Physics for Scientists and Engineers with Modern Physics

Douglas C. Giancoli

Chapter 5

Using Newton's Laws: Friction, Circular Motion, Drag Forces - all with Video Answers

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Chapter Questions

Problem 1

The Problems in this Section are ranked $1,$ II, or III according to
estimated difficulty, with $(1)$ Problems being easiest. Level (III)
Problems are meant mainly as a challenge for the best students, for
"extra credit." The Problems are arranged by Sections, meaning that
the reader should have read up to and inciuding that Section, but
this Chapter also has a group of General Problems that are not
arranged by Section and not ranked.
$\begin{array}{l}{\text { (1) If the coefficient of kinetic friction between a } 22 \text { -kg crate }} \\ {\text { and the floor is } 0.30 \text { , what horizontal force is required to }} \\ {\text { move the crate at a steady speed across the floor? What }} \\ {\text { horizontal force is required if } \mu_{k} \text { is zero? }}\end{array}$

Kristela Garcia

Kristela Garcia

Numerade Educator

Problem 2

(I) A force of 35.0 $\mathrm{N}$ is required to start a 6.0 -kg box moving
across a horizontal concrete floor. $(a)$ What is the coefficient
of static friction between the box and the floor? (b) If the
35.0 -N force continues, the box accelerates at 0.60 $\mathrm{m} / \mathrm{s}^{2}$ .
What is the coefficient of kinetic friction?

Abhishek Jana

Numerade Educator

Problem 3

(I) Suppose you are standing on a train accelerating at 0.20 $\mathrm{g}$ .
What minimum coefficient of static friction must exist
between your feet and the floor if you are not to slide?

Kristela Garcia

Kristela Garcia

Numerade Educator

Problem 4

(I) The coefficient of static friction between hard rubber
and normal street pavement is about $0.90 .$ On how steep a
hill (maximum angle) can you leave a car parked?

Abhishek Jana

Numerade Educator

Problem 5

(I) What is the maximum acceleration a car can undergo if
the coefficient of static friction between the tires and the
ground is 0.90$?$

Kristela Garcia

Kristela Garcia

Numerade Educator

Problem 6

(II) $(a)$ A box sits at rest on a rough $33^{\circ}$ inclined plane.
Draw the free-body diagram, showing all the forces acting
on the box. (b) How would the diagram change if the box
were sliding down the plane? (c) How would it change if the
box were sliding up the plane after an initial shove?

Abhishek Jana

Numerade Educator

Problem 7

(II) A $25.0-\mathrm{kg}$ box is released on a $27^{\circ}$ incline and accelerates
down the incline at 0.30 $\mathrm{m} / \mathrm{s}^{2}$ . Find the friction force impeding
its motion. What is the coefficient of kinetic friction?

Kristela Garcia

Kristela Garcia

Numerade Educator

Problem 8

(II) A car can decelerate at $-3.80 \mathrm{m} / \mathrm{s}^{2}$ without skidding
when coming to rest on a level road. What would its decel-
eration be if the road is inclined at $9.3^{\circ}$ and the car moves
uphill? Assume the same static friction coefficient.

Abhishek Jana

Numerade Educator

Problem 9

(II) A skier moves down a $27^{\circ}$ slope at constant speed. What
can you say about the coefficient of friction, $\mu_{k} ?$ Assume
the speed is low enough that air resistance can be ignored.

Kristela Garcia

Kristela Garcia

Numerade Educator

Problem 10

(II) A wet bar of soap slides freely down a ramp 9.0 $\mathrm{m}$ long
inclined at $8.0^{\circ} .$ How long does it take to reach the bottom?
Assume $\mu_{\mathrm{k}}=0.060 .$

Abhishek Jana

Numerade Educator

Problem 11

(II) A box is given a push so that it slides across the floor.
How far will it go, given that the coefficient of kinetic friction
is 0.15 and the push imparts an initial speed of 3.5 $\mathrm{m} / \mathrm{s} ?$

Pk

Pankaj Kumawat

Numerade Educator

Problem 12

(II) $(a)$ Show that the minimum stopping distance for an auto-
mobile traveling at speed $v$ is equal to $v^{2} / 2 \mu_{\mathrm{s}} g,$ where $\mu_{\mathrm{s}}$ is
the coefficient of static friction between the tires and the road,
and $g$ is the acceleration of gravity. $(b)$ What is this distance for
a $1200-\mathrm{kg}$ car traveling 95 $\mathrm{km} / \mathrm{h}$ if $\mu_{\mathrm{s}}=0.65 ?(c)$ What would
it be if the car were on the Moon (the acceleration of gravity
on the Moon is about $g / 6$ ) but all else stayed the same?

Abhishek Jana

Numerade Educator

Problem 13

(II) A 1280 -kg car pulls a 350 -kg trailer. The car exerts a hori-
zontal force of $3.6 \times 10^{3} \mathrm{N}$ against the ground in order to
accelerate. What force does the car exert on the trailer?
Assume an effective friction coefficient of 0.15 for the trailer.

Kristela Garcia

Kristela Garcia

Numerade Educator

Problem 14

(1I) Police investigators, examining the scene of an accident
involving two cars, measure 72 -m-long skid marks of one of
the cars, which nearly came to a stop before colliding. The
coefficient of kinetic friction between rubber and the pave-
ment is about $0.80 .$ Estimate the initial speed of that car
assuming a level road.

Abhishek Jana

Numerade Educator

Problem 15

(II) Piles of snow on slippery roofs can become dangerous
projectiles as they melt. Consider a chunk of snow at the
ridge of a roof with a slope of $34^{\circ} .(a)$ What is the minimum
value of the coefficient of static friction that will keep the
snow from sliding down? (b) As the snow begins to melt the
coefficient of static friction decreases and the snow finally
slips. Assuming that the distance from the chunk to the edge
of the roof is 6.0 $\mathrm{m}$ and the coefficient of kinetic friction is
$0.20,$ calculate the speed of the snow chunk when it slides off
the roof. (c) If the edge of the roof is 10.0 $\mathrm{m}$ above ground,
estimate the speed of the snow when it hits the ground.

Kristela Garcia

Kristela Garcia

Numerade Educator

Problem 16

(II) A small box is held in place against a rough vertical wall by
someone pushing on it with a force directed upward at $28^{\circ}$
above the horizontal. The coefficients of static and kinetic
friction between the box and wall are 0.40 and 0.30 , respec-
tively. The box slides down unless the applied force has
magnitude 23 $\mathrm{N}$ . What is the mass of the box?

Abhishek Jana

Numerade Educator

Problem 17

(II) Two crates, of mass 65 $\mathrm{kg}$ and 125 $\mathrm{kg}$ , are in contact and at
rest on a horizontal surface (Fig, $32 ) . \mathrm{A} 650$ -N force is exerted
on the 65 -kg crate. If the coefficient of kinetic friction is $0.18,$
calculate $(a)$ the acceleration of the system, and $(b)$ the force
that each crate exerts on the other. (c) Repeat with the crates
reversed.

Kristela Garcia

Kristela Garcia

Numerade Educator

Problem 18

(II) The crate shown in Fig. 33 lies on a plane tilted at
an angle $\theta=25.0^{\circ}$ to the horizontal, with $\mu_{\mathrm{k}}=0.19$ .
(a) Determine the acceleration of the
crate as it slides down the plane. (b)
If the crate starts from rest 8.15 $\mathrm{m}$ up
the plane from its base, what will be
the crate's speed when it reaches
the bottom of the incline?

Abhishek Jana

Numerade Educator

Problem 19

(II) A crate is given an initial speed of 3.0 $\mathrm{m} / \mathrm{s}$ up the
$25.0^{\circ}$ plane shown in Fig. $33 .(a)$ How far up the plane will it
go? (b) How much time elapses before it returns to its
starting point? Assume $\mu_{\mathrm{k}}=0.17$

Kristela Garcia

Kristela Garcia

Numerade Educator

Problem 20

(II) Two blocks made of different materials connected together
by a thin cord, slide down a plane ramp inclined at an angle $\theta$
to the horizontal as shown in Fig. 34 (block $B$ is above
block $A$ ). The masses of the blocks are $m_{A}$ and $m_{B},$ and the
coefficients of friction are $\mu_{\mathrm{A}}$ and $\mu_{\mathrm{B}} .$ If $m_{\mathrm{A}}=m_{\mathrm{B}}=5.0 \mathrm{kg}$
and $\mu_{\mathrm{A}}=0.20$ and $\mu_{\mathrm{B}}=0.30,$ deter-
mine $(a)$ the acceleration of the
blocks and $(b)$ the tension in
the cord, for an angle
$\theta=32^{\circ}$

Donald Albin

Numerade Educator

Problem 21

(II) For two blocks, connected by a cord and sliding down
the incline shown in Fig. 34 (see Problem 20$)$ , describe the
motion $(a)$ if $\mu_{A}<\mu_{B},$ and $(b)$ if $\mu_{A}>\mu_{B} .(c)$ Determine
a formula for the acceleration of each block and the tension
$F_{\mathrm{T}}$ in the cord in terms of $m_{\mathrm{A}}, m_{\mathrm{B}},$ and $\theta ;$ interpret your
results in light of your answers to $(a)$ and $(b) .$

Keshav Singh

Numerade Educator

Problem 22

(II) A flatbed truck is carrying a heavy crate. The coefficient
of static friction between the crate and the bed of the truck
is $0.75 .$ What is the maximum rate at which the driver can
decelerate and still avoid having the crate slide against the
cab of the truck?

Abhishek Jana

Numerade Educator

Problem 23

(II) In Fig. 35 the coefficient of static friction between mass
$m_{\mathrm{A}}$ and the table is $0.40,$ whereas the coefficient of kinetic
friction is 0.30$(a)$ What
minimum value of $m_{A}$
will keep the system
from starting to move?
(b) What value(s) of $m_{A}$
will keep the system
moving at constant
speed?

Meghan Miholics

Meghan Miholics

Numerade Educator

Problem 24

(II) Determine a formula for the acceleration of the system
shown in Fig. 35 in terms of $m_{A}, m_{B},$ and the mass of the
cord, $m_{C} .$ Define any other variables needed.

Abhishek Jana

Numerade Educator

Problem 25

(II) A small block of mass $m$ is given an initial speed $v_{0}$ up
a ramp inclined at angle $\theta$ to the horizontal. It travels a
distance $d$ up the ramp and comes to rest. (a) Determine
a formula for the coefficient of kinetic friction between
block and ramp. (b) What can you say about the value of
the coefficient of static friction?

Kristela Garcia

Kristela Garcia

Numerade Educator

Problem 26

(II) $\mathrm{A}$ 75-kg snowboarder has an initial velocity of
5.0 $\mathrm{m} / \mathrm{s}$ at the top of a $28^{\circ}$ incline (Fig. $36 ) .$ After sliding
down the 110 -m long incline (on which the coefficient of
kinetic friction is $\mu_{k}=0.18$ ), the snowboarder has
attained a velocity $v .$ The snowboarder then slides along
a flat surface (on which $\mu_{k}=0.15$ and comes to rest
after a distance $x .$ Use Newton's second law to find the
snowboarder's acceleration while on the incline and
while on the flat surface. Then use these accelerations to
determine $x .$

Abhishek Jana

Numerade Educator

Problem 27

(II) A package of mass $m$ is dropped vertically onto a hori-
zontal conveyor belt whose speed is $v=1.5 \mathrm{m} / \mathrm{s},$ and the
coefficient of kinetic friction between the package and the
belt is $\mu_{k}=0.70$ . (a) For how much time does the package
slide on the belt (until it is at rest relative to the belt)?
(b) How far does the package move during this time?

Kristela Garcia

Kristela Garcia

Numerade Educator

Problem 28

(II) Two masses $m_{A}=2.0 \mathrm{kg}$ and $m_{B}=5.0 \mathrm{kg}$ are on
inclines and are connected together by a string as shown in
Fig. $37 .$ The coefficient of kinetic friction between each mass
and its incline is $\mu_{\mathrm{k}}=0.30 .$ If $m_{\mathrm{A}}$ moves up, and $m_{\mathrm{B}}$ moves
down, determine their acceleration.

Abhishek Jana

Numerade Educator

Problem 29

(II) A child slides down a slide with a $34^{\circ}$ incline, and at the
bottom her speed is precisely half what it would have been
if the slide had been frictionless. Calculate the coefficient of
kinetic friction between the slide and the child.

Kristela Garcia

Kristela Garcia

Numerade Educator

Problem 30

(II) $(a)$ Suppose the coefficient of kinetic friction between
$m_{\mathrm{A}}$ and the plane in Fig. 38 is $\mu_{\mathrm{k}}=0.15,$ and that
$m_{\mathrm{A}}=m_{\mathrm{B}}=2.7 \mathrm{kg} .$ As $m_{\mathrm{B}}$ moves down, determine the
magnitude of the acceleration of $m_{\mathrm{A}}$ and $m_{\mathrm{B}},$ given
$\theta=34^{\circ} .(b)$ What smallest value of $\mu_{\mathrm{k}}$ will keep the system
from accelerating?

Abhishek Jana

Numerade Educator

Problem 31

(III) A 3.0 -kg block sits on top of a $5.0-\mathrm{kg}$ block which is on
a horizontal surface. The 5.0 -kg block is pulled to the right
with a force $\vec{\mathbf{F}}$ as shown in Fig. $39 .$ The coefficient of static
friction between all surfaces is 0.60 and the kinetic coeffi-
cient is $0.40 .$ (a) What is the minimum value of $F$ needed to
move the two blocks? (b) If the force is 10$\%$ greater than
your answer for $(a),$ what is the acceleration of each block?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 32

(III) A 4.0 -kg block is stacked on top of a 12.0 -kg block,
which is accelerating along a horizontal table at $a=5.2 \mathrm{m} / \mathrm{s}^{2}$
(Fig. $40 ) .$ Let $\mu_{\mathrm{k}}=\mu_{\mathrm{s}}=\mu .$ (a) What minimum coefficient of
friction $\mu$ between the two blocks will prevent the 4.0 -kg
block from sliding off? (b) If $\mu$ is only half this minimum
value, what is the acceleration of the 4.0 -kg block with respect
to the table, and $(c)$ with respect to the 12.0 -kg block?
(d) What is the force
that must be applied to
the 12.0 -kg block in $(a)$
and in $(b),$ assuming that
the table is frictionless?

Abhishek Jana

Numerade Educator

Problem 33

(III) A small block of mass $m$ rests on the rough, sloping side
of a triangular block of mass $M$ which itself rests on a hori-
zontal frictionless table as shown in Fig. $41 .$ If the coefficient
of static friction is $\mu,$ determine the minimum horizontal

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 34

(I) What is the maximum speed with which a 1200 -kg car
can round a turn of radius 80.0 $\mathrm{m}$ on a flat road if the coeffi-
cient of friction between tires and road is 0.65$?$ Is this result
independent of the mass of the car?

Abhishek Jana

Numerade Educator

Problem 35

(I) A child sitting 1.20 $\mathrm{m}$ from the center of a merry-
go-around moves with a speed of 1.30 $\mathrm{m} / \mathrm{s}$ . Calculate
$(a)$ the centripetal acceleration of the child and $(b)$ the
net horizontal force exerted on the child (mass $=22.5 \mathrm{kg} )$ .

Rahul Nikhar

Numerade Educator

Problem 36

(I) A jet plane traveling 1890 $\mathrm{km} / \mathrm{h}(525 \mathrm{m} / \mathrm{s})$ pulls out of a
dive by moving in an arc of radius 4.80 $\mathrm{km} .$ What is the
plane's acceleration in $g^{\prime} s ?$

Abhishek Jana

Numerade Educator

Problem 37

(II) Is it possible to whirl a bucket of water fast enough in a
vertical circle so that the water won't fall out? If so, what is
the minimum speed? Define all quantities needed.

Kristela Garcia

Kristela Garcia

Numerade Educator

Problem 38

(II) How fast (in rpm) must a centrifuge rotate if a particle
8.00 $\mathrm{cm}$ from the axis of rotation is to experience an acceler-
ation of $125,000 g^{\prime} s ?$

Abhishek Jana

Numerade Educator

Problem 39

(II) Highway curves are marked with a suggested speed. If
this speed is based on what would be safe in wet weather,
estimate the radius of curvature for a curve marked 50 $\mathrm{km} / \mathrm{h}$ .
Use Table $1 .$

Kristela Garcia

Kristela Garcia

Numerade Educator

Problem 40

(II) At what minimum
speed must a roller
coaster be traveling
when upside down at
the top of a circle
(Fig. 42$)$ so that the
passengers do that fall
out? Assume a radius
of curvature of 7.6 $\mathrm{m} .$

Abhishek Jana

Numerade Educator

Problem 41

(II) A sports car crosses the bottom of a valley with a radius
of curvature equal to 95 $\mathrm{m}$ . At the very bottom, the normal
force on the driver is twice his weight. At what speed was
the car traveling?

Kristela Garcia

Kristela Garcia

Numerade Educator

Problem 42

(II) How large must the coefficient of static friction be
between the tires and the road if a car is to round a level
curve of radius 85 $\mathrm{m}$ at a speed of 95 $\mathrm{km} / \mathrm{h} ?$

Abhishek Jana

Numerade Educator

Problem 43

(II) Suppose the space shuttle is in orbit 400 $\mathrm{km}$ from the
Earth's surface, and circles the Earth about once every
90 min. Find the centripetal acceleration of the space
shuttle in its orbit. Express your answer in terms of $g,$ the
gravitational acceleration at the Earth's surface.

Kristela Garcia

Kristela Garcia

Numerade Educator

Problem 44

(II) A bucket of mass 2.00 $\mathrm{kg}$ is whirled in a vertical circle of
radius 1.10 $\mathrm{m}$ . At the lowest point of its motion the tension
in the rope supporting the bucket is 25.0 $\mathrm{N}$ . (a) Find the
speed of the bucket. (b) How fast must the bucket move at
the top of the circle so that the rope does not go slack?

Abhishek Jana

Numerade Educator

Problem 45

(II) How many revolutions per minute would a $22-\mathrm{m}-$
diameter Ferris wheel need to make for the passengers to
feel "weightless" at the topmost point?

Shoukat Ali

Other Schools

Problem 46

(II) Use dimensional analysis to obtain the form for the
centripetal acceleration, $a_{\mathrm{R}}=v^{2} / r$ .

Abhishek Jana

Numerade Educator

Problem 47

(II) A jet pilot takes his aircraft in a vertical loop
(Fig. $43 ) .(a)$ If the jet is moving at a speed of 1200 $\mathrm{km} / \mathrm{h}$ at
the lowest point of the loop, determine the minimum radius
of the circle so that the centripetal acceleration at the
lowest point does not exceed 6.0 $\mathrm{g}^{\prime}$ s. $(b)$ Calculate the 78 -kg
pilot's effective weight (the force with which the seat pushes
up on him at the bottom of the circle, and $(c)$ at the top of
the circle (assume the same speed).

Kristela Garcia

Kristela Garcia

Numerade Educator

Problem 48

(II) A proposed space station consists of a circular tube that
will rotate about its center (like a tubular bicycle tire),
Fig. $44 .$ The circle formed by the tube has a diameter of
about 1.1 $\mathrm{km} .$ What must be the rotation speed (revolutions
per day if an effect equal to
gravity at the surface of the
Earth $(1.0 g)$ is to be felt?

Abhishek Jana

Numerade Educator

Problem 49

(II) On an ice rink two skaters of equal mass grab hands
and spin in a mutual circle once every 2.5 $\mathrm{s}$ . If we assume
their arms are each 0.80 $\mathrm{m}$ long and their individual masses
are $60.0 \mathrm{kg},$ how hard are they pulling on one another?

Kristela Garcia

Kristela Garcia

Numerade Educator

Problem 50

(II) Redo Example 11 of "Using Newton's Laws: Friction,
Circular Motion, Drag Forces", precisely this time, by not
ignoring the weight of the ball which revolves on a string
0.600 $\mathrm{m}$ long. In particular, find the magnitude of $\vec{\mathbf{F}}_{\mathrm{T}},$ and the
angle it makes with the horizontal. [Hint: Set the horizontal
component of $\vec{\mathbf{F}}_{\mathrm{T}}$ equal to $\mathrm{ma}_{\mathrm{R}} ;$ also, since there is no vertical
motion, what can you say about the vertical component of
$\vec{\mathbf{F}}_{\mathrm{T}} ? ]$

Abhishek Jana

Numerade Educator

Problem 51

(II) A coin is placed 12.0 $\mathrm{cm}$ from the axis of a rotating
turntable of variable speed. When the speed of the turntable
is slowly increased, the coin remains fixed on the turntable
until a rate of 35.0 rpm (revolutions per minute) is reached,
at which point the coin slides off. What is the coefficient of
static friction between the coin and the turntable?

Kristela Garcia

Kristela Garcia

Numerade Educator

Problem 52

(II) The design of a new road includes a straight stretch that
is horizontal and flat but that suddenly dips down a steep
hill at $22^{\circ} .$ The
transition should
be rounded with
what minimum
radius so that cars
traveling 95 $\mathrm{km} / \mathrm{h}$
will not leave the
road (Fig. 45$) ?$

Abhishek Jana

Numerade Educator

Problem 53

(II) A $975-\mathrm{kg}$ sports car (including driver) crosses the
rounded top of a hill (radius = 88.0 $\mathrm{m} )$ at 12.0 $\mathrm{m} / \mathrm{s}$ .
Determine $(a)$ the normal force exerted by the road on the
car, $(b)$ the normal force exerted by the car on the $72.0-\mathrm{kg}$
driver, and $(c)$ the car speed at which the normal force on
the driver equals zero.

Kristela Garcia

Kristela Garcia

Numerade Educator

Problem 54

(II) Two blocks, with masses $m_{A}$ and $m_{B},$ are connected to
each other and to a central post by cords as shown in
Fig. $46 .$ They rotate about the post at frequency $f$
(revolutions per second) on a frictionless horizontal surface
at distances $r_{A}$ and $r_{B}$ from the post. Derive an algebraic
expression for the tension in each segment of the cord
(assumed massless).

Abhishek Jana

Numerade Educator

Problem 55

(II) Tarzan plans to cross a gorge by swinging in an arc from
a hanging vine (Fig. $47 ) .$ If his arms are capable of exerting
a force of 1350 $\mathrm{N}$ on the rope,
what is the maximum speed he
can tolerate at the lowest point
of his swing? His mass is 78 $\mathrm{kg}$
and the vine is 5.2 $\mathrm{m}$ long.

Kristela Garcia

Kristela Garcia

Numerade Educator

Problem 56

(II) A pilot performs an evasive maneuver by diving verti-
cally at 310 $\mathrm{m} / \mathrm{s}$ . If he can withstand an acceleration of
9.0 $\mathrm{g}$ s without blacking out, at what altitude must he begin
to pull out of the dive to avoid crashing into the sea?

Abhishek Jana

Numerade Educator

Problem 57

(III) The position of a particle moving in the $x y$ plane is
given by $\quad \vec{\mathbf{r}}=2.0 \cos (3.0 \mathrm{rad} / \mathrm{s} t) \hat{\mathbf{i}}+2.0 \sin (3.0 \mathrm{rad} / \mathrm{s} t) \hat{\mathbf{j}},$
where $r$ is in meters and $t$ is in seconds. $(a)$ Show that this
represents circular motion of radius 2.0 $\mathrm{m}$ centered at the
origin. (b) Determine the velocity and acceleration vectors as
functions of time. (c) Determine the speed and magnitude of
the acceleration. $(d)$ Show that $a=v^{2} / r .$ (e) Show that the
acceleration vector always points toward the center of the
circle.

DF

Dr. Frank

Numerade Educator

Problem 58

(III) If a curve with a radius of 85 $\mathrm{m}$ is properly banked for a
car traveling 65 $\mathrm{km} / \mathrm{h}$ , what must be the coefficient of static
friction for a car not to skid when traveling at 95 $\mathrm{km} / \mathrm{h} ?$

Abhishek Jana

Numerade Educator

Problem 59

(III) A curve of radius 68 $\mathrm{m}$ is banked for a design speed of
85 $\mathrm{km} / \mathrm{h} .$ If the coefficient of static friction is 0.30 (wet pave-
ment), at what range of speeds can a car safely make the
curve? [Hint: Consider the direction of the friction force
when the car goes too slow or too fast.]

Vishal Gupta

Numerade Educator

Problem 60

(II) A particle starting from rest revolves with uniformly
increasing speed in a clockwise circle in the $x y$ plane. The
center of the circle is at the origin of an $x y$ coordinate
system. At $t=0,$ the particle is at $x=0.0, y=2.0 \mathrm{m} .$ At
$t=2.0 \mathrm{s},$ it has made one-quarter of a revolution and is at
$x=2.0 \mathrm{m}, y=0.0 .$ Determine $(a)$ its speed at $t=2.0 \mathrm{s}$
(b) the average velocity vector, and $(c)$ the average acceler-
ation vector during this interval.

Abhishek Jana

Numerade Educator

Problem 61

(II) In Problem 60 assume the tangential acceleration is
constant and determine the components of the instantaneous
acceleration at $(a) t=0.0,(b) t=1.0 \mathrm{s},$ and $(c) t=2.0 \mathrm{s}$

Check back soon!

Problem 62

(II) An object moves in a circle of radius 22 $\mathrm{m}$ with its speed
given by $v=3.6+1.5 t^{2},$ with $v$ in meters per second and $t$
in seconds. At $t=3.0 \mathrm{s},$ find $(a)$ the tangential acceleration
and $(b)$ the radial acceleration.

Abhishek Jana

Numerade Educator

Problem 63

(III) A particle rotates in a circle of radius 3.80 $\mathrm{m}$ . At a
particular instant its acceleration is 1.15 $\mathrm{m} / \mathrm{s}^{2}$ in a direction
that makes an angle of $38.0^{\circ}$ to its direction of motion.
Determine its speed $(a)$ at this moment and $(b) 2.00$ s later,
assuming constant tangential acceleration.

Kristela Garcia

Kristela Garcia

Numerade Educator

Problem 64

(III) An object of mass $m$ is constrained to move in a circle of
radius $r .$ Its tangential acceleration as a function of time is given
by $a_{\text { tan }}=b+c t^{2},$ where $b$ and $c$ are constants. If $v=v_{0}$ at
$t=0,$ determine the tangential and radial components of the
force, $F_{\text { tan }}$ and $F_{\mathrm{R}},$ acting on the object at any time $t>0$

Abhishek Jana

Numerade Educator

Problem 65

(I) Use dimensional analysis in Example 17 of "Using
Newton's Laws: Friction, Circular Motion, Drag Forces" to
determine if the time constant $\tau$ is $\tau=m / b$ or $\tau=b / m$ .

Check back soon!

Problem 66

(II) The terminal velocity of a $3 \times 10^{-5} \mathrm{kg}$ raindrop is about
9 $\mathrm{m} / \mathrm{s} .$ Assuming a drag force $F_{\mathrm{D}}=-b v,$ determine $(a)$ the
value of the constant $b$ and $(b)$ the time required for such a
drop, starting from rest, to reach 63$\%$ of terminal velocity.

Abhishek Jana

Numerade Educator

Problem 67

(II) An object moving vertically has $\vec{\mathbf{v}}=\vec{\mathbf{v}}_{0}$ at $t=0$ .
Determine a formula for its velocity as a function of time
assuming a resistive force $F=-b v$ as well as gravity for
two cases: $(a) \vec{\mathbf{v}}_{0}$ is downward and $(b) \vec{\mathbf{v}}_{0}$ is upward.

Meghan Miholics

Meghan Miholics

Numerade Educator

Problem 68

(III) The drag force on large objects such as cars, planes, and
sky divers moving through air is more nearly $F_{D}=-b v^{2}$ .
(a) For this quadratic dependence on $v,$ determine a
formula for the terminal velocity $v_{\mathrm{T}},$ of a vertically falling
object. $(b)$ A 75 -kg sky diver has a terminal velocity of
about $60 \mathrm{m} / \mathrm{s} ;$ determine the value of the constant $b$ .
(c) Sketch a curve like that of Fig. 27 $\mathrm{b}$ for this case of
$F_{\mathrm{D}} \propto v^{2} .$ For the same terminal velocity, would this curve lie
above or below that in Fig. 27$?$ Explain why.

Abhishek Jana

Numerade Educator

Problem 69

(III) A bicyclist can coast down a $7.0^{\circ}$ hill at a steady
9.5 $\mathrm{km} / \mathrm{h}$ . If the drag force is proportional to the square of
the speed $v,$ so that $F \mathrm{D}=-c v^{2},$ calculate $(a)$ the value of
the constant $c$ and $(b)$ the average force that must be applied
in order to descend the hill at 25 $\mathrm{km} / \mathrm{h}$ . The mass of the
cyclist plus bicycle is 80.0 $\mathrm{kg}$ . Ignore other types of friction.

Kristela Garcia

Kristela Garcia

Numerade Educator

Problem 70

(III) Two drag forces act on a bicycle and rider: $F_{\mathrm{D} 1}$ due to
rolling resistance, which is essentially velocity independent;
and $F_{\mathrm{D} 2}$ due to air resistance, which is proportional to $v^{2}$ .
For a specific bike plus rider of total mass 78 $\mathrm{kg}$ ,
$F_{\mathrm{D} 1} \approx 4.0 \mathrm{N}$ ; and for a speed of $2.2 \mathrm{m} / \mathrm{s}, F_{\mathrm{D} 2} \approx 1.0 \mathrm{N}$
$$\begin{array}{c}{\text { (a) Show that the total drag force is }} \\ {F_{\mathrm{D}}=4.0+0.21 v^{2}}\end{array}$$
where $v$ is in $\mathrm{m} / \mathrm{s},$ and $F_{\mathrm{D}}$ is in $\mathrm{N}$ and opposes the motion.
$$\begin{array}{l}{\text { (b) Determine at what slope angle } \theta \text { the bike and rider can }} \\ {\text { coast downhill at a constant speed of } 8.0 \mathrm{m} / \mathrm{s} \text { . }}\end{array}$$

Abhishek Jana

Numerade Educator

Problem 71

(III) Determine a formula for the position and acceleration
of a falling object as a function of time if the object starts
from rest at $t=0$ and undergoes a resistive force
$F=-b v,$ as in Example 17 of "Using Newton's Laws:
Friction, Circular Motion, Drag Forces".

Kristela Garcia

Kristela Garcia

Numerade Educator

Problem 72

(III) A block of mass $m$ slides along a horizontal surface
lubricated with a thick oil which provides a drag force
proportional to the square root of velocity:
$$F_{\mathrm{D}}=-b v^{\frac{1}{2}}$$
If $v=v_{0}$ at $t=0,$ determine $v$ and $x$ as functions of
time.

Abhishek Jana

Numerade Educator

Problem 73

(III) Show that the maximum distance the block in Problem 72
can travel is 2$m v_{0}^{3 / 2} / 3 b .$

Check back soon!

Problem 74

(III) You dive straight down into a pool of water. You hit the
water with a speed of $5.0 \mathrm{m} / \mathrm{s},$ and your mass is 75 $\mathrm{kg}$ . Assuming
a drag force of the form $F_{\mathrm{D}}=-\left(1.00 \times 10^{4} \mathrm{kg} / \mathrm{s}\right) v,$ how
long does it take you to reach 2$\%$ of your original speed?
(Ignore any effects of buoyancy.)

Abhishek Jana

Numerade Educator

Problem 75

(III) A motorboat traveling at a speed of 2.4 $\mathrm{m} / \mathrm{s}$ shuts off
its engines at $t=0 .$ How far does it travel before coming
to rest if it is noted that after 3.0 $\mathrm{s}$ its speed has dropped to
half its original value? Assume that the drag force of the
water is proportional to $v$ .

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 76

A coffee cup on the horizontal dashboard of a car slides
forward when the driver decelerates from 45 $\mathrm{km} / \mathrm{h}$ to rest
in 3.5 $\mathrm{s}$ or less, but not if she decelerates in a longer time.
What is the coefficient of static friction between the cup
and the dash? Assume the road and the dashboard are
level (horizontal).

Abhishek Jana

Numerade Educator

Problem 77

A 2.0 -kg silverware drawer does not slide readily. The
owner gradually pulls with more and more force, and when
the applied force reaches $9.0 \mathrm{N},$ the drawer suddenly
opens, throwing all the utensils to the floor. What is the
coefficient of static friction between the drawer and the
cabinet?

Kristela Garcia

Kristela Garcia

Numerade Educator

Problem 78

A roller coaster reaches the top of the steepest hill with a
speed of 6.0 $\mathrm{km} / \mathrm{h}$ . It then descends the hill, which is at an
average angle of $45^{\circ}$ and is 45.0 $\mathrm{m}$ long. What will its
speed be when it reaches the bottom? Assume $\mu_{\mathrm{k}}=0.12$ .

Abhishek Jana

Numerade Educator

Problem 79

An $18.0-\mathrm{kg}$ box is released on a $37.0^{\circ}$ incline and accelerates
down the incline at 0.220 $\mathrm{m} / \mathrm{s}^{2} .$ Find the friction force
impeding its motion. How large is the coefficient of friction?

Kristela Garcia

Kristela Garcia

Numerade Educator

Problem 80

A flat puck (mass $M$ ) is revolved in a circle on a frictionless
air hockey table top, and is held in this orbit by a light cord
which is connected to a dangling mass (mass $m )$ through a
central hole as shown in Fig. 48 . Show that the speed of the
puck is given by $v=\sqrt{m g R / M} .$

Abhishek Jana

Numerade Educator

Problem 81

A motorcyclist is coasting with the engine off at a steady
speed of 20.0 $\mathrm{m} / \mathrm{s}$ but enters a sandy stretch where the coeffi-
cient of kinetic friction is $0.70 .$ Will the cyclist emerge from the
sandy stretch without having to start the engine if the sand
lasts for 15 $\mathrm{m}$ ? If so, what will be the speed upon emerging?

Kristela Garcia

Kristela Garcia

Numerade Educator

Problem 82

In a "Rotor-ride" at a carnival, people rotate in a vertical
cylindrically walled "room." (See Fig. 49). If the room radius
was 5.5 $\mathrm{m}$ , and the rotation frequency 0.50 revolutions per
second when the floor drops out, what minimum coefficient
of static friction keeps the people from slipping down?
People on this ride said they were "pressed against the
wall." Is there really an outward force pressing them against the
the wall? If so, what is its source? If not, what is the proper
description of their situation (besides nausea)? [Hint: Draw
a free-body diagram for a person.]

Abhishek Jana

Numerade Educator

Problem 83

A device for training astronauts and jet fighter pilots is
designed to rotate the trainee in a horizontal circle of radius
11.0 $\mathrm{m} .$ If the force felt by the trainee is 7.45 times her own
weight, how fast is she rotating? Express your answer in
both $\mathrm{m} / \mathrm{s}$ and rev/s.

Kristela Garcia

Kristela Garcia

Numerade Educator

Problem 84

A $1250-\mathrm{kg}$ car rounds a curve of radius 72 $\mathrm{m}$ banked at an
angle of $14^{\circ} .$ If the car is traveling at 85 $\mathrm{km} / \mathrm{h}$ , will a friction
force be required? If so, how much and in what direction?

Abhishek Jana

Numerade Educator

Problem 85

Determine the tangential and centripetal components of the
net force exerted on a car (by the ground) when its speed is
$27 \mathrm{m} / \mathrm{s},$ and it has accelerated to this speed from rest in 9.0 $\mathrm{s}$
on a curve of radius 450 $\mathrm{m} .$ The car's mass is 1150 $\mathrm{kg}$ .

Kristela Garcia

Kristela Garcia

Numerade Educator

Problem 86

The 70.0 -kg climber in Fig. 50 is supported in the "chimney"
by the friction forces exerted on his
shoes and back. The static coeffi-
cients of friction between his shoes
and the wall, and between his back
and the wall, are 0.80 and 0.60,
respectively. What is the minimum
normal force he must exert?
Assume the walls are vertical and
that the static friction forces are
both at their maximum. Ignore his
grip on the rope.

Abhishek Jana

Numerade Educator

Problem 87

A small mass $m$ is set on the surface of a sphere, Fig. $51 .$ If
the coefficient of static fric-
tion is $\mu_{s}=0.70,$ at what
angle $\phi$ would the mass
start sliding?

Kristela Garcia

Kristela Garcia

Numerade Educator

Problem 88

A 28.0 -kg block is connected to an empty 2.00 -kg bucket by a
cord running over a frictionless pulley (Fig. $52 ) .$ The coefficient
of static friction between the table and the block is 0.45 and the
coefficient of kinetic friction between the table and the block
is 0.32 . Sand is gradually
added to the bucket until the
system just begins to move.
(a) Calculate the mass of
sand added to the bucket.
(b) Calculate the acceleration
of the system.

Abhishek Jana

Numerade Educator

Problem 89

A car is heading down a slippery road at a speed of 95 $\mathrm{km} / \mathrm{h}$ .
The minimum distance within which it can stop without
skidding is 66 $\mathrm{m} .$ What is the sharpest curve the car can
negotiate on the icy surface at the same speed without
skidding?

Kristela Garcia

Kristela Garcia

Numerade Educator

Problem 90

What is the acceleration experienced by the tip of the
1.5 -cm-long sweep second hand on your wrist watch?

Abhishek Jana

Numerade Educator

Problem 91

An airplane traveling at 480 $\mathrm{km} / \mathrm{h}$ needs to reverse its course.
The pilot decides to accomplish this by banking the wings at
an angle of $38^{\circ} .(a)$ Find the time needed to reverse course.
(b) Describe any additional force the
passengers experience during the
turn. [Hint: Assume an aerodynamic
"lift" force that acts perpendicularly
to the flat wings; see Fig. $53 . ]$

Kristela Garcia

Kristela Garcia

Numerade Educator

Problem 92

A banked curve of radius $R$ in a new highway is designed so
that a car traveling at speed $v_{0}$ can negotiate the turn safely
on glare ice (zero friction). If a car travels too slowly, then it
will slip toward the center of the circle. If it travels too fast,
it will slip away from the center of the circle. If the
coefficient of static friction increases, it becomes possible for
a car to stay on the road while traveling at a speed within
a range from $v_{\min }$ to $v_{\text { max }}$ . Derive formulas for $v_{\text { min }}$ and
$v_{\text { max }}$ as functions of $\mu_{\mathrm{s}}, v_{0},$ and $R.$

Abhishek Jana

Numerade Educator

Problem 93

A small bead of mass $m$ is constrained to slide without
friction inside a circular vertical hoop of radius $r$ which
rotates about a vertical axis
(Fig. 54 at a frequency $f$ .
(a) Determine the angle $\theta$
where the bead will be in
equilibrium - that is, where
it will have no tendency to
move up or down along the
hoop. $(b)$ If $f=2.00 \mathrm{rev} / \mathrm{s}$
and $r=22.0 \mathrm{cm},$ what is $\theta$ ?
(c) Can the bead ride as
high as the center of the
circle $\left(\theta=90^{\circ}\right) ?$ Explain.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 94

Earth is not quite an inertial frame. We often make measure-
ments in a reference frame fixed on the Earth, assuming
Earth is an inertial reference frame. But the Earth rotates, so
this assumption is not quite valid. Show that this assumption
is off by 3 parts in 1000 by calculating the acceleration of an
object at Earth's equator due to Earth's daily rotation, and
compare to $g=9.80 \mathrm{m} / \mathrm{s}^{2},$ the acceleration due to gravity.

Abhishek Jana

Numerade Educator

Problem 95

While fishing, you get bored and start to swing a sinker weight
around in a circle below you on a 0.45 -m piece of fishing line.
The weight makes a complete circle every 0.50 s. What is the
angle that the fishing line makes with the vertical?

Kristela Garcia

Kristela Garcia

Numerade Educator

Problem 96

Consider a train that rounds a curve with a radius of 570 $\mathrm{m}$
at a speed of 160 $\mathrm{km} / \mathrm{h}$ (approximately 100 $\mathrm{mi} / \mathrm{h} ) .(a)$ Calcu-
late the friction force needed on a train passenger of mass
75 $\mathrm{kg}$ if the track is not banked and the train does not tilt.
(b) Calculate the friction force on the passenger if the train
tilts at an angle of $8.0^{\circ}$ toward the center of the curve.

Abhishek Jana

Numerade Educator

Problem 97

A car starts rolling down a $1-\mathrm{in}-4$ hill $(1-\mathrm{in}-4$ means that for
each 4 $\mathrm{m}$ traveled along the road, the elevation change is
1 $\mathrm{m} ) .$ How fast is it going when it reaches the bottom after
traveling 55 $\mathrm{m}$ ? (a) Ignore friction. (b) Assume an effective
coefficient of friction equal to $0.10 .$

Kristela Garcia

Kristela Garcia

Numerade Educator

Problem 98

The sides of a cone make an angle $\phi$ with the vertical. A
small mass $m$ is placed on the inside of the cone and the cone,
with its point down, is revolved at a frequency $f$ (revolutions
per second) about its symmetry axis. If the coefficient of static
friction is $\mu_{s},$ at what positions on the cone can the mass be
placed without sliding on the cone? (Give the maximum and
minimum distances, $r$ , from the axis).

Abhishek Jana

Numerade Educator

Problem 99

A 72 -kg water skier is being accelerated by a ski boat on a
flat $($ "glassy") lake. The coefficient of kinetic friction
between the skier's skis and the water surface is $\mu_{k}=0.25$
(Fig. 55 ). (a) What is the skier's acceleration if the rope
pulling the skier behind the boat applies a horizontal tension
force of magnitude $F_{T}=240 \mathrm{N}$ to the skier $\left(\theta=0^{\circ}\right) ?$
(b) What is the skier's horizontal acceleration if the rope
pulling the skier exerts a force of $F_{T}=240 \mathrm{N}$ on the skier
at an upward angle $\theta=12^{\circ} ?$ (c) Explain why the skier's
acceleration in part $(b)$ is greater than that in part $(a)$ .

Kristela Garcia

Kristela Garcia

Numerade Educator

Problem 100

A ball of mass $m=1.0 \mathrm{kg}$ at the end of a thin cord of length
$r=0.80 \mathrm{m}$ revolves in a vertical circle about point $\mathrm{O},$ as
shown in Fig. $56 .$ During the time we observe it, the only
forces acting on the ball are gravity and the tension in the
cord. The motion is circular but not uniform because of the
force of gravity. The ball increases in speed as it descends and
decelerates as it rises on the other side of the circle. At the
moment the cord makes an angle $\theta=30^{\circ}$ below the
horizontal, the ball's
speed is 6.0 $\mathrm{m} / \mathrm{s}$ . At
this point, determine
the tangential accel-
eration, the radial
acceleration, and the
tension in the cord,
$F_{\mathrm{T}} .$ Take $\theta$ increasing
downward as shown.

Abhishek Jana

Numerade Educator

Problem 101

A car drives at a constant speed around a banked circular
track with a diameter of 127 $\mathrm{m}$ . The motion of the car can
be described in a coordinate system with its origin at the
center of the circle. At a particular instant the car's accel-
eration in the horizontal plane is given by
$$\vec{\mathbf{a}}=(-15.7 \hat{\mathbf{i}}-23.2 \hat{\mathbf{j}}) \mathrm{m} / \mathrm{s}^{2}$$
(a) What is the car's speed? (b) Where $(x$ and $y)$ is the car
at this instant?

Kristela Garcia

Kristela Garcia

Numerade Educator

Problem 102

(III) The force of air resistance (drag force) on a rapidly
falling body such as a skydiver has the form $F_{\mathrm{D}}=-k v^{2},$ so
that Newton's second law applied to such an object is
$$m \frac{d v}{d t}=m g-k v^{2},$$
where the downward direction is taken to be positive.
(a) Use numerical integration to estimate (within 2$\% )$ the
position, speed, and acceleraton, from $t=0$ up to
$t=15.0 \mathrm{s},$ for a $75-\mathrm{kg}$ skydiver who starts from rest,
assuming $k=0.22 \mathrm{kg} / \mathrm{m} .$ (b) Show that the diver eventually
reaches a steady speed, the terminal speed, and explain why
this happens. (c) How long does it take for the skydiver to
reach 99.5$\%$ of the terminal speed?

Abhishek Jana

Numerade Educator

Problem 103

(III) The coefficient of kinetic friction $\mu_{\mathrm{k}}$ between two
surfaces is not strictly independent of the velocity of the
object. A possible expression for $\mu_{\mathrm{k}}$ for wood on wood is
$$\mu_{\mathrm{k}}=\frac{0.20}{\left(1+0.0020 v^{2}\right)^{2}},$$
where $v$ is in $\mathrm{m} / \mathrm{s} .$ A wooden block of mass 8.0 $\mathrm{kg}$ is at rest
on a wooden floor, and a constant horizontal force of 41 $\mathrm{N}$
acts on the block. Use numerical integration to determine
and graph $(a)$ the speed of the block, and $(b)$ its position, as
a function of time from 0 to 5.0 $\mathrm{s}$ (c) Determine the
percent difference for the speed and position at 5.0 $\mathrm{s}$
if $\mu_{\mathrm{k}}$ is constant and equal to $0.20 .$

Check back soon!

Problem 104

(III) Assume a net force $F=-m g-k v^{2}$ acts during the
upward vertical motion of a $250-\mathrm{kg}$ rocket, starting at
the moment $(t=0)$ when the fuel has burned out and the
rocket has an upward speed of 120 $\mathrm{m} / \mathrm{s}$ . Let $k=0.65 \mathrm{kg} / \mathrm{m}$ .
Estimate $v$ and $y$ at 1.0 -s intervals for the upward motion
only, and estimate the maximum height reached. Compare
to free-flight conditions without air resistance $(k=0)$

Abhishek Jana

Numerade Educator