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Force and Motion Part IIForce and Motion Part IICircular DynamicsCircular Dynamics
February 15, 2006February 15, 2006
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ScheduleSchedule
• Wednesday– A few problems from Chapter 5 (very few) … probably
only one…. if that.– Force and Motion while spinning around in a circle.
• Friday– More running around in a circle
• Monday– Quiz on end of chapter 5 and what we cover this
week in chapter 6.– Review of problems.
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Screwed this up a bit!Everything that we did was ok but Ihad the frictional force going in the wrongdirection.
SHAME ON ME!
)tan()cos(
)sin(
0)sin()cos(
0)sin(
)cos(
mgmg
mamgf
mgN
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Problem
What horizontal force must be applied to the cart shown in Figure P5.61 in order that the blocks remain stationary relative to the cart? Assume all surfaces, wheels, and pulley are frictionless. (Hint: Note that the force exerted by the string accelerates m1.)
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Another Problem
An object of mass M is held in place by an applied force F and a pulley system as shown in Figure P5.55. The pulleys are massless and frictionless. Find (a) the tension in each section of rope, T1, T2, T3, T4, and T5 and (b) the magnitude of F. Suggestion: Draw a free-body diagram for each pulley.
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Chapter 6
Circular Motion
and
Other Applications of Newton’s Laws
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Uniform Circular Motion
A force, Fr , is directed toward the center of the circle
This force is associated with an acceleration, ac
Applying Newton’s Second Law along the radial direction gives
2
c
vF ma m
r
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Uniform Circular Motion, cont
A force causing a centripetal acceleration acts toward the center of the circle
It causes a change in the direction of the velocity vector
If the force vanishes, the object would move in a straight-line path tangent to the circle
CENTRIPETALFORCE
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Conical Pendulum
The object is in equilibrium in the vertical direction and undergoes uniform circular motion in the horizontal direction
v is independent of m
sin tanv Lg
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Horizontal (Flat) Curve
The force of static friction supplies the centripetal force
The maximum speed at which the car can negotiate the curve is
Note, this does not depend on the mass of the car
v gr
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Banked Curve
These are designed with friction equaling zero
There is a component of the normal force that supplies the centripetal force
2
tanv
rg
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Loop-the-Loop
This is an example of a vertical circle
At the bottom of the loop (b), the upward force experienced by the object is greater than its weight
2
1bot
vn mg
rg
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Loop-the-Loop, Part 2
At the top of the circle (c), the force exerted on the object is less than its weight
2
1top
vn mg
rg
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Non-Uniform Circular Motion
The acceleration and force have tangential components
Fr produces the centripetal acceleration
Ft produces the tangential acceleration
F = Fr + Ft
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Vertical Circle with Non-Uniform Speed The gravitational
force exerts a tangential force on the object Look at the
components of Fg
The tension at any point can be found
2
cosv
T m gR
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Top and Bottom of Circle
The tension at the bottom is a maximum
The tension at the top is a minimum
If Ttop = 0, then
topv gR
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Motion in Accelerated Frames
A fictitious force results from an accelerated frame of reference A fictitious force appears to act on an object in
the same way as a real force, but you cannot identify a second object for the fictitious force
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“Centrifugal” Force
From the frame of the passenger (b), a force appears to push her toward the door
From the frame of the Earth, the car applies a leftward force on the passenger
The outward force is often called a centrifugal force It is a fictitious force due to the
acceleration associated with the car’s change in direction
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“Coriolis Force”
This is an apparent force caused by changing the radial position of an object in a rotating coordinate system
The result of the rotation is the curved path of the ball
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Fictitious Forces, examples
Although fictitious forces are not real forces, they can have real effects
Examples: Objects in the car do slide You feel pushed to the outside of a rotating
platform The Coriolis force is responsible for the rotation
of weather systems and ocean currents
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Fictitious Forces in Linear Systems
The inertial observer (a) sees
The noninertial observer (b) sees
sin
cos 0
x
y
F T ma
F T mg
' sin
' cos 0
x fictitious
y
F T F ma
F T mg
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Fictitious Forces in a Rotating System
According to the inertial observer (a), the tension is the centripetal force
The noninertial observer (b) sees
2mvT
r
2
0fictitious
mvT F T
r
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Motion with Resistive Forces
Motion can be through a medium Either a liquid or a gas
The medium exerts a resistive force, R, on an object moving through the medium
The magnitude of R depends on the medium The direction of R is opposite the direction of
motion of the object relative to the medium R nearly always increases with increasing speed
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Motion with Resistive Forces, cont
The magnitude of R can depend on the speed in complex ways
We will discuss only two R is proportional to v
Good approximation for slow motions or small objects
R is proportional to v2
Good approximation for large objects
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R Proportional To v
The resistive force can be expressed as R = - b v
b depends on the property of the medium, and on the shape and dimensions of the object
The negative sign indicates R is in the opposite direction to v
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R Proportional To v, Example
Analyzing the motion results in
dvmg bv ma m
dtdv b
a g vdt m
What would you do with this???
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R Proportional To v, Example, cont
Initially, v = 0 and dv/dt = g
As t increases, R increases and a decreases
The acceleration approaches 0 when R mg
At this point, v approaches the terminal speed of the object
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Terminal Speed
To find the terminal speed, let a = 0
Solving the differential equation gives
is the time constant and = m/b
T
mgv
b
1 1bt m tT
mgv e v e
b
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For objects moving at high speeds through air, the resistive force is approximately equal to the square of the speed
R = ½ DAv2
D is a dimensionless empirical quantity that called the drag coefficient
is the density of air A is the cross-sectional area of the object v is the speed of the object
R Proportional To v2
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R Proportional To v2, example
Analysis of an object falling through air accounting for air resistance
2
2
1
2
2
F mg D Av ma
D Aa g v
m
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R Proportional To v2, Terminal Speed
The terminal speed will occur when the acceleration goes to zero
Solving the equation gives
2T
mgv
D A
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Some Terminal Speeds
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NUMERICAL MODELINGif included
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Numerical Modeling
In many cases, the analytic method is not sufficient for solving “real” problems
Numerical modeling can be used in place of the analytic method for these more complicated situations
The Euler method is one of the simplest numerical modeling techniques
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Euler Method
In the Euler Method, derivatives are approximated as ratios of finite differences
t is assumed to be very small, such that the change in acceleration during the time interval is also very small
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Equations for the Euler Method
( ) ( )
( ) ( ) ( )
and
( ) ( )( )
( ) ( )
v v t t v ta t
t tv t t v t a t t
x x t t x tv t
t tx t t x t v t t
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Euler Method Continued
It is convenient to set up the numerical solution to this kind of problem by numbering the steps and entering the calculations into a table
Many small increments can be taken, and accurate results can be obtained by a computer
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Euler Method Set Up
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Euler Method Final
One advantage of the method is that the dynamics are not obscured The relationships among acceleration, force, velocity
and position are clearly shown The time interval must be small
The method is completely reliable for infinitesimally small time increments
For practical reasons a finite increment must be chosen A time increment can be chosen based on the initial
conditions and used throughout the problem In certain cases, the time increment may need to be
changed within the problem
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Accuracy of the Euler Method
The size of the time increment influences the accuracy of the results
It is difficult to determine the accuracy of the result without knowing the analytical solution
One method of determining the accuracy of the numerical solution is to repeat the solution with a smaller time increment and compare the results If the results agree, the results are correct to the
precision of the number of significant figures of agreement
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Euler Method, Numerical Example
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Euler Method, Numerical Example cont.