontal (Fig. $38 ) .$ The spring is then released. (a) If the mass is not attached to the spring, how far up the slope will the mass move before coming to rest? (b) If the mass is attached to the spring, how far up the slope will the mass move before coming to rest? (c) Now the incline has a coefficient of kinetic friction $\mu_{k}$ . If the block, attached to the spring, is observed to stop just as it reaches the spring's equilibrium position, what is the coefficient of friction $\mu_{k} ?$

Farhanul Hasan

Numerade Educator

Problem 43

(III) $\mathrm{A} 2.0$ -kg block slides along a horizontal surface with a
coefficient of kinetic friction $\mu_{\mathrm{k}}=0.30 .$ The block has a
speed $v=1.3 \mathrm{m} / \mathrm{s}$ when it strikes a massless spring head-on.
(a) If the spring has force constant $k=120 \mathrm{N} / \mathrm{m},$ how far
is the spring compressed? (b) What minimum value of the coefficient of static friction, $\mu_{\mathrm{S}},$ will assure that the spring remains compressed at the maximum compressed position? (c) If $\mu_{\mathrm{s}}$ is less than this, what is the speed of the block when it detaches from the decompressing spring? [Hint: Detach-
ment occurs when the spring reaches its natural length $(x=0) :$ explain why 1

Ravi Lall

Numerade Educator

Problem 44

(III) Early test flights for the space shuttle used a "glider"
(mass of 980 kg including pilot). After a horizontal launch at
480 $\mathrm{km} / \mathrm{h}$ at a height of $3500 \mathrm{m},$ the glider eventually landed at a speed of 210 $\mathrm{km} / \mathrm{h}$ . (a) What would its landing speed
have been in the absence of air resistance? (b) What was the
average force of air resistance exerted on it if it came in at a
constant glide angle of $12^{\circ}$ to the Earth's surface?

Farhanul Hasan

Numerade Educator

Problem 45

(I) For a satellite of mass $m_{\mathrm{s}}$ in a circular orbit of radius $r_{\mathrm{S}}$
around the Earth, determine $(a)$ its kinetic energy $K,$ (b) its
potential energy $U(U=0$ at infinity $),$ and $(c)$ the ratio $K / U$

Ravi Lall

Numerade Educator

Problem 46

(1) Jill and her friends have built a small rocket that soon
after lift-off reaches a speed of 850 $\mathrm{m} / \mathrm{s} .$ How high above
the Earth can it rise? Ignore air friction.

Farhanul Hasan

Numerade Educator

Problem 47

(1) The escape velocity from planet A is double that for
planet B. The two planets have the same mass. What is the
ratio of their radii, $r_{A} / r_{\mathrm{B}} ?$

Ravi Lall

Numerade Educator

Problem 48

(II) Show that Eq. 16 for gravitational potential energy reduces to Eq. $2, \Delta U=m g\left(y_{2}-y_{1}\right),$ for objects near the surface of the Earth.
$$\Delta U=U_{2}-U_{1}=-\frac{G m M_{\mathrm{E}}}{r_{2}}+\frac{G m M_{\mathrm{E}}}{r_{1}}$$
$$\Delta U=U_{2}-U_{1}=-W_{\mathrm{G}}=m g\left(y_{2}-y_{1}\right)$$

Farhanul Hasan

Numerade Educator

Problem 49

(II) Determine the escape velocity from the Sun for an
object $(a)$ at the Sun's surface $\left(r=7.0 \times 10^{5} \mathrm{km}$ , \right.
$M=2.0 \times 10^{30} \mathrm{kg} ),$ and $(b)$ at the average distance of the
Earth $\left(1.50 \times 10^{8} \mathrm{km}\right)$ . Compare to the speed of the Earth
in its orbit.

Ravi Lall

Numerade Educator

Problem 50

(II) Two Earth satellites, A and B, each of mass $m=950 \mathrm{kg}$ ,
are launched into circular orbits around the Earth's center. Satellite A orbits at an altitude of $4200 \mathrm{km},$ and satellite $\mathrm{B}$ orbits at an altitude of $12,600 \mathrm{km}$ . (a) What are the potential energies of the two satellites? (b) What are the kinetic energies of the two satellites? (c) How much work would it require of change the orbit of satellite A to match that of satellite B?

Farhanul Hasan

Numerade Educator

Problem 51

(II) Show that the escape velocity for any satellite in a
circular orbit is $\sqrt{2}$ times its velocity.

Ravi Lall

Numerade Educator

Problem 52

(1I) $(a)$ Show that the total mechanical energy of a satellite mass $m$ orbiting at a distance $r$ from the center of the Earth (mass $M_{E} )$ is
$$E=-\frac{1}{2} \frac{G m M_{\mathrm{E}}}{r}$$
if $U=0$ at $r=\infty .(b)$ Show that although friction causes
the value of $E$ to decrease slowly, kinetic energy must actu-
ally increase if the orbit remains a circle.

Farhanul Hasan

Numerade Educator

Problem 53

(II) Take into account the Earth's rotational speed ( 1 rev/day)
and determine the necessary speed, with respect to Earth, for a
rocket to escape if fired from the Earth at the equator in a
direction $(a)$ eastward; $(b)$ westward; $(c)$ vertically upward.

Ravi Lall

Numerade Educator

Problem 54

(II) $(a)$ Determine a formula for the maximum height $h$ that a rocket will reach if launched vertically from the Earth's surface with speed $v_{0}\left(<v_{\text { esc }}\right)$ . Express in terms of $v_{0}, r_{\mathrm{E}}, M_{\mathrm{E}}$ and $G .(b)$ How high does a rocket go if $v_{0}=8.35 \mathrm{km} / \mathrm{s} ?$ Ignore air resistance and the Earth's rotation.

Farhanul Hasan

Numerade Educator

Problem 55

II) $(a)$ Determine the rate at which the escape velocity from the Earth changes with distance from the center of the Earth, $d v_{\text { esc }} / d r,$ (b) Use the approximation $\Delta v \approx(d v / d r) \Delta r$ to determine the escape velocity for a spacecraft orbiting the Earth at a height of 320 $\mathrm{km} .$

Ravi Lall

Numerade Educator

Problem 56

(II) A meteorite has a speed of 90.0 $\mathrm{m} / \mathrm{s}$ when 850 $\mathrm{km}$ above the Earth. It is falling vertically (ignore air resistance) and
strikes a bed of sand in which it is brought to rest in 3.25 $\mathrm{m}$ .
(a) What is its speed just before striking the sand? (b) How much
work does the sand do to stop the meteorite (mass $=575 \mathrm{kg}$ ?
(c) What is the average force exerted by the sand on the
meteorite? (d) How much thermal energy is produced?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 57

(II) How much work would be required to move a satellite
of mass $m$ from a circular orbit of radius $r_{1}=2 r_{\mathrm{E}}$ about
the Earth to another circular orbit of radius $r_{2}=3 r_{\mathrm{E}} ?$
$\left(r_{\mathrm{E}}$ is the radius of the Earth.) \right.

Ravi Lall

Numerade Educator

Problem 58

(II) $(a)$ Suppose we have three masses, $m_{1}, m_{2},$ and $m_{3},$ that
initially are infinitely far apart from each other. Show that the work needed to bring them to the positions shown in Fig. 39 is
$$W=-G\left(\frac{m_{1} m_{2}}{r_{12}}+\frac{m_{1} m_{3}}{r_{13}}+\frac{m_{2} m_{3}}{r_{23}}\right)$$
(b) Can we say that this formula also gives the potential energy
of the system, or the potential energy of one or two of the objects? $(c)$ Is $W$ equal to
the binding energy of the system-that is, equal to the energy required to separate the components by an infinite distance? Explain.

Farhanul Hasan

Numerade Educator

Problem 59

(II) A NASA satellite has just observed an asteroid that is on a collision course with the Earth. The asteroid has an estimated mass, based on its size, of $5 \times 10^{9} \mathrm{kg}$ . It is approaching the Earth on a head-on course with a velocity of 660 $\mathrm{m} / \mathrm{s}$ relative to the Earth and is now $5.0 \times 10^{6} \mathrm{km}$ away. With what speed will it hit the Earth's surface, neglecting friction with the atmosphere?

Ravi Lall

Numerade Educator

Problem 60

(II) A sphere of radius $r_{1}$ has a concentric spherical cavity of radius $r_{2}$ (Fig. 40$)$ . Assume this spherical shell of thickness $r_{1}-r_{2}$ is uniform and has a total mass $M .$ Show that the gravitational potential energy of a mass $m$ at
a distance $r$ from the center of the shell
$\left(r>r_{1}\right)$ is given by
$$U=-\frac{G m M}{r}$$

Farhanul Hasan

Numerade Educator

Problem 61

(III) To escape the solar system, an interstellar spacecraft must overcome the gravitational attraction of both the Earth and Sun. Ignore the effects of other bodies in the solar system. $(a)$ Show that the escape velocity is
$$v=\sqrt{v_{\mathrm{E}}^{2}+\left(v_{\mathrm{S}}-v_{0}\right)^{2}}=16.7 \mathrm{km} / \mathrm{s}$$ where: $v_{\mathrm{E}}$ is the escape velocity from the Earth (Eq. 19);
$v_{\mathrm{S}}=\sqrt{2 G M_{\mathrm{S}} / r_{\mathrm{SE}}}$ is the escape velocity from the gravitational field of the Sun at the orbit of the Earth but far from
the Earth's influence $\left(r_{\mathrm{SE}}$ is the Sun-Earth distance); and $v_{0}$ is \right. the Earth's orbital velocity about the Sun. (b) Show that the energy required is $1.40 \times 10^{3} \mathrm{J}$ per kilogram of spacecraft mass [Hint. Write the energy equation for escape from Earth with $v^{\prime}$ as the velocity, relative to Earth, but far from Earth; then let $v^{\prime}+v_{0}$ be the escape velocity from the Sun.
$$v_{\mathrm{esc}}=\sqrt{2 G M_{\mathrm{E}} / r_{\mathrm{E}}}=1.12 \times 10^{4} \mathrm{m} / \mathrm{s}$$

Check back soon!

Problem 62

$\begin{array}{l}{\text { (I) How long will it take a } 1750-\mathrm{W} \text { motor to lift a } 335 \text { -kg }} \\ {\text { piano to a sixth-story window } 16.0 \mathrm{m} \text { above? }}\end{array}$

Farhanul Hasan

Numerade Educator

Problem 63

(I) If a car generates 18 $\mathrm{hp}$ when traveling at a steady
$95 \mathrm{km} / \mathrm{h},$ what must be the average force exerted on the car
due to friction and air resistance?

Ravi Lall

Numerade Educator

Problem 64

(I) An 85 -kg football player traveling 5.0 $\mathrm{m} / \mathrm{s}$ is stopped in
1.0 $\mathrm{s}$ by a tackler. (a) What is the original kinetic energy of
the player? $(b)$ What average power is required to stop him?

Farhanul Hasan

Numerade Educator

Problem 65

(II) A driver notices that her $1080-\mathrm{kg}$ car slows down from
95 $\mathrm{km} / \mathrm{h}$ to 65 $\mathrm{km} / \mathrm{h}$ in about 7.0 $\mathrm{s}$ on the level when it is in
neutral. Approximately what power (watts and hp) is
needed to keep the car traveling at a constant 80 $\mathrm{km} / \mathrm{h} ?$

AM

Adam Moon

Numerade Educator

Problem 66

(II) How much work can a 3.0 -hp motor do in 1.0 $\mathrm{h} ?$

Farhanul Hasan

Numerade Educator

Problem 67

(II) An outboard motor for a boat is rated at 55 hp. If it can
move a particular boat at a steady speed of $35 \mathrm{km} / \mathrm{h},$ what is
the total force resisting the motion of the boat?

Ravi Lall

Numerade Educator

Problem 68

(II) A 1400 -kg sports car accelerates from rest to 95 $\mathrm{km} / \mathrm{h}$ in
7.4 $\mathrm{s} .$ What is the average power delivered by the engine?

Farhanul Hasan

Numerade Educator

Problem 69

(II) During a workout, football players ran up the stadium
stairs in 75 s the stairs are 78 m long and inclined at an angle
of $33^{\circ} .$ If a player has a mass of 92 $\mathrm{kg}$ . estimete his average
power output on the way up. Ignore friction and air resistance.

Ravi Lall

Numerade Educator

Problem 70

(II) A pump lifts 21.0 $\mathrm{kg}$ of water per minute through a height of 3.50 $\mathrm{m}$ . What minimum output rating (watts) must the pump motor have?

Farhanul Hasan

Numerade Educator

Problem 71

(1I) A ski area claims that its can move $47,000$ people per hour. If the average lift carries people about 200 $\mathrm{m}$ (vertically) higher, estimate the maximum total power needed.

Ravi Lall

Numerade Educator

Problem 72

(II) $\mathrm{A} 75$ -kg skier grips a moving rope that is powered by an engine and is pulled at constant speed to the top of a $23^{\circ}$ hill. The skier is pulled a distance $x=220 \mathrm{m}$ along the incline and it takes 2.0 min to reach the top of the hill. If the
coefficient of kinetic friction between the snow and skis is $\mu_{k}=0.10,$ what horsepower engine is required if 30 such skiers $(\max )$ are on the rope at one time?

Farhanul Hasan

Numerade Educator

Problem 73

(III) The position of a $280-$ g object is given (in meters) by $x=5.0 t^{3}-8.0 t^{2}-44 t$ , where $t$ is in seconds. Determine the net rate of work done on this object $(a)$ at $t=2.0 \mathrm{s}$ and (b) at $t=4.0 \mathrm{s}$ (c) What is the average net power input during the interval from $t=0$ s to $t=2.0 \mathrm{s},$ and in the interval from $t=2.0 \mathrm{s}$ to 4.0 $\mathrm{s} ?$

Ravi Lall

Numerade Educator

Problem 74

(1II) A bicyclist coasts down a $6.0^{\circ}$ hill at a steady speed of
4.0 $\mathrm{m} / \mathrm{s} .$ Assuming a total mass of 75 $\mathrm{kg}$ (bicycle plus rider), what must be the cyclist's power output to climb the same hill at the same speed?

Farhanul Hasan

Numerade Educator

Problem 75

(1I) Draw a potential energy diagram, $U$ vs. $x,$ and analyze the motion of a mass $m$ resting on a frictionless horizontal table and connected to a horizontal spring with stiffness
constant $k$ . The mass is pulled a distance to the right so that the spring is stretched a distance $x_{0}$ initially, and then the mass is released from rest.

Ravi Lall

Numerade Educator

Problem 76

(II) The spring of Problem 75 has a stiffness constant
$k=160 \mathrm{N} / \mathrm{m} .$ The mass $m=5.0 \mathrm{kg}$ is released from rest
when the spring is stretched $x_{0}=1.0 \mathrm{m}$ from equilibrium. Determine $(a)$ the total energy of the system; $(b)$ the kinetic energy when $x=\frac{1}{2} x_{0} ;$ (c) the maximum kinetic energy; (d) the maximum speed and at what positions it occurs;
(e) the maximum acceleration and where it occurs.

Farhanul Hasan

Numerade Educator

Problem 77

(III) The potential energy of the two atoms in a diatomic (two-atom) molecule can be written
$$U(r)=-\frac{a}{r^{6}}+\frac{b}{r^{12}}$$
where $r$ is the distance between the two atoms and $a$ and $b$ are positive constants. $(a)$ At what values of $r$ is $U(r)$ a minimum? A maximum? $(b)$ At what values of $r$ is $U(r)=0 ?(c)$ Plot $U(r)$ as a function of $r$ from $r=0$ to $r$ at a value large enough for all the features in $(a)$ and $(b)$ to show. (d) Describe the motion of one atom with respect to the second atom when $E<0,$ and when $E>0 .(e)$ Let $F$
be the force one atom exerts on the other. For what values of $r$ is $F>0, F<0, F=0 ?$ (f) Determine $F$ as a function of $r .$

Bret Rosen

Numerade Educator

Problem 78

(III) The binding energy of a two-particle system is defined as the energy required to separate the two particles from their state of lowest energy to $r=\infty$ . Determine
the binding energy for the molecule discussed in Problem $77 .$

Farhanul Hasan

Numerade Educator

Problem 79

What is the average power output of an elevator that lifts
885 kg a vertical height of 32.0 $\mathrm{m}$ in 11.0 $\mathrm{s}$ ?

Ravi Lall

Numerade Educator

Problem 80

A projectile is fired at an upward angle of $48.0^{\circ}$ from the
top of a $135-$ m-high cliff with a speed of 165 $\mathrm{m} / \mathrm{s}$ . What will be its speed when it strikes the ground below? (Use conservation of energy.)

Farhanul Hasan

Numerade Educator

Problem 81

Water flows over a dam at the rate of 580 $\mathrm{kg} / \mathrm{s}$ and falls
vertically 88 $\mathrm{m}$ before striking the turbine blades. Calculate
(a) the speed of the water just before striking the turbine blades (neglect air resistance), and (b) the rate at which mechanical energy is transferred to the turbine blades,
assuming 55$\%$ efficiency.

Ravi Lall

Numerade Educator

Problem 82

A bicyclist of mass 75 $\mathrm{kg}$ (including the bicycle) can coast
down a $4.0^{\circ}$ hill at a steady speed of 12 $\mathrm{km} / \mathrm{h}$ . Pumping hard, the cyclist can descend the hill at a speed of 32 $\mathrm{km} / \mathrm{h}$ . Using
the same power, at what speed can the cyclist climb the same hill? Assume the force of friction is proportional to the square of the speed $v ;$ that is, $F_{\text { tr }}=b v^{2},$ where $b$ is a constant.

Farhanul Hasan

Numerade Educator

Problem 83

A 62 -kg skier starts from rest at the top of a ski jump, point $A$ in Fig. $41,$ and travels down the ramp. If friction and air resistance can be neglected, $(a)$ determine her speed $v_{B}$ when he reaches the horizontal end of the ramp at B. $B$ (b) Determine the distance $s$ to where she strikes the ground

Ravi Lall

Numerade Educator

Problem 84

Repeat Problem $83,$ but now assume the ski jump turns
upward at point $B$ and gives her a vertical component of
velocity (at $B )$ of 3.0 $\mathrm{m} / \mathrm{s}$ .

Farhanul Hasan

Numerade Educator

Problem 85

A ball is attached to a horizontal cord of length $\ell$ whose other end is fixed, Fig. $42 .(a)$ If the ball is released, what will be its speed at the lowest point of its path? (b) A peg is
located a distance $h$ directly below the point of attachment of the cord. If $h=0.80 \ell,$ what will be the speed of the ball when it reaches the top of its circular path about the peg?

Ravi Lall

Numerade Educator

Problem 86

Show that $h$ must be greater than 0.60$\ell$ if the ball in Fig. 42 is
to make a complete circle about the peg.

Farhanul Hasan

Numerade Educator

Problem 87

Show that on a roller coaster with a circular vertical loop (Fig, 43), the difference in your apparent weight at the top of the loop and the bottom of the loop is $6 g^{\prime} s-$ that is six times your weight. Ignore friction. Show also that as long as your speed is above the
minimum needed, this answer doesn't depend on the size of the loop or
how fast you go through it.

Ravi Lall

Numerade Educator

Problem 88

If you stand on a bathroom scale, the spring inside the
scale compresses $0.50 \mathrm{mm},$ and it tells you your weight is
760 $\mathrm{N} .$ Now if you jump on the scale from a height of $1.0 \mathrm{m},$
what does the scale read at its peak?

Farhanul Hasan

Numerade Educator

Problem 89

A 65 -kg hiker climbs to the top of a 4200 -m-high mountain. The climb is made in 5.0 h starting at an elevation of 2800 m. Calculate $(a)$ the work done by the hiker against
gravity, $(b)$ the average power output in watts and in horsepower, and $(c)$ assuming the body is 15$\%$ efficient, what rate of energy input was required.

Ravi Lall

Numerade Educator

Problem 90

The small mass $m$ sliding without friction along the looped track shown in Fig. 44 is to remain on the track at all times, even at the very top of the loop of radius $r .$ (a) In terms of the given quantities, determine the minimum release height $h .$ Next, if the actual release height is 2$h$ , calculate the normal orce exerted $(b)$ by the track at the bottom of the loop, $c )$ by the track at the top of the loop, and $(d)$ by the track after the block exits the loop onto the flat section.

Farhanul Hasan

Numerade Educator

Problem 91

A 56 -kg student runs at $5.0 \mathrm{m} / \mathrm{s},$ grabs a hanging rope, and swings out over a lake (Fig. $45 ) .$ He releases the rope when his velocity is zero, $(a)$ What is
the angle $\theta$ when he releases the rope? (b) What is the tension in the rope just before he releases it? ( c) What is the maximum tension in the rope?

Ravi Lall

Numerade Educator

Problem 92

The nuclear force between two neutrons in a nucleus is described roughly by the Yukawa potential
$$\quad U(r)=-U_{0} \frac{r_{0}}{r} e^{-r / r_{s}}$$
where $r$ is the distance between the neutrons and $U_{0}$ and
$r_{0}\left(\approx 10^{-15} \mathrm{m}\right)$ are constants. (a) Determine the force $F(r)$ .
(b) What is the ratio $F\left(3 r_{0}\right) / F\left(r_{0}\right) ?(c)$ Calculate this same
ratio for the force between two electrically charged parti-
cles where $U(r)=-C / r,$ with $C$ a constant. Why is the Yukawa force referred to as a short-range force ? .

Farhanul Hasan

Numerade Educator

Problem 93

A fire hose for use in urban areas must be able to shoot a stream of water to a maximum height of 33 $\mathrm{m}$ . The water leaves the hose at ground level in a circular stream 3.0 $\mathrm{cm}$ in diameter. What minimum power is required to create such a stream of water? Every cubic meter of water has a mass of $1.00 \times 10^{3} \mathrm{kg}$ .

Ravi Lall

Numerade Educator

Problem 94

A 16 -kg sled starts up a $28^{\circ}$ incline with a speed of 2.4 $\mathrm{m} / \mathrm{s}$ . The coefficient of kinetic friction is $\mu_{\mathrm{k}}=0.25 .$ (a) How far up the incline does the sled travel? (b) What condition must you put on the coefficient of static friction if the sled s not to get stuck at the point determined in part $(a) ?$ c) If the sled slides back down, what is its speed when it
returns to its starting point?

Farhanul Hasan

Numerade Educator

Problem 95

The Lunar Module could make a safe landing if its vertical velocity at impact is 3.0 $\mathrm{m} / \mathrm{s}$ or less. Suppose that you want to determine the greatest height $h$ at which the pilot could shut off the engine if the velocity of the lander relative to the surface is $(a)$ zero; $(b) 2.0 \mathrm{m} / \mathrm{s}$ downward; $(c) 2.0 \mathrm{m} / \mathrm{s}$ upward. Use conservation of energy to determine $h$ in
each case. The acceleration due to gravity at the surface of the Moon is 1.62 $\mathrm{m} / \mathrm{s}^{2}$ .

Ravi Lall

Numerade Educator

Problem 96

Proper design of automobile braking systems must account for heat buildup under heavy braking. Calculate the thermal energy dissipated from brakes in a 1500 -kg car
that descends a $17^{\circ}$ hill. The car begins braking when its speed is 95 $\mathrm{km} / \mathrm{h}$ and slows to a speed of 35 $\mathrm{km} / \mathrm{h}$ in a
distance of 0.30 $\mathrm{km}$ measured along the road.

Farhanul Hasan

Numerade Educator

Problem 97

Some electric power companies use water to store energy. Water is pumped by reversible turbine pumps from a low reservoir to a high reservoir. To store the energy produced in
1.0 hour by a $180-\mathrm{M} \mathrm{W}$ electric power plant, how many cubic meters of water will have to be pumped from the lower to the upper reservoir? Assume the upper reservoir is 380 $\mathrm{m}$ above the lower one, and we can neglect the small change in depths of each. Water has a mass of $1.00 \times 10^{3}$ kg for every
1.0 $\mathrm{m}^{3} .$

Ravi Lall

Numerade Educator

Problem 98

Estimate the energy required from fuel to launch a 1465 -kg
satellite into orbit 1375 $\mathrm{km}$ above the Earth's surface.
Consider two cases: $(a)$ the satellite is launched into an
equatorial orbit from a point on the Earth's equator, and
(b) it is launched from the North Pole into a polar orbit.

Farhanul Hasan

Numerade Educator

Problem 99

A satellite is in an elliptic orbit around the Earth (Fig, 46). Its speed at the perigee $A$ is 8650 $\mathrm{m} / \mathrm{s}$ . (a) Use conservation of energy to determine its speed at B. The radius of the Earth is 6380 $\mathrm{km}$ . (b) Use conservation of energy to determine the speed at the apogee $\mathrm{C}$ .

Ravi Lall

Numerade Educator

Problem 100

Suppose the gravitational potential energy of an object of
mass $m$ at a distance $r$ from the center of the Earth is given by
$$U(r)=-\frac{G M m}{r} e^{-\alpha r}$$
where $\alpha$ is a positive constant and $e$ is the exponential function.
(Newton's law of universal gravitation has $\alpha=0 ) .(a)$ What
would be the force on the object as a function of $r ?(b)$ What
would be the object's escape velocity in terms of the Earth's
radius $R_{E} ?$

Farhanul Hasan

Numerade Educator

Problem 101

(a) If the human body could convert a candy bar directly into work, how high could a 76 -kg man climb a ladder if he were fueled by one bar $(=1100 \mathrm{kJ}) ?$ (b) If the man
then jumped off the ladder, what will be his speed when he reaches the bottom?

Ravi Lall

Numerade Educator

Problem 102

Electric energy units are often expressed in the form of kilowatt-hours (a) Show that one kilowatt-hour (kWh) is equal to $3.6 \times 10^{6} \mathrm{J}$ . (b) If a typical family of four uses electric energy at an average rate of $580 \mathrm{W},$ how many kWh would
their electric bill show for one month, and (c) how many joules would this be? (d) At a cost of $\$ 0.12$ per $\mathrm{kWh}$ , what would their monthly bill be in dollars? Does the monthly bill depend on the rate at which they use the electric energy?

Farhanul Hasan

Numerade Educator

Problem 103

Chris jumps off a bridge with a bungee cord (a heavy stretchable cord) tied around his ankle, Fig. $47 .$ He falls for 15 before the bungee cord begins to stretch. Chris's mass
is 75 $\mathrm{kg}$ and we assume the cord obeys Hooke's law, $F=-k x,$ with $k=50 \mathrm{N} / \mathrm{m}$ . If we neglect air resistance, estimate how far below the bridge Chris's foot will be before coming to a stop. Ignore the mass of the cord (not realistic,
however) and treat Chris as a particle.

Ravi Lall

Numerade Educator

Problem 104

In a common test for cardiac function (the stress test), the patient walks on an inclined treadmill (Fig. 48$)$ . Estimate the power required from a 75 -kg patient when
the treadmill is sloping at an angle of $12^{\circ}$ and the velocity
is 3.3 $\mathrm{km} / \mathrm{h} .$ How does this power compare to the power
rating of a lightbulb?

Farhanul Hasan

Numerade Educator

Problem 105

(a) If a volcano spews a $450-\mathrm{kg}$ rock vertically upward a distance of $320 \mathrm{m},$ what was its velocity when it left the volcano? $(b)$ If the volcano spews 1000 rocks of this size every minute, estimate its power output.

Ravi Lall

Numerade Educator

Problem 106

A film of Jesse Owens's famous long jump (Fig. 49) in the 1936 Olympics shows that his
center of mass rose 1.1 $\mathrm{m}$ from launch point to the top of the
arc. What minimum speed did he need at launch if he was
traveling at 6.5 $\mathrm{m} / \mathrm{s}$ at the top of the arc?

Farhanul Hasan

Numerade Educator

Problem 107

An elevator cable breaks when a $920-\mathrm{kg}$ elevator is 24 $\mathrm{m}$
above a huge spring $\left(k=2.2 \times 10^{5} \mathrm{N} / \mathrm{m}\right)$ at the bottom of the shaft. Calculate (a) the work done by gravity on the elevator before it hits the spring, $(b)$ the speed of the elevator just before striking the spring, and $(c)$ the amount
the spring compresses (note that work is done by both the spring and gravity in this part).

Ravi Lall

Numerade Educator

Problem 108

A particle moves where its potential energy is given by
$U(r)=U_{0}\left[\left(2 / r^{2}\right)-(1 / r)\right] .(a)$ Plot $U(r)$ versus $r .$ Where does the curve cross the $U(r)=0$ axis? At what value of $r$ does the minimum value of $U(r)$ occur? $(b)$ Suppose that the particle has an energy of $E=-0.050 U_{0 .}$ Sketch in
the approximate turning points of the motion of the particle on your diagram. What is the maximum kinetic energy of the particle, and for what value of $r$ does this occur?

Farhanul Hasan

Numerade Educator

Problem 109

A particle of mass $m$ moves under the influence of a
potential energy
$$U(x)=\frac{a}{x}+b x$$
where $a$ and $b$ are positive constants and the particle is
restricted to the region $x>0 .$ Find a point of equilibrium
for the particle and demonstrate that it is stable.

Ravi Lall

Numerade Educator

Problem 110

(III) The two atoms in a diatomic molecule exert an attractive force on each other at large distances and a repulsive force at short distances. The magnitude of the
force between two atoms in a diatomic molecule can be approximated by the Lennard-Jones force, or $F(r)=F_{0}\left[2(\sigma / r)^{13}-(\sigma / r)^{7}\right],$ where $r$ is the separation between the two atoms, and $\sigma$ and $F_{0}$ are constant. For an
oxygen molecule (which is diatomic) $F_{0}=9.60 \times 10^{-11} \mathrm{N}$
and $\sigma=3.50 \times 10^{-11} \mathrm{m} .$ (a) Integrate the equation for
$F(r)$ to determine the potential energy $U(r)$ of the oxygen nolecule. (b) Find the equilibrium distance $r_{0}$ between the two atoms $(c)$ Graph $F(r)$ and $U(r)$ between 0.9$r_{0}$ and 2.5$r_{0}$ .

Farhanul Hasan

Numerade Educator