Transcript 4.1 The Concepts of Force and Mass

Dynamics of Uniform Circular Motion
Chapter 5
Learning ObjectivesCircular 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 particle’s 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-goround, 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).
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
Chapter 5:
Dynamics of Uniform Circular Motion
Section 1:
Uniform Circular Motion
Other Effects of Forces
 Up until now, we’ve 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.
DEFINITION OF
UNIFORM CIRCULAR MOTION
Uniform circular motion is the motion of an object
traveling at a constant speed on a circular path.
Let T be the time it takes for the object to
travel once around the circle.
r
2 r
v
T
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.
1
 1.2 10 3 min revolution
830 revolution s min
T  1.2 10 3 min  0.072 s
2 r 2 0.29 m 
v

 25 m s
T
0.072 s
Newton’s 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.
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
Chapter 5:
Dynamics of Uniform Circular Motion
Section 2:
Centripetal Acceleration
In uniform circular motion, the speed is constant, but the
direction of the velocity vector is not constant.
    90

    90
 
v vt

v
r
v v

t
r
2
2
v
ac 
r
The direction of the centripetal acceleration is towards the
center of the circle; in the same direction as the change in
velocity.
2
v
ac 
r
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
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
2
multiples of g  9.8 m s .
ac  v 2 r
ac

34 m s 

 35 m s 2  3.6 g
ac

34 m s 

 48 m s 2  4.9 g
2
33 m
2
24 m
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.
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.
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
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
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
Chapter 5:
Dynamics of Uniform Circular Motion
Section 3:
Centripetal Force
Recall Newton’s 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.

a


F
m



F  ma
Recall Newton’s 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.
2
v
Fc  mac  m
r

Problem Solving Strategy
– Horizontal Circles
R
1. Draw a free-body diagram of the curving object(s).
2. 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).
3. 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
4. Do the math of two equations with two unknowns.

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
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.
2
v
Fc  T  m
r

19 m s 
T  0.90 kg 
2
17 m
 19 N
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.
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
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 wasn’t 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
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
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.
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
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
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.
Chapter 5:
Dynamics of Uniform Circular Motion
Section 4:
Banked Curves
Unbanked curve
On an unbanked curve, the static frictional force
provides the centripetal force.
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 car’s weight.
2
v
Fc  FN sin   m
r
FN cos   mg
2
v
tan  
rg
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?
2
v
tan  
rg
v
v  rg tan 
316 m 9.8 m

s 2 tan 31
v  43 m s 96 mph 
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.
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
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
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
Chapter 5:
Dynamics of Uniform Circular Motion
Section 5:
Satellites in Circular Orbits
Don’t worry, it’s 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.
2
v
mM E
Fc  m  G 2
r
r
GM E
v
r
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 earth’s surface.
GM E
v
r
v
6.67 10
11

N  m kg 5.98 10 kg
6
3
6.38 10 m  598 10 m
2
2
v  7.56 103 m s  16,900 mph
v  MACH 22
24

Period to orbit the Earth
GM E 2 r
v

r
T
2 r
T
GM E
32
Geosynchronous Orbit
2 r
T
GM E
32
T 2GM E
r 3
2
2 
r 3
T 2GM E
2 2
T  24 hours
3

11 m 
 5.9742 x10 24 kg
86,400s   6.67 x10
kg s 
3

r
2 2

2
r  4.22 x10 m
7

rE  6.38 x106 m
h  3.58 x107 m  22,000 miles
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) v  1
b) v  1
c) v 
d) v 
gRE
3
e) v 
2 gRE
1 gR
E
3
2
gRE
1 gR
E
2
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
station’s orbital radius about the Sun to that of the Earth’s orbital
radius about the Sun. Assume that the Earth’s obit about the Sun is
circular.
a) 2.4
b) 3.3
c) 4.0
d) 5.2
e) 6.0
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 probe’s acceleration in the new orbit?
a) a/2
b) a
c) 2a
d) 4a
e) 8a
Chapter 5:
Dynamics of Uniform Circular Motion
Section 6:
Apparent Weightlessness and Artificial Gravity
Conceptual Example 12: Apparent Weightlessness and
Free Fall
In each case, what is the weight recorded by the scale?
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.
2
v
Fc  m  mg
r
v  rg

1700 m9.80 m s
v  130 m s
2

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
Chapter 5:
Dynamics of Uniform Circular Motion
Section 7:
Vertical Circular Motion
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.
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.
Problem Solving Strategy for Vertical Circles
1. Draw a free-body diagram for the curving objects.
2. 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.
3. 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
4. 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.
5. 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.
6. Do the math with 3(a) and 4 or perhaps 3(a) and 3(b).
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.
FC  FW
mv 2
 mg
r
v  rg
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?
2
1
v
FN 1  mg  m
r
FN 2
FN 4
2
2
v
m
r
2
4
v
m
r
2
3
v
FN 3  mg  m
r
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) v 
2 gr
c) v = gR
e) v = MgR
b) v 
gr
d) v = 2gR
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.
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
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
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