4.1 The Concepts of Force and Mass

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Dynamics of Uniform Circular Motion
Chapter 5
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Learning Objectives- Circular motion and
rotation

• Uniform circular motion
• Students should understand the uniform circular
  motion of a particle, so they can
• Relate the radius of the circle and the speed or
  rate of revolution of the particle to the
  magnitude of the centripetal acceleration.
• Describe the direction of the particles velocity
  and acceleration at any instant during the
  motion.
• Determine the components of the velocity and
  acceleration vectors at any instant, and sketch
  or identify graphs of these quantities.
• Analyze situations in which an object moves with
  specified acceleration under the influence of one
  or more forces so they can determine the
  magnitude and direction of the net force, or of
  one of the forces that makes up the net force, in
  situations such as the following
• Motion in a horizontal circle (e.g., mass on a
  rotating merry-go-round, or car rounding a banked
  curve).
• Motion in a vertical circle (e.g., mass swinging
  on the end of a string, cart rolling down a
  curved track, rider on a Ferris wheel).

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Table Of Contents
5.1 Uniform Circular Motion 5.2 Centripetal
Acceleration 5.3 Centripetal Force 5.4 Banked
Curves 5.5 Satellites in Circular Orbits 5.6
Apparent Weightlessness and Artificial
Gravity 5.7 Vertical Circular Motion
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Chapter 5Dynamics of Uniform Circular Motion

• Section 1
• Uniform Circular Motion

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Other Effects of Forces

• Up until now, weve focused on forces that speed
  up or slow down an object.
• We will now look at the third effect of a force
• Turning
• We need some other equations as the object will
  be accelerating without necessarily changing
  speed.

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DEFINITION OF UNIFORM CIRCULAR MOTION
Uniform circular motion is the motion of an
object traveling at a constant speed on a
circular path.
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Let T be the time it takes for the object
to travel once around the circle.
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Example 1 A Tire-Balancing Machine The wheel
of a car has a radius of 0.29m and it being
rotated at 830 revolutions per minute on a
tire-balancing machine. Determine the speed at
which the outer edge of the wheel is moving.
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Newtons Laws

• 1st
• When objects move along a straight line the
  sideways/perpendicular forces must be balanced.
• 2nd
• When the forces directed perpendicular to
  velocity become unbalanced the object will curve.
• 3rd
• The force that pulls inward on the object,
  causing it to curve off line provides the action
  force that is centripetal in nature. The object
  will in return create a reaction force that is
  centrifugal in nature.

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5.1.1. An airplane flying at 115 m/s due east
makes a gradual turn while maintaining its speed
and follows a circular path to fly south. The
turn takes 15 seconds to complete. What is the
radius of the circular path? a) 410 m b) 830
m c) 1100 m d) 1600 m e) 2200 m
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Chapter 5Dynamics of Uniform Circular Motion

• Section 2
• Centripetal Acceleration

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In uniform circular motion, the speed is
constant, but the direction of the velocity
vector is not constant.
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The direction of the centripetal acceleration is
towards the center of the circle in the same
direction as the change in velocity.
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Conceptual Example 2 Which Way Will the Object
Go? An object is in uniform circular motion.
At point O it is released from its circular
path. Does the object move along the
straight path between O and A or along the
circular arc between points O and P ?
Straight path
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Example 3 The Effect of Radius on Centripetal
Acceleration The bobsled track contains turns
with radii of 33 m and 24 m. Find the
centripetal acceleration at each turn for a
speed of 34 m/s. Express answers as multiples
of
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5.2.1. A ball is whirled on the end of a string
in a horizontal circle of radius R at constant
speed v. By which one of the following means can
the centripetal acceleration of the ball be
increased by a factor of two? a) Keep the
radius fixed and increase the period by a factor
of two. b) Keep the radius fixed and decrease
the period by a factor of two. c) Keep the
speed fixed and increase the radius by a factor
of two. d) Keep the speed fixed and decrease
the radius by a factor of two. e) Keep the
radius fixed and increase the speed by a factor
of two.
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5.2.2. A steel ball is whirled on the end of a
chain in a horizontal circle of radius R with a
constant period T. If the radius of the circle
is then reduced to 0.75R, while the period
remains T, what happens to the centripetal
acceleration of the ball? a) The centripetal
acceleration increases to 1.33 times its initial
value. b) The centripetal acceleration
increases to 1.78 times its initial value. c)
The centripetal acceleration decreases to 0.75
times its initial value. d) The centripetal
acceleration decreases to 0.56 times its initial
value. e) The centripetal acceleration does not
change.
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5.2.3. While we are in this classroom, the Earth
is orbiting the Sun in an orbit that is nearly
circular with an average radius of 1.50 1011 m.
Assuming that the Earth is in uniform circular
motion, what is the centripetal acceleration of
the Earth in its orbit around the Sun? a) 5.9
10?3 m/s2 b) 1.9 10?5 m/s2 c) 3.2 10?7
m/s2 d) 7.0 10?2 m/s2 e) 9.8 m/s2
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5.2.4. A truck is traveling with a constant speed
of 15 m/s. When the truck follows a curve in the
road, its centripetal acceleration is 4.0 m/s2.
What is the radius of the curve? a) 3.8 m b)
14 m c) 56 m d) 120 m e) 210 m
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5.2.5. Consider the following situations (i) A
minivan is following a hairpin turn on a mountain
road at a constant speed of twenty miles per
hour. (ii) A parachutist is descending at a
constant speed 10 m/s. (iii) A heavy crate has
been given a quick shove and is now sliding
across the floor. (iv) Jenny is swinging back
and forth on a swing at the park. (v) A football
that was kicked is flying through the goal
posts. (vi) A plucked guitar string vibrates at
a constant frequency. In which one of these
situations does the object or person experience
zero acceleration? a) i only b) ii only c)
iii and iv only d) iv, v, and vi only e) all
of the situations
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Chapter 5Dynamics of Uniform Circular Motion

• Section 3
• Centripetal Force

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Recall Newtons Second Law
When a net external force acts on an object of
mass m, the acceleration that results is
directly proportional to the net force and has a
magnitude that is inversely proportional to the
mass. The direction of the acceleration is the
same as the direction of the net force.
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Recall Newtons Second Law

• Thus, in uniform circular motion there must be a
  net force to produce the centripetal
  acceleration.
• The centripetal force is the name given to the
  net force required to keep an object moving on a
  circular path.
• The direction of the centripetal force always
  points toward the center of the circle and
  continually changes direction as the object
  moves.

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Problem Solving Strategy Horizontal Circles
?
R

• Draw a free-body diagram of the curving
  object(s).
• Choose a coordinate system with the following two
  axes.
• a) One axis will point inward along the radius
  (inward is positive direction).
• b) One axis will point perpendicular to the
  circular path (up is positive direction).
• Sum the forces along each axis to get two
  equations for two unknowns.
• a) ? FRADIUS FIN ? FOUT m(v2)/ r
• b) ? F? FUP ? FDOWN 0
• Do the math of two equations with two unknowns.

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Just in case
?
tan
R

• The third dimension in these problems would be a
  direction tangent to the circle and in the plane
  of the circle.
• We choose to ignore this direction for objects
  moving at constant speed.
• If an object moves along the circle with changing
  speed then the forces tangent to the circle have
  become unbalanced.
• You can sum the tangential forces to find the
  rate at which speed changes with time, aTAN.
• The linear kinematics equations can then be used
  to describe motion along or tangent to the
  circle.
• ? FTAN FFORWARD ? FBACKWARD m aTAN

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Example 5 The Effect of Speed on Centripetal
Force The model airplane has a mass of 0.90 kg
and moves at constant speed on a circle that is
parallel to the ground. The path of the airplane
and the guideline lie in the same horizontal
plane because the weight of the plane is
balanced by the lift generated by its wings.
Find the tension in the 17 m guideline for a
speed of 19 m/s.
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5.3.1. A boy is whirling a stone at the end of a
string around his head. The string makes one
complete revolution every second, and the tension
in the string is FT. The boy increases the speed
of the stone, keeping the radius of the circle
unchanged, so that the string makes two complete
revolutions per second. What happens to the
tension in the sting? a) The tension increases
to four times its original value. b) The
tension increases to twice its original
value. c) The tension is unchanged. d) The
tension is reduced to one half of its original
value. e) The tension is reduced to one fourth
of its original value.
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5.3.2. An aluminum rod is designed to break when
it is under a tension of 600 N. One end of the
rod is connected to a motor and a 12-kg spherical
object is attached to the other end. When the
motor is turned on, the object moves in a
horizontal circle with a radius of 6.0 m. If the
speed of the motor is continuously increased, at
what speed will the rod break? Ignore the mass
of the rod for this calculation. a) 11 m/s b)
17 m/s c) 34 m/s d) 88 m/s e) 3.0 102 m/s
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5.3.3. A ball is attached to a string and whirled
in a horizontal circle. The ball is moving in
uniform circular motion when the string separates
from the ball (the knot wasnt very tight).
Which one of the following statements best
describes the subsequent motion of the ball? a)
The ball immediately flies in the direction
radially outward from the center of the circular
path the ball had been following. b) The ball
continues to follow the circular path for a short
time, but then it gradually falls away. c) The
ball gradually curves away from the circular path
it had been following. d) The ball immediately
follows a linear path away from, but not tangent
to the circular path it had been following. e)
The ball immediately follows a line that is
tangent to the circular path the ball had been
following
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5.3.4. A rancher puts a hay bail into the back of
her SUV. Later, she drives around an unbanked
curve with a radius of 48 m at a speed of 16 m/s.
What is the minimum coefficient of static
friction for the hay bail on the floor of the SUV
so that the hay bail does not slide while on the
curve? a) This cannot be determined without
knowing the mass of the hay bail. b) 0.17 c)
0.33 d) 0.42 e) 0.54
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5.3.5. Imagine you are swinging a bucket by the
handle around in a circle that is nearly level
with the ground (a horizontal circle). What is
the force, the physical force, holding the bucket
in a circular path? a) the centripetal
force b) the centrifugal force c) your hand
on the handle d) gravitational force e) None
of the above are correct.
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5.3.6. Imagine you are swinging a bucket by the
handle around in a circle that is nearly level
with the ground (a horizontal circle). Now
imagine there's a ball in the bucket. What keeps
the ball moving in a circular path? a) contact
force of the bucket on the ball b) contact
force of the ball on the bucket c)
gravitational force on the ball d) the
centripetal force e) the centrifugal force
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5.3.7. The moon, which is approximately 4 109 m
from Earth, has a mass of 7.4 1022 kg and a
period of 27.3 days. What must is the magnitude
of the gravitational force between the Earth and
the moon? a) 1.8 1018 N b) 2.1 1022 N c)
1.7 1013 N d) 5.0 1022 N e) 4.2 1020 N
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5.3.8. The Rapid Rotor amusement ride is spinning
fast enough that the floor beneath the rider
drops away and the rider remains in place. If
the Rotor speeds up until it is going twice as
fast as it was previously, what is the effect on
the frictional force on the rider? a) The
frictional force is reduced to one-fourth of its
previous value. b) The frictional force is the
same as its previous value. c) The frictional
force is reduced to one-half of its previous
value. d) The frictional force is increased to
twice its previous value. e) The frictional
force is increased to four times its previous
value.
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Chapter 5Dynamics of Uniform Circular Motion

• Section 4
• Banked Curves

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Unbanked curve
On an unbanked curve, the static frictional
force provides the centripetal force.
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Banked Curve
On a frictionless banked curve, the centripetal
force is the horizontal component of the normal
force. The vertical component of the normal
force balances the cars weight.
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Example 8 The Daytona 500 The turns at the
Daytona International Speedway have a maximum
radius of 316 m and are steely banked at
31 degrees. Suppose these turns were
frictionless. As what speed would the cars have
to travel around them?
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5.4.1. Complete the following statement The
maximum speed at which a car can safely negotiate
an unbanked curve depends on all of the following
factors except a) the coefficient of kinetic
friction between the road and the tires. b) the
coefficient of static friction between the road
and the tires. c) the acceleration due to
gravity. d) the diameter of the curve. e) the
ratio of the static frictional force between the
road and the tires and the normal force exerted
on the car.
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5.4.2. A 1000-kg car travels along a straight
portion of highway at a constant velocity of 10
m/s, due east. The car then encounters an
unbanked curve of radius 50 m. The car follows
the curve traveling at a constant speed of 10 m/s
while the direction of the car changes from east
to south. What is the magnitude of the
acceleration of the car as it travels the
unbanked curve? a) zero m/s2 b) 2 m/s2 c) 5
m/s2 d) 10 m/s2 e) 20 m/s2
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5.4.3. A 1000-kg car travels along a straight
portion of highway at a constant velocity of 10
m/s, due east. The car then encounters an
unbanked curve of radius 50 m. The car follows
the curve traveling at a constant speed of 10 m/s
while the direction of the car changes from east
to south. What is the magnitude of the
frictional force between the tires and the road
as the car negotiates the unbanked curve? a)
500 N b) 1000 N c) 2000 N d) 5000 N e) 10
000 N
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5.4.4. You are riding in the forward passenger
seat of a car as it travels along a straight
portion of highway. The car continues traveling
at a constant speed as it follows a sharp,
unbanked curve to the left. You feel the door
pushing on the right side of your body. Which of
the following forces in the horizontal direction
are acting on you? a) a static frictional force
between you and the seat b) a normal force of
the door c) a force pushing you toward the
door d) answers a and b e) answers a and c
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Chapter 5Dynamics of Uniform Circular Motion

• Section 5
• Satellites in Circular Orbits

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Dont worry, its only rocket science
There is only one speed that a satellite can have
if the satellite is to remain in an orbit with a
fixed radius.
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Example 9 Orbital Speed of the Hubble Space
Telescope Determine the speed of the Hubble
Space Telescope orbiting at a height of 598 km
above the earths surface.
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Period to orbit the Earth
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Geosynchronous Orbit
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5.5.1. A satellite is in a circular orbit around
the Earth. If it is at an altitude equal to
twice the radius of the Earth, 2RE, how does its
speed v relate to the Earth's radius RE, and the
magnitude g of the acceleration due to gravity on
the Earth's surface? a) b) c) d) e)
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5.5.2. It is the year 2094 and people are
designing a new space station that will be placed
in a circular orbit around the Sun. The orbital
period of the station will be 6.0 years.
Determine the ratio of the stations orbital
radius about the Sun to that of the Earths
orbital radius about the Sun. Assume that the
Earths obit about the Sun is circular. a)
2.4 b) 3.3 c) 4.0 d) 5.2 e) 6.0
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5.5.3. A space probe is orbiting a planet on a
circular orbit of radius R and a speed v. The
acceleration of the probe is a. Suppose rockets
on the probe are fired causing the probe to move
to another circular orbit of radius 0.5R and
speed 2v. What is the magnitude of the probes
acceleration in the new orbit? a) a/2 b)
a c) 2a d) 4a e) 8a
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Chapter 5Dynamics of Uniform Circular Motion

• Section 6
• Apparent Weightlessness and Artificial Gravity

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Conceptual Example 12 Apparent Weightlessness
and Free Fall In each case, what is the weight
recorded by the scale?
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Example 13 Artificial Gravity At what speed
must the surface of the space station move so
that the astronaut experiences a push on his feet
equal to his weight on earth? The radius is
1700 m.
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5.6.1. A space station is designed in the shape
of a large, hollow donut that is uniformly
rotating. The outer radius of the station is 460
m. With what period must the station rotate so
that a person sitting on the outer wall
experiences artificial gravity, i.e. an
acceleration of 9.8 m/s2? a) 43 s b) 76 s c)
88 s d) 110 s e) 230 s
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Chapter 5Dynamics of Uniform Circular Motion

• Section 7
• Vertical Circular Motion

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Circular Motion

• In the previous lesson the radial and the
  perpendicular forces were emphasized while the
  tangential forces were ignored. Each class of
  forces serves a different function for objects
  moving along a circle.

Class of Force Purpose of the Force
Radial Forces Curves the object off a straight-line path.
Perpendicular Forces Holds the object in the plane of the circle.
Tangential Forces Changes the speed of the object along the circle.
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Circular Motion

• Most of the horizontal, circular problems
  occurred at constant speed so that we could
  ignore the tangential forces. The vertical,
  circular problems have objects moving with and
  against gravity so that speed changes.
  Tangential forces become significant. The good
  news is that perpendicular forces can now be
  ignored unless hurricanes are present.

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Problem Solving Strategy for Vertical Circles

• Draw a free-body diagram for the curving objects.
• Choose a coordinate system with the following two
  axes.
• a) One axis will point inward along the radius.
• b) One axis points tangent to the circle in the
  circular plane, along the direction of motion.
• Sum the forces along each axis to get two
  equations for two unknowns.
• a) ? FRADIUS FIN ? FOUT m(v2)/ r b) ?
  FTAN FFORWARD ? FBACKWARDS ma
• You can generally expect the weight of the object
  to have components in both equations unless the
  object is exactly at the top, bottom or sides of
  the circle.
• If the object changes height along the circle you
  may need to write a conservation of energy
  statement. This goes well with centripetal
  forces since there is an mv2 in both kinetic
  energy terms and in centripetal force terms.
• Do the math with 3(a) and 4 or perhaps 3(a) and
  3(b).

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Minimum/Maximum Speed Problems

• Sometimes the problem addresses the minimum
  speed that an object can move through the top of
  the circle or maximum speed that an object can
  move along the top of the circle.
• If the bucket of water turns too slowly you get
  wet.
• If a car tops a hill too quickly it leaves the
  ground.
• Allowing v2/r to equal g can solve many of these
  questions.
• By solving for v you will find a critical speed.

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Conceptual Example A Trapeze Act In a circus, a
man hangs upside down from a trapeze, legs bent
over and arms downward, holding his partner. Is
it harder for the man to hold his partner when
the partner hangs straight down and is
stationary of when the partner is
swinging through the straight-down position?
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5.7.1. At a circus, a clown on a motorcycle with
a mass M travels along a horizontal track and
enters a vertical circle of radius r. Which one
of the following expressions determines the
minimum speed that the motorcycle must have at
the top of the track to remain in contact with
the track? a) b) c) v gR d) v
2gR e) v MgR
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5.7.2. A ball on the end of a rope is moving in a
vertical circle near the surface of the earth.
Point A is at the top of the circle C is at the
bottom. Points B and D are exactly halfway
between A and C. Which one of the following
statements concerning the tension in the rope is
true? a) The tension is the same at points A and
C. b) The tension is smallest at point C. c)
The tension is smallest at both points B and
D. d) The tension is smallest at point A. e)
The tension is the same at all four points.
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5.7.3. An aluminum rod is designed to break when
it is under a tension of 650 N. One end of the
rod is connected to a motor and a 12-kg spherical
object is attached to the other end. When the
motor is turned on, the object moves in a
vertical circle with a radius of 6.0 m. If the
speed of the motor is continuously increased,
what is the maximum speed the object can have at
the bottom of the circle without breaking the
rod? Ignore the mass of the rod for this
calculation. a) 4.0 m/s b) 11 m/s c) 16
m/s d) 128 m/s e) 266 m/s
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5.7.4. A girl is swinging on a swing in the park.
As she wings back and forth, she follows a path
that is part of a vertical circle. Her speed is
maximum at the lowest point on the circle and
temporarily zero m/s at the two highest points of
the motion as her direction changes. Which of
the following forces act on the girl when she is
at the lowest point on the circle? a) the
force of gravity, which is directed downward b)
the force which is directed radially outward from
the center of the circle c) the tension in the
chains of the swing, which is directed upward d)
answers b and c only e) answers a and c only
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5.7.5 Which of the following parameters determine
how fast you need to swing a water bucket
vertically so that water in the bucket will not
fall out? a) radius of swing b) mass of
bucket c) mass of water d) a and b e) a and
c