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8/19/2019 5. Application of Newtons Laws
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5.
Newton’s Laws and FrictionPhysics 50
Peter Beyersdorf
1
F o x t r o t b y
B i l l A m e n d
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5.
Class Outline
Problem solving strategy
Example Problems
Description of Friction
Static
Kinetic
Example Problems
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5.
Solving “Force” Problems
1. Identify the object of interest
2. Draw a free-body diagram for that object
3. Add up all forces in x, in y
4. Use F=ma, in x and in y directions
5. Solve for quantity of interest
6. Repeat with another object if necessary tosolve for another unknown
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5.
Objects in Equilibrium
Equilibrium implies no net force on object
Forces in x-direction balance
Forces in y-direction balance
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5.
Equilibrium Example 1
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A car engine is suspended from a chain linked at O to two
other chains. Which of the following forces should be
included in the free-body diagram for the engine?
1. tension T 1
2. tension T 2
3. tension T 3
4. two of the above
5. all of T 1, T 2, and T 3
Q5.1
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Equilibrium Example 2
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A cable attached to the
car holds the car at
rest on the frictionless
ramp (angle ! ).
1. n = w
2. n > w
3. n < w
4. not enough information given to decide
Q5.2
The ramp exerts a normal force on the car. How does the magnitude
n of the normal force compare to the weight w of the car?
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Modified Atwood’s Machine Example
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A cart (mass m1, weight w1)
is attached by a lightweight
cable to a bucket (mass m2,
weight w2) as shown. The
ramp is frictionless.
Q5.3
When released, the cart accelerates up the ramp.
Which of the following is a correct free-body diagram for the cart?
T
w1
n
T
w1
n
T
w1
n
m a1
w1
n
T
m a1
1. 2. 3. 4.
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5.
A cart (mass m1, weight w1)
is attached by a lightweight
cable to a bucket (mass m2,
weight w2) as shown.
Q5.4
The ramp is frictionless. The pulley is frictionless and does
not rotate. When released, the cart accelerates up the ramp
and the bucket accelerates downward.
1. T = w2
2. T > w2
3. T < w2
4. not enough information given to decide
Which statement about the cable tension (magnitude T ) is correct?
Modified Atwood’s Machine Example
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Friction Example
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You are pushing a 1.00-kg
food tray through the
cafeteria line with a
constant 9.0-N force. As
the tray moves, it pushes ona 0.50-kg milk carton.
1. the weight of the milk carton
2. the force of the milk carton on the tray
3. the force you exert with your hand
4. the acceleration of the milk carton
5. more than one of the above
Q5.5
Which of the following forces should be included in a free-body
diagram for the milk carton?
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Blocks A and C are connected
by a string as shown. When
released, block A accelerates
to the right and block C
accelerates downward.
1. block B accelerating to the right
2. block B accelerating to the left3. block B moving at constant speed to the right
4. block B moving at constant speed to the left
5. none of the above
Q5.6
Although there is friction between blocks A and B,there is not enough to prevent block B from slipping.
If you stood next to the table during the time that
block B is slipping on top of block A, you would see
Modified Atwood’s Machine Example 2
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You are driving your car
(mass m) at a constant speed v
around a flat, unbanked curve
of radius R.
Which of the following forcesshould be included in a free-
body diagram for the car?
1. an outward centrifugal force of magnitude mv2/ R
2. an inward centripetal force of magnitude mv2/ R
3. the force of the cars acceleration
4. two of the above
5. none of the above
Q5.7
R
Unbanked Curve Example
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You are driving your car
(mass m) at a constant speed v
around a banked curve of
radius R and bank angle !
(measured from thehorizontal).
1. a normal force that points vertically upward
2. a normal force that points at an angle ! from the vertical3. a normal force that points at an angle ! from the horizontal
4. an outward centrifugal force of magnitude mv2/ R
5. more than one of the above
Q5.8
Which of the following forces should be included in a free-body
diagram for the car?
!
R
Banked Curve Example
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5.
Equilibrium example
An engine of mass M hangs inan autoshop as shown in thediagram. What is the tensionin each of the three chains?
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Equilibrium example
Free-bodydiagram for
Engine
Free-bodydiagram for
suspension ring
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Equilibrium example
T1=mg T2=T3 cos 60° = mg/tan(60°) T3 = mg/sin(60°)
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Elevator example
An elevator acceleratesdownwards at 2m/s2. Awoman of mass M is on ascale, what weight does the
scale read?
Before stopping thedownward traveling elevatordecelerates at 2m/s2, what
weight does the scale read?
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Elevator example
An elevator acceleratesdownwards at 2m/s2. Awoman of mass M is on ascale, what weight does the
scale read?
N=m 7.8 m/s^2 = 0.8 Mg
Before stopping thedownward traveling elevatordecelerates at 2m/s2, whatweight does the scale read?
N=m 11.8m/s^2 = 1.2 Mg
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Elevator example 2
An elevator of mass m iscounterbalanced by a weight withmass M. If the motor and brakeare not engaged what will be the
acceleration of the elevator?
m M
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Elevator example 2
An elevator of mass m iscounterbalanced by a weight withmass M. If the motor and brakeare not engaged what will be the
acceleration of the elevator?
a=(m-M)/(M+m) g
m M
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Example
A tightrope walker has a massof 50 kg. The rope has abreaking strength of 2.5 kN. Ifthe rope is 10m long, What is the
least amount it can sag when theacrobat is in the middle of therope?
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5.
2(T sin θ)−mg = may
y
21
L/2
d
p d 2 +
( L/ 2 ) 2
mg
TT
in y-direction
x
0
d
sin θ = d
p d2 + (L/2)2
2Td = mgp d2 + (L/2)2
2Tdp d2 + (L/2)2
= mg
(2Td)2 = (mg)2 d2 + (L/2)2
d2
(2T )2 − (mg)2
= (mg)2(L/2)2
d2 = (mgL/2)2
((2T )2 − (mg)2)
d = (mgL/2)
p (2T )
2−
(mg)2
d = L
2
1p
(2T/mg)2 − 1
when T=Tmax=2500 N
d = m
2 p
(2(2500 N)/(50 kg)(9.8 m/s2))2 − 1
d = m
2 p
(2(2500 N)/(50 kg)(9.8 m/s2))2 − 1
= 0.
50 m
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5.
Frictional forces
Friction is a contact force
Opposes motion
Proportional to the Normal force
Depends on the materials in contact
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Friction as a contact force
Comes from roughness ofsurfaces
Opposes motion
Friction ranges from zero toa maximum value for“stationary objects”
Friction has a constant value
for” moving objects”
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Friction depends on theNormal force
Max value proportional tothe normal force
Constant of
proportionality dependson the materials incontact
direction of force vectoropposes (would-be) motion
N
N
-N
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Some values for ! s
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Free-body diagram
N
mg
f
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Free-body diagram
N
mg sinf
mg cos
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Static friction example
A brass and an aluminum block sit on an inclined planemade of steel. Over what range of inclination angleswill one block stay stationary while the other slidesdown the slope?
Which one will slide down the slope?
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Static friction example
A brass and an aluminum block sit on an inclined planemade of steel. Over what range of inclination angleswill one block stay stationary while the other slidesdown the slope?
Which one will slide down the slope?
The angle when a block will slide is =tan1() so for aluminum-steel = 0.61
brass-steel=0.51 27°
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Kinetic friction
Constant force opposingmotion for a “movingobject”
Constant of proportionalitydepends on the materials incontact
Is always less or equal tomaximum static friction
direction of force vectoropposes motion
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Some values for ! k
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Kinetic friction example
A brass and an aluminum block sit on an inclined planemade of steel. When the plane is inclined just enoughso that they both begin to slide, what is theiracceleration?
How much further apart are they are sliding for 1second.
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5.
Kinetic friction example
A brass and an aluminum block sit on an inclined planemade of steel. When the plane is inclined just enoughso that they both begin to slide, what is theiracceleration?
How much further apart are they are sliding for 1second.
With =31° (from earlier problem) both will slide
Brass (k=0.44) so a=1.35m/s^2
Aluminum (
k=0.47) a=1.10 m/s^2
and x(1 s)=12.5 cm 33
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Friction in real life
What force causes a car to turn?
What is the maximum value for that force?
How can that be increased? (i.e. how do you design a
car that corners better?)
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Example #1
On the “Gravitron” riderstravel in a circle atconstant speed. Theperiod of revolution is
4.0s, the radius is 5m.What is the minimumcoefficient of frictionnecessary to keep themfrom sliding down the
walls when the floordrops out?
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Example #1
On the “Gravitron” riderstravel in a circle atconstant speed. Theperiod of revolution is
4.0s, the radius is 5m.What is the minimumcoefficient of frictionnecessary to keep themfrom sliding down the
walls when the floordrops out?
circumference=31.4m
speed=7.9 m/s a=12.3 m/s^2 s>0.79
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Example #2
Draw free body diagramsfor the following situations.
Under what conditions willthe masses slide?
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Example #2
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x y
mgMg
N1
N2
f 2
f 1
Mg cos
m g
c o s
M g s i n
m g s i n
Fy=0:
N2-Mg cos=0
N1-mg cos=0
Fx=0:
f 2-Mg sin=0
f 1-mg sin=0
For static friction f 2"sN2
f 1"sN1
s(Mg cos)-Mg sin=0
s(mg cos)-mg sin=0 s(cos)-sin=0 =tan-1
When friction is at its maximum value
Beyond this the blocks will slide, i.e. they will slide for>tan-1
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Example #2
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x y
mg Mg
N1
N2
f 2
f 1
Mg cos
m g
c o s
M g s i n
m g s i n
Fy=0:
N1-m1g cos=0
N2=m1g cos
Fx=0:
T-f 1-m1g sin=0
For static friction: f 1"sN1
m1g sin+sm1g cos"T
T
Tf 1N1
Block 1:
thus:
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Example #2
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x y
mg Mg
N1
N2
f 2
f 1
Mg cos
m g
c o s
M g s i n
m g s i n
Fy=0:
N2 —N1-m2g cos=0
N2 —m1g cos-m2g cos=0
N2=(m1+m2)g cos
Fx=0:
T+f 1+f 2-m2g sin=0
For static friction: f 2"sN2
T+s(m1g cos)+s(m1+m2)g cos-m2g sin"0
T
Tf 1N1
Block 2:
thus:
T"-s(m1g cos)-s(m1+m2)g cos+m2g sin
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Example #2
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x y
mg Mg
N1
N2
f 2
f 1
Mg cos
m g
c o s
M g s i n
m g s i n
T
Tf 1N1
Block 2: T"-s(m1g cos)-s(m1+m2)g cos+m2g sin
m1g sin+sm1g cos"T Block 1:
m1g sin+sm1g cos"-s(m1g cos)-s(m1+m2)g cos+m2g sin
m1g sin-m2g sin"-s(m1g cos)-s(m1+m2)g cos-sm1g cos
(m1-m2)g sin "s(-m1-(m1+m2)-m1)g cos
tanθ ≤ µs
−3m1 + m2
m1 −m2
θ ≤ tan−1µs
3m1 + m2
m2 −m1
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Example 3
A crate of mass m rests on a horizontal floor withcoefficients of friction µs and µk. A woman pushes on itwith a force of magnitude F at an angle below the
horizontal. What magnitude of force F is required to
keep it moving at a steady rate.
Show that there is a critical value for µs beyond whichshe cannot get the crate to move no matter how hardshe pushes.
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Example 3
A crate of mass m rests on a horizontal floor withcoefficients of friction µs and µk. A woman pushes on itwith a force of magnitude F at an angle below the
horizontal. What magnitude of f is required to keep it
moving at a steady rate.
Show that there is a critical value for µs beyond whichshe cannot get the crate to move no matter how hard
she pushes.
f = k(mg) /(cos -k sin)
for k=tan-1, f #. Beyond this the crate cannot be moved!
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Summary
Problems involving forces can be solved systematically
In equilibrium
horizontal forces balance
vertical forces balance
Friction is a contact force that opposes motion
static friction is less than or equal to some maximum value
kinetic friction is a constant
coefficient of friction is a constant of proportionality betweenthe maximum frictional force and the normal force at thesurface
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