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# Ch_8

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```Chapter 8. Dynamics II: Motion in a Plane
Chapter Goal: To learn how to solve problems about
motion in a plane.
Slide 8-2
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Slide 8-3
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Slide 8-5
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Slide 8-6
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Slide 8-7
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Slide 8-8
Dynamics in Two Dimensions
п‚§ Suppose the x- and y-components of acceleration are
independent of each other.
п‚§ That is, ax does not depend on y or vy, and ay does not
depend on x or vx .
п‚§ Your problem-solving strategy is to:
1. Draw a pictorial representation: a motion diagram
(if needed) and a free-body diagram.
2. Use NewtonвЂ™s second law in component form:
The force components (including proper signs) are found
from the free-body diagram
Slide 8-21
Dynamics in Two Dimensions
Dynamics in Two Dimensions
3. Solve for the acceleration. If the acceleration is constant,
use the two-dimensional kinematic equations of Chapter
4 to find velocities and positions:
Slide 8-22
Example 8.1 Rocketing in the Wind
Slide 8-23
Example 8.1 Rocketing in the Wind
Slide 8-24
Example 8.1 Rocketing in the Wind
Slide 8-25
Example 8.1 Rocketing in the Wind
Slide 8-26
Example 8.1 Rocketing in the Wind
Slide 8-27
Example 8.1 Rocketing in the Wind
Slide 8-28
Example 8.1 Rocketing in the Wind
Slide 8-29
Projectile Motion: Review
п‚§ In the absence of air
resistance, a projectile
moves under the influence
of only gravity.
п‚§ If we choose a coordinate
system with a vertical
y-axis, then
п‚§ Consequently, from NewtonвЂ™s second law, the
acceleration is
Slide 8-30
Projectile Motion: Review
п‚§ Consider a projectile with
initial speed v0, and a
launch angle of пЃ± above
the horizontal.
п‚§ In Chapter 4 (eq 4.16) we
found that the distance it
travels before it returns to the
same elevation from which it
was launched (the range) is:
Trajectories of a projectile launched at different
angles with a speed of 99 m/s.
п‚§ The maximum range occurs for пЃ± пЂЅ 45п‚°.
п‚§ All of these results neglect the effect of air resistance.
Slide 8-31
Projectile Motion
п‚§ For low-mass projectiles on
earth, the effects of air
resistance, or drag, are too
large to ignore.
п‚§ When drag is included, the
angle for maximum range of a
projectile depends both on its
size and mass.
п‚§ The optimum angle is roughly
35п‚° for baseballs.
п‚§ The flight of a golf ball is even more complex, because
of the dimples and effects of spin.
п‚§ Professional golfers achieve their maximum range at
launch angles of barely 15п‚°!
Slide 8-32
Projectile Motion
п‚§ The acceleration of a typical projectile subject to drag
force from the air is:
п‚§ The components of
acceleration are not
independent of each other.
п‚§ These equations can
only be solved numerically.
п‚§ The figure shows the
numerical solution for
a 5-g plastic ball.
Slide 8-33
Uniform Circular Motion
п‚§ When describing circular
motion, it is convenient
to define a moving
rtz-coordinate system.
п‚§ The origin moves along
with a certain particle
moving in a circular path.
п‚§ The r-axis (radial) points from the particle toward the
center of the circle.
п‚§ The t-axis (tangential) is tangent to the circle, pointing
in the ccw direction.
п‚§ The z-axis is perpendicular to the plane of motion.
Slide 8-34
Uniform Circular Motion
п‚§ A particle in uniform circular
motion with angular velocity П‰
has velocity v = П‰r, in the
tangential direction.
п‚§ The acceleration of uniform
circular motion points to the
center of the circle.
п‚§ The rtz-components of
and are:
Slide 8-35
QuickCheck 8.1
The diagram shows three
points of a motion diagram.
The particle changes direction
with no change of speed. What
is the acceleration at point 2?
Slide 8-36
QuickCheck 8.1
The diagram shows three
points of a motion diagram.
The particle changes direction
with no change of speed. What
is the acceleration at point 2?
Acceleration of
changing direction
Slide 8-37
QuickCheck 8.2
A toy car moves around a circular track at constant
speed. It suddenly doubles its speed вЂ” a change of a
factor of 2. As a result, the centripetal acceleration
changes by a factor of
A. 1/4.
B. 1/2.
C. No change since the radius doesnвЂ™t change.
D. 2.
E. 4.
Slide 8-38
QuickCheck 8.2
A toy car moves around a circular track at constant
speed. It suddenly doubles its speedвЂ”a change of a
factor of 2. As a result, the centripetal acceleration
changes by a factor of
A. 1/4.
B. 1/2.
C. No change since the radius doesnвЂ™t change.
D. 2.
E. 4.
Slide 8-39
Example 8.2 The Ultracentrifuge
Slide 8-40
Example 8.2 The Ultracentrifuge
Slide 8-41
Dynamics of Uniform Circular Motion
п‚§ An object in uniform circular motion is not traveling at a
constant velocity in a straight line.
п‚§ Consequently, the particle must have a net force acting
on it
п‚§ Without such a force,
the object would move
off in a straight line
tangent to the circle.
п‚§ The car would end
up in the ditch!
Highway and racetrack curves are banked to allow
the normal force of the road to provide the
centripetal acceleration of the turn.
Slide 8-42
Dynamics of Uniform Circular Motion
п‚§ The figure shows a particle
in uniform circular motion.
п‚§ The net force must point in
the center of the circle.
п‚§ This centripetal force is not
a new force; it must be
provided by familiar forces.
Slide 8-43
QuickCheck 8.3
An ice hockey puck is tied by a string to
a stake in the ice. The puck is then
swung in a circle. What force or forces
does the puck feel?
A.
B.
C.
D.
E.
A new force: the centripetal force.
A new force: the centrifugal force.
One or more of our familiar forces pushing outward.
One or more of our familiar forces pulling inward.
I have no clue.
Slide 8-44
QuickCheck 8.3
An ice hockey puck is tied by a string to
a stake in the ice. The puck is then
swung in a circle. What force or forces
does the puck feel?
A.
B.
C.
D.
E.
A new force: the centripetal force.
A new force: the centrifugal force.
One or more of our familiar forces pushing outward.
One or more of our familiar forces pulling inward.
I have no clue.
The rules about what is or is not a force havenвЂ™t changed.
1. Force must be exerted at a point of contact (except for gravity).
2. Force must have an identifiable agent doing the pushing or pulling.
3. The net force must point in the direction of acceleration (NewtonвЂ™s second law).
Slide 8-45
QuickCheck 8.4
An ice hockey puck is tied by a
string to a stake in the ice. The puck
is then swung in a circle. What force
is producing the centripetal
acceleration of the puck?
A.
Gravity
B.
Air resistance
C.
Friction
D.
Normal force
E.
Tension in the string
Slide 8-46
QuickCheck 8.4
An ice hockey puck is tied by a
string to a stake in the ice. The puck
is then swung in a circle. What force
is producing the centripetal
acceleration of the puck?
A.
Gravity
B.
Air resistance
C.
Friction
D.
Normal force
E.
Tension in the string
Slide 8-47
Example 8.4 Turning the Corner I
Slide 8-48
Example 8.4 Turning the Corner I
VISUALIZE
п‚§ The second figure below shows the top view of a tire as it turns a
corner.
п‚§ The force that prevents the tire from sliding across a surface is static
friction.
п‚§ Static friction pushes sideways on the tire, perpendicular to the
velocity, since the car is not speeding up or slowing down.
п‚§ The free-body diagram, drawn from behind the car, shows the static
friction pointing toward the center of the circle.
Slide 8-49
Example 8.4 Turning the Corner I
Slide 8-50
Example 8.4 Turning the Corner I
Slide 8-51
Example 8.4 Turning the Corner I
Slide 8-52
Banked Curves
п‚§ Real highway curves are banked by being tilted up at the
outside edge of the curve.
п‚§ The radial component of the normal force can provide
centripetal acceleration needed to turn the car. (Ex 8.5)
п‚§ For a curve of radius r banked at an angle пЃ±, the exact
speed at which a car must take the curve without
assistance from friction is
.
Slide 8-53
Banked Curves
п‚§ Consider a car going around a banked curve at a speed
higher than
.
п‚§ In this case, static friction must prevent the car from
slipping up the hill.
Slide 8-54
Banked Curves
п‚§ Consider a car going around a banked curve at a speed
slower than
.
п‚§ In this case, static friction must prevent the car from
slipping down the hill.
Slide 8-55
Circular Orbits
п‚§ The figure shows a
perfectly smooth,
spherical, airless planet
with one tower of height h.
п‚§ A projectile is launched
parallel to the ground with
speed v0 .
п‚§ If v0 is very small, as in
trajectory A, it simply falls
to the ground along a
parabolic trajectory.
п‚§ This is the вЂњflat-earth approximation.вЂќ
Slide 8-58
Circular Orbits
п‚§ As the initial speed v0 is
increased, the range of the
projectile increases as the
ground curves away from it.
п‚§ Trajectories B and C are of
this type.
п‚§ If v0 is sufficiently large,
there comes a point where
the trajectory and the curve
of the earth are parallel.
п‚§ In this case, the projectile
вЂњfallsвЂќ but it never gets any closer to the ground!
п‚§ This is trajectory D, called an orbit.
Slide 8-59
Circular Orbits
п‚§ In the flat-earth approximation,
shown in figure (a), the
gravitational force on an object
of mass m is:
п‚§ Since actual planets are
spherical, the real force
of gravity is toward the
center of the planet, as
shown in figure (b).
Slide 8-60
Circular Orbits
п‚§ An object in a low circular
orbit has acceleration:
п‚§ If the object moves in a circle
of radius r at speed vorbit the
centripetal acceleration is:
п‚§ The required speed for a circular orbit near a planetвЂ™s
surface, neglecting air resistance, is:
Slide 8-61
Circular Orbits
п‚§ The period of a low-earth-orbit
satellite is:
п‚§ If r is approximately the radius
of the earth Re = 6400 km, then
п‚§ An orbiting spacecraft is constantly in free fall, falling
under the influence only of the gravitational force.
п‚§ This is why astronauts feel weightless in space.
Slide 8-62
Fictitious Forces
п‚§ If you are riding in a car that
makes a sudden stop, you
seem to be hurled forward.
п‚§ You can describe your
experience in terms of
fictitious forces.
п‚§ Fictitious forces are not real
because no agent is exerting
them.
п‚§ Fictitious forces describe
noninertial reference frame.
Slide 8-64
Centrifugal Force?
п‚§ The figure shows a birdвЂ™seye view of you riding in a
car as it makes a left turn.
п‚§ From the perspective of an
inertial reference frame, the
normal force from the door
points inward, keeping you
on the road with the car.
п‚§ Relative to the noninertial reference frame of the car,
you feel pushed toward the outside of the curve.
п‚§ The fictitious force which seems to push an object to
the outside of a circle is called the centrifugal force.
п‚§ There really is no such force in an inertial reference frame.
Slide 8-65
QuickCheck 8.6
A coin sits on a turntable as the
free-body diagrams below show the
coin from behind, moving away from
you. Which is the correct diagram?
Slide 8-66
QuickCheck 8.6
A coin sits on a turntable as the
free-body diagrams below show the
coin from behind, moving away from
you. Which is the correct diagram?
Slide 8-67
QuickCheck 8.7
A coin sits on a turntable as the
force or forces act in the plane of
the turntable?
Slide 8-68
QuickCheck 8.7
A coin sits on a turntable as the
force or forces act in the plane of
the turntable?
Slide 8-69
QuickCheck 8.8
Two coins are on a turntable that
rest, with a ccw rotation. Which
coin flies off the turntable first?
A.
Coin 1 flies off first.
B.
Coin 2 flies off first.
C.
Both coins fly off at the same time.
D.
We canвЂ™t say without knowing their
masses.
Slide 8-70
QuickCheck 8.8
Two coins are on a turntable that
rest, with a ccw rotation. Which
coin flies off the turntable first?
A.
Coin 1 flies off first.
B.
Coin 2 flies off first.
C.
Both coins fly off at the same time.
D.
We canвЂ™t say without knowing their
masses.
Slide 8-71
Loop-the-Loop
п‚§ The figure shows a rollercoaster going around a vertical
п‚§ Note this is not uniform circular
motion; the car slows down
going up one side, and speeds
up going down the other.
п‚§ At the very top and very bottom points, only the carвЂ™s
direction is changing, so the acceleration is purely
centripetal.
п‚§ Because the car is moving in a circle, there must be
a net force toward the center of the circle.
Slide 8-73
Loop-the-Loop
п‚§ The figure shows the roller-coaster
free-body diagram at the bottom
of the loop.
п‚§ Since the net force is toward the
center (upward at this point), n > FG .
п‚§ This is why you вЂњfeel heavyвЂќ at the
bottom of the valley on a roller coaster.
п‚§ The normal force at the bottom is larger than mg.
Slide 8-74
Loop-the-Loop
п‚§ The figure shows the roller-coaster
free-body diagram at the top of
the loop.
п‚§ The track can only push on the
wheels of the car, it cannot pull,
therefore presses downward.
п‚§ The car is still moving in a circle, so the net force is also
downward:
п‚§ The normal force at the at the top can exceed mg if vtop
is large enough.
Slide 8-75
Loop-the-Loop
п‚§ At the top of the roller coaster,
the normal force of the track on
the car is:
п‚§ As vtop decreases, there comes
a point when n reaches zero.
п‚§ The speed at which n = 0 is called the critical speed:
п‚§ This is the slowest speed at which the car can complete
the circle without falling off the track near the top.
Slide 8-76
Loop-the-Loop
A roller-coaster car at the top of the loop.
Slide 8-77
QuickCheck 8.9
A physics textbook swings back
and forth as a pendulum. Which is
the correct free-body diagram
when the book is at the bottom
and moving to the right?
Slide 8-78
QuickCheck 8.9
A physics textbook swings back
and forth as a pendulum. Which is
the correct free-body diagram
when the book is at the bottom
and moving to the right?
Centripetal acceleration
requires an upward force.
Slide 8-79
QuickCheck 8.10
A car thatвЂ™s out of gas coasts
over the top of a hill at a steady
20 m/s. Assume air resistance
is negligible. Which free-body
diagram describes the car at
this instant?
Slide 8-80
QuickCheck 8.10
A car thatвЂ™s out of gas coasts
over the top of a hill at a steady
20 m/s. Assume air resistance
is negligible. Which free-body
diagram describes the car at
this instant?
Now the centripetal
acceleration points down.
Slide 8-81
QuickCheck 8.11
A roller coaster car does a loopthe-loop. Which of the free-body
diagrams shows the forces on
the car at the top of the loop?
Rolling friction can be neglected.
Slide 8-82
QuickCheck 8.11
A roller coaster car does a loopthe-loop. Which of the free-body
diagrams shows the forces on
the car at the top of the loop?
Rolling friction can be neglected.
The track is above the car, so
the normal force of the track
pushes down.
Slide 8-83
Nonuniform Circular Motion: Review
п‚§ The particle in the figure
is speeding up as it moves
around the circle.
п‚§ The tangential acceleration is:
п‚§ The centripetal acceleration is:
ar = v2/r = П‰2r
Slide 8-84
Nonuniform Circular Motion: Review
п‚§ In terms of angular quantities, the equations
of constant-acceleration kinematics are:
Slide 8-85
Dynamics of Nonuniform Circular Motion
п‚§ A net force
is applied to
a particle moving in a circle.
п‚§
is likely to be a
superposition of several
forces, such as tension,
thrust, friction, etc.
п‚§ The tangential component of
the net force (Fnet)t creates a
tangential acceleration and
causes the particle to change speed.
п‚§ The radial component (Fnet)r is directed toward the center,
creates a centripetal acceleration, and causes the particle
to change direction.
Slide 8-86
Dynamics of Nonuniform Circular Motion
п‚§ Force and acceleration are related through
NewtonвЂ™s second law:
Slide 8-87
QuickCheck 8.12
A ball rolls ccw around the inside
of a horizontal pipe. The ball is
fastest at the lowest point, slowest
at the highest point. At the point
shown, with the ball moving down,
what is the direction of the net
force on the ball?
Slide 8-90
QuickCheck 8.12
A ball rolls ccw around the inside
of a horizontal pipe. The ball is
fastest at the lowest point, slowest
at the highest point. At the point
shown, with the ball moving down,
what is the direction of the net
force on the ball?