Lecture 5 Chapter 5 Using Newton’s Laws Friction Circular motion Drag Forces Numerical integration...
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Lecture 5Chapter 5
Using Newton’s LawsFriction
Circular motionDrag Forces
Numerical integrationMisconceptions
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Assignment 2
• Chp 4 4-52, 4-67, 4-75 • Chp 5 5-30, 5-44
Due Friday 12:00 Mar 11
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Friction
You are standing still, then begin to walk.What was the external forced that causedyou to accelerate?
Hint: It is very hard to start walking if youare standing on ice.
What force causes a car to accelerate whena traffic light turns green?
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Frictional Forces
Models of frictionSee Chabay and SherwoodMatter and IneractionsVolume 1ISBN 0-471-35491-0
Model of dry friction 3 or 4 asperites support top block. Temporarily weld together
The Friction and Lubrication of SolidsF. P. Bowden and D. Tabor, Oxford University Press1964
Friction is an attractive force between two surfaces that is a result of the vector sum of many electrical forces betweenthe surface atoms of the two different bodies. Only about 10-4 of the surface atoms actually contribute.
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No motion and no horizontal forces
What is the free body diagram without friction?
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No Motion
What is the free body diagram with friction but no motion?
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Still No Motion with larger Force F
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Increase F more and now you get acceleration
Static friction force decreases to kinetic friction and its value remans constant and now you get acceleration
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Constant velocity
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How is the frictional force related to the normal?
The maximum value equals is where N is the normal force. Above we would have The coefficient of static friction ranges from 0 to 1.2
Kinetic Friction: If we increase F until the block starts to move, the friction force decreases to
mg
F
fs
Fixed block N
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Problem Solving with Newton’s 2nd Lawinvolving friction
• Vector sum of external forces in x direction = max
• Vector sum of external forces in y direction = may
• If no acceleration, then set sum equal to 0
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04/21/23 Physics 631
.
θθ
N
mg
+x
+y
θ
Mass on incline plane at rest with impending motion. Find the coefficient of static frictionWhat is the free body diagram?
hd
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04/21/23 Physics 631
.Free Body Diagram
Apply Newton's 2nd law in the x directions and using fs = μs N
x direction
θ
N
mg
+x
+y
θ
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04/21/23 Physics 631
.Free Body Diagram
Apply Newton's 2nd law in the y direction
θ
N
mg
+x
+y
θ
Apply Newtons 2nd Law
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04/21/23 Physics 631
.
θ
N
mg
+x
+y
θ
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Find a and T for the Atwood’s machine with friction between M and surface.
TT
T
fk
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Find a and T for the Atwood’s machine with friction between M and surface.
T
mg
-y
Mg
NTf
TT
T
fk
+y
+x
Solve for a and then T
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Find a and T for the Atwood’s machine with friction between M and surface.
TT
T
fk
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Question: What is the minimum magnitude force required to start the crate moving?
Tφ
N
fs
mg
+x
+y
Tcosφ
x componentsy components
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Question: What is the initial acceleration if μk
Newton's 2nd law
Tφ
N
fs
mg
+x
+y
cos φ
x components y components
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Inertial Drag Force and Terminal Velocity
Drag force: Whenever you have a body like a ball moving through a medium that behaves like a fluid, there will be a drag force opposing the motion.
Imagine a falling ball slowed down due to elastic collisionswith air molecules. Simply pushing the air out of the way.
air molecules
ballv A
dy.
Inertial drag
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Inertial Drag Force and Terminal Velocity
air molecules
ballv A
dy.
Inertial drag
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Terminal speeds in air
where m is the mass of the falling ball
Solve for v0
Using Newton’s 2nd law,
Stokes-Napier Law
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TERMINAL SPEEDS IN AIR
Object ` Speed (m/s) Speed (mph)
Feather 0.4 0.9Snowflake 1 2.2BB 9 20Mouse 13 29Tennis ball 31 66Baseball 42 86Sky diver 60 -120 134 -268Cannonball 250 560
Show demo of falling feather in vacuum
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How to solve this equation?Two ways
One way is to use a spread sheet in Exel.
Ball
Air
Gravity
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Use Newtons 2nd Law
Using 2nd Law
Initial component of momentum:
Initial force on ball:
Find new p
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Newtons 2nd Law
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Go to Excel Spread Sheet
631 Website: Lecture 3 Materials
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=C16+(g-(b_1/m_1)*C16*C16)*delta_t
=D16+1/2*(C16+C17)*delta_t
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We can also solve the equation to get the velocity as a function of time before it reaches terminal velocity. Let b = 1/2C
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Solving equation continued
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Now show comparison of this solution with numerical integration with Excel.
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Comparison
The curve modeled by velocity squared for terminal velocityDiffers from the true equation due to a large delta t.When delta t becomes small enough the two curves are Indistinguishable.
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ma = - kv
Whenever you have a body moving through a liquid there will be a drag force opposing the motion. Here the drag force is proportional to - kv. Viscous drag.
Water Resistance and Drag Forces
A 1000 kg boat in the water shuts off its engine at 90 km/hr. Find the time required to slow down to 45 km/hr due to a water drag force equal to -70v Newtons, where v is the speed of the boat. Let k = 70 N/s/m.
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v/v0 = 45/90 =1/2
t = m/k ln 2=1000/70 ln 2 = 9.9 s
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UNIFORM CIRCULAR MOTION
• Centripetal Acceleration: accelerates a body by changing the direction of the body’s velocity without changing the speed. There must be a force also pointing radially inward to make this motion.
• Examples: – Ball on a string : show demo: Force is produced by the weight
of the mass and transmitted by the tension in the string.– Moon in orbit around the earth: gravitational force– A car making a sharp turn: friction– A carousel; friction and contact forces
• Demo: pushing bowling ball with broom in a circle
39
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Uniform circular motion
Centripetal force is really not a new force like gravity, tension, friction.
Motion of earth around sun – centripetal force is a result of gravityRock whirled around on a string – centripetal force is a result of tensionSometimes it is a result of friction or the normal force
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CENTRIPETAL ACCELERATION:
. Δv points radially inward
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CENTRIPETAL ACCELERATION
.
Triangles are similar
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Centripetal Acceleration
.And, so
Magnitude of
Period of the motion
Magnitudes are related by due to similar triangles
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What is the magnitude of ac and its direction for a radius of r = 0.5 m and a
period of T= 2 s,
• Need to find v
• What is the direction of ac ?
44
INWARD
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A ball is being whirled around on a string.The string breaks. Which path does the ball take?
v
a
c
e
d b
45
QUALITATIVE QUIZ
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VERTICAL CIRCULAR MOTIONDown is negative, Up is positive
N
N
At the top:
Minimum v for N = 0:(apparent weightlessness)
At the bottom:Apparent weight = N = mv2/r + mgWeigh more
r
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What do we mean by Fictitious ForcesFf = - ma (the fictitious force always acts in the opposite
direction of acceleration)
T
ma
mg
mgθ
TT
T
Ff
θ
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In a vertically accelerated reference frame, eg. an elevator, what is your apparent weight? Apparent weight = N
Upwards is positiveDownwards is negative
Example of fictitious force (Ff = - ma)
Upward acceleration
Downward acceleration
Free fall a = gN = 0 Weightless condition
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Figure 5.34
Find the T in the cord between A and B and find the acceleration a
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Wednesday Agenda
• Any questions about homework Assignment 2
• See Section 2-8 page 38 and problem 5-102 on numerical integration
• Also see 5-87, 5-92, 5-100
• More about centripetal acceleration
• Conceptual questions from Chapter 5
• Go on to Chapter 7- Skip Chapter 6 for now
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Figure 5.18
What happens in between?
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Figure 5.56
What is the tension T in the cord and the tangential at and centripetal ac acceleration as a function of m,g, r, v and theta?
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Figure 5.20
Find v velocity and T period in terms of g, l, and angle theta.
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Figure 5.24
f
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Conceptual QuestionsChapter 5
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