Lecture 5Chapter 5

Using Newton’s LawsFriction

Circular motionDrag Forces

Numerical integrationMisconceptions

Assignment 2

• Chp 4 4-52, 4-67, 4-75 • Chp 5 5-30, 5-44

Due Friday 12:00 Mar 11

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?

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.

No motion and no horizontal forces

What is the free body diagram without friction?

No Motion

What is the free body diagram with friction but no motion?

Still No Motion with larger Force F

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

Constant velocity

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

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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.

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

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

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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θ

N

mg

+x

+y

θ

Find a and T for the Atwood’s machine with friction between M and surface.

TT

T

fk

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

Find a and T for the Atwood’s machine with friction between M and surface.

TT

T

fk

Question: What is the minimum magnitude force required to start the crate moving?

Tφ

N

fs

mg

+x

+y

Tcosφ

x componentsy components

Question: What is the initial acceleration if μk

Newton's 2nd law

Tφ

N

fs

mg

+x

+y

cos φ

x components y components

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

Inertial Drag Force and Terminal Velocity

air molecules

ballv A

dy.

Inertial drag

Terminal speeds in air

where m is the mass of the falling ball

Solve for v0

Using Newton’s 2nd law,

Stokes-Napier Law

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

How to solve this equation?Two ways

One way is to use a spread sheet in Exel.

Ball

Air

Gravity

Use Newtons 2nd Law

Using 2nd Law

Initial component of momentum:

Initial force on ball:

Find new p

Newtons 2nd Law

Go to Excel Spread Sheet

631 Website: Lecture 3 Materials

=C16+(g-(b_1/m_1)*C16*C16)*delta_t

=D16+1/2*(C16+C17)*delta_t

We can also solve the equation to get the velocity as a function of time before it reaches terminal velocity. Let b = 1/2C

Solving equation continued

Now show comparison of this solution with numerical integration with Excel.

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.

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.

v/v0 = 45/90 =1/2

t = m/k ln 2=1000/70 ln 2 = 9.9 s

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

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

41

CENTRIPETAL ACCELERATION:

. Δv points radially inward

42

CENTRIPETAL ACCELERATION

.

Triangles are similar

43

Centripetal Acceleration

.And, so

Magnitude of

Period of the motion

Magnitudes are related by due to similar triangles

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

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

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

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

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

Figure 5.34

Find the T in the cord between A and B and find the acceleration a

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

Figure 5.18

What happens in between?

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?

Figure 5.20

Find v velocity and T period in terms of g, l, and angle theta.

Figure 5.24

f

Conceptual QuestionsChapter 5