Download Equilibrium and Elasticity

Equilibrium and Elasticity
• Many bodies, such as bridges,
aqueducts, and ladders, are
designed so they do not
accelerate.
• Real materials are not truly
rigid. They are elastic and do
deform to some extent.
• We consider stress and strain to
understand the deformation of
real bodies.
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Tuesday, October 18, 11
Tuesday, October 18, 11
Acts like a spring
Tuesday, October 18, 11
Tuesday, October 18, 11
Quiz
Hooke’s law describes the force of
A.
B.
C.
D.
E.
gravity.
a spring.
collisions.
tension.
none of the above.
© 2010 Pearson Education, Inc.
Tuesday, October 18, 11
Slide 8-9
Answer
Hooke’s law describes the force of
A.
B.
C.
D.
E.
gravity.
a spring.
collisions.
tension.
none of the above.
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Tuesday, October 18, 11
Slide 8-10
Tuesday, October 18, 11
Checking Understanding
Which spring has the largest spring constant?
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Tuesday, October 18, 11
Slide 8-23
Answer
Which spring has the largest spring constant?
A
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Tuesday, October 18, 11
Slide 8-24
Checking Understanding
The same spring is stretched or compressed as shown below. In
which case does the force exerted by the spring have the largest
magnitude?
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Tuesday, October 18, 11
Slide 8-25
Answer
The same spring is stretched or compressed as shown below. In
which case does the force exerted by the spring have the largest
magnitude?
E. Not enough information to tell.
© 2010 Pearson Education, Inc.
Tuesday, October 18, 11
Slide 8-26
Torque and Static Equilibrium
For an extended object to be in equilibrium, the net force
and the net torque must be zero.
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Tuesday, October 18, 11
Slide 8-11
Choosing the Pivot Point
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Tuesday, October 18, 11
Slide 8-12
Conditions for equilibrium
• First condition: The sum of all the
forces is equal to zero:
ΣFx = 0
ΣFy = 0
ΣFz = 0
• Second condition: The sum of all
torques about any given point is
equal to zero.
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Tuesday, October 18, 11
Center of gravity
• We can treat a body’s
weight as though it all
acts at a single point—
the center of gravity.
• If we can ignore the
variation of gravity with
altitude, the center of
gravity is the same as
the center of mass.
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Tuesday, October 18, 11
Solving Static Equilibrium Problems
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Tuesday, October 18, 11
Slide 8-13
Checking Understanding
What does the scale read?
A.
B.
C.
D.
500 N
1000 N
2000 N
4000 N
© 2010 Pearson Education, Inc.
Tuesday, October 18, 11
Slide 8-14
Answer
What does the scale read?
A.
B.
C.
D.
500 N
1000 N
2000 N
4000 N
© 2010 Pearson Education, Inc.
Tuesday, October 18, 11
Slide 8-15
Balance
Line of action
Gravity acts at the center
of gravity.
This force exerts no
torque about her toes.
Base of support
For an object to balance, its center of gravity must reside over its
base of support. That way gravity does not exert a torque.
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Tuesday, October 18, 11
Slide 8-18
Stability of a Car
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Tuesday, October 18, 11
Slide 8-19
Tiptoeing
Why can’t you stand on tiptoes if your toes are against a wall?
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Tuesday, October 18, 11
Slide 8-20
Tiptoeing
Why can’t you stand on tiptoes if your toes are against a wall?
Center of gravity has to be over toes – the
base of support – to balance. That requires
shifting your body slightly forward. But you
can’t shift your body forward if your toes are
against the wall.
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Tuesday, October 18, 11
Slide 8-20
Tensile and compressive stress and strain
• Tensile stress = F⊥ /A and tensile strain = Δl/l0.
• Compressive stress and compressive strain are defined in a similar
way.
• Young’s modulus is tensile stress divided by tensile strain, and is
given by Y = (F⊥/A)(l0/Δl).
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Tuesday, October 18, 11
Some values of elastic moduli
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Tuesday, October 18, 11
Tensile stress and strain
• In many cases, a body can experience both tensile and
compressive stress at the same time
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Tuesday, October 18, 11
Bulk stress and strain
• Pressure in a fluid is force per
unit area: p = F⊥/A.
• Bulk stress is pressure change
Δp and bulk strain is fractional
volume change ΔV/V0.
• Bulk modulus is bulk stress
divided by bulk strain and is
given by B = –Δp/(ΔV/V0).
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Tuesday, October 18, 11
Sheer stress and strain
• Sheer stress is F||/A and
sheer strain is x/h
• Sheer modulus is sheer
stress divided by sheer
strain, and is given by S
= (F||/A)(h/x).
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Tuesday, October 18, 11
Elasticity and plasticity
• Hooke’s law applies up to point a
• Table 11.3 shows some approximate breaking stresses.
Copyright © 2012 Pearson Education Inc.
Tuesday, October 18, 11
Q11.1
Which of the following situations satisfies both the first condition
for equilibrium (net force = 0) and the second condition for
equilibrium (net torque = 0)?
A. an automobile crankshaft turning at an increasing angular
speed in the engine of a parked car
B. a seagull gliding at a constant angle below the horizontal
and at a constant speed
C. a thrown baseball that does not rotate as it sails through
the air
D. more than one of the above
E. none of the above
© 2012 Pearson Education, Inc.
Tuesday, October 18, 11
A11.1
Which of the following situations satisfies both the first condition
for equilibrium (net force = 0) and the second condition for
equilibrium (net torque = 0)?
A. an automobile crankshaft turning at an increasing angular
speed in the engine of a parked car
B. a seagull gliding at a constant angle below the horizontal
and at a constant speed
C. a thrown baseball that does not rotate as it sails through
the air
D. more than one of the above
E. none of the above
© 2012 Pearson Education, Inc.
Tuesday, October 18, 11
Q11.2
A rock is attached to the left
end of a uniform meter stick
that has the same mass as the
rock. How far from the left end
of the stick should the
triangular object be placed so
that the combination of meter
stick and rock is in balance?
A.less than 0.25 m
B. 0.25 m
C. between 0.25 m and 0.50 m
D. 0.50 m
E. more than 0.50 m
© 2012 Pearson Education, Inc.
Tuesday, October 18, 11
A11.2
A rock is attached to the left
end of a uniform meter stick
that has the same mass as the
rock. How far from the left end
of the stick should the
triangular object be placed so
that the combination of meter
stick and rock is in balance?
A.less than 0.25 m
L
B. 0.25 m
C. between 0.25 m and 0.50 m
D. 0.50 m
E. more than 0.50 m
© 2012 Pearson Education, Inc.
Tuesday, October 18, 11
0.5-L
L × m = (0.5 − L) × m
L = (0.5 − L)
2L = 0.5
L = 0.25
Q11.3
A metal advertising sign (weight
w) is suspended from the end of
a massless rod of length L. The
rod is supported at one end by a
hinge at point P and at the other
end by a cable at an angle θ from
the horizontal.
What is the tension in the cable?
A. T = w sin θ
B. T = w cos θ
C. T = w/(sin θ)
D. T = w/(cos θ)
E. none of the above
© 2012 Pearson Education, Inc.
Tuesday, October 18, 11
A11.3
A metal advertising sign (weight
w) is suspended from the end of
a massless rod of length L. The
rod is supported at one end by a
hinge at point P and at the other
end by a cable at an angle θ from
the horizontal.
What is the tension in the cable?
A. T = w sin θ
B. T = w cos θ
C. T = w/(sin θ)
D. T = w/(cos θ)
E. none of the above
© 2012 Pearson Education, Inc.
Tuesday, October 18, 11
Q11.5
Two rods are made of the
same kind of steel and
have the same diameter.
F
F
length L
length 2L
A force of magnitude F is applied to the end of each rod.
Compared to the rod of length L, the rod of length 2L has
A. more stress and more strain.
B. the same stress and more strain.
C. the same stress and less strain.
D. less stress and less strain.
E. the same stress and the same strain.
© 2012 Pearson Education, Inc.
Tuesday, October 18, 11
F
F
A11.5
Two rods are made of the
same kind of steel and
have the same diameter.
F
F
length L
length 2L
A force of magnitude F is applied to the end of each rod.
Compared to the rod of length L, the rod of length 2L has
A. more stress and more strain.
B. the same stress and more strain.
C. the same stress and less strain.
D. less stress and less strain.
E. the same stress and the same strain.
© 2012 Pearson Education, Inc.
Tuesday, October 18, 11
F
F
Q11.6
Two rods are made of the
same kind of steel. The
longer rod has a greater
diameter.
F
F
length L
length 2L
A force of magnitude F is applied to the end of each rod.
Compared to the rod of length L, the rod of length 2L has
A. more stress and more strain.
B. the same stress and more strain.
C. the same stress and less strain.
D. less stress and less strain.
E. the same stress and the same strain.
© 2012 Pearson Education, Inc.
Tuesday, October 18, 11
F
F
A11.6
Two rods are made of the
same kind of steel. The
longer rod has a greater
diameter.
F
F
length L
length 2L
A force of magnitude F is applied to the end of each rod.
Compared to the rod of length L, the rod of length 2L has
A. more stress and more strain.
B. the same stress and more strain.
C. the same stress and less strain.
D. less stress and less strain.
E. the same stress and the same strain.
© 2012 Pearson Education, Inc.
Tuesday, October 18, 11
F
F
Energy and Work
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What Energy Transfer is Occurring Here?
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How do we transfer energy?
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How do we transfer energy?
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How do we transfer energy?
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Power: The rate at which energy is
transformed from one kind to
another.
Tuesday, October 18, 11
Quiz
1. If a system is isolated, the total energy of the system
A.
B.
C.
D.
E.
Tuesday, October 18, 11
increases constantly.
decreases constantly.
is constant.
depends on work into the system.
depends on work out of the system.
Answer
1. If a system is isolated, the total energy of the system
A.
B.
C.
D.
E.
Tuesday, October 18, 11
increases constantly.
decreases constantly.
is constant.
depends on work into the system.
depends on work out of the system.
Quiz
• If you raise an object to a greater height, you are increasing
A.
B.
C.
D.
E.
kinetic energy.
heat.
potential energy.
chemical energy.
thermal energy.
Slide 10-10
Tuesday, October 18, 11
Answer
• If you raise an object to a greater height, you are increasing
A.
B.
C.
D.
E.
Tuesday, October 18, 11
kinetic energy.
heat.
potential energy.
chemical energy.
thermal energy.
Forms of Energy
Mechanical Energy
Thermal
Energy
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Other forms include
The Basic Energy Model
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Energy Transformations
Kinetic energy K = energy of motion
Potential energy U = energy of position
Thermal energy Eth = energy associated with
temperature
System energy E = K + U + Eth + Echem + ...
Energy can be transformed within the system
without loss.
Energy is a property of a system.
Tuesday, October 18, 11
Energy Transformations
Kinetic energy K = energy of motion
Potential energy U = energy of position
Thermal energy Eth = energy associated with
temperature
System energy E = K + U + Eth + Echem + ...
Energy can be transformed within the system
without loss.
Energy is a property of a system.
Tuesday, October 18, 11
Some Energy Transformations
Tuesday, October 18, 11
Some Energy Transformations
Echem → Ug
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Some Energy Transformations
Echem → Ug
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Some Energy Transformations
Echem → Ug
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K → Eth
Some Energy Transformations
Echem → Ug
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K → Eth
Some Energy Transformations
Echem → Ug
Echem → Ug
Echem → Eth
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K → Eth
Some Energy Transformations
Echem → Ug
Echem → Ug
Echem → Eth
Tuesday, October 18, 11
K → Eth
Some Energy Transformations
Echem → Ug
K → Eth
Echem → Ug
Us → K → Ug
Echem → Eth
Tuesday, October 18, 11
Checking Understanding
A child is on a playground swing, motionless at the highest point of
his arc. As he swings back down to the lowest point of his motion,
what energy transformation is taking place?
A. K → Ug
B. Ug→ Eth
C.
Us → Ug
D. Ug → K
E. K → Eth
Tuesday, October 18, 11
Answer
A child is on a playground swing, motionless at the highest point of
his arc. As he swings back down to the lowest point of his motion,
what energy transformation is taking place?
A. K → Ug
B. Ug→ Eth
C.
Us → Ug
D. Ug → K
E. K → Eth
Tuesday, October 18, 11
Energy Transfers
These change the energy of the system.
Interactions with the environment.
Work is the mechanical transfer of energy to or
from a system via pushes and pulls.
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Energy Transfers: Work
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Energy Transfers: Work
W→ K
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Energy Transfers: Work
W→ K
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Energy Transfers: Work
W→ K
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W→ Eth
Energy Transfers: Work
W→ K
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W→ Eth
Energy Transfers: Work
W→ K
W→ Eth
W→ Us
Elastic Potential Energy
Tuesday, October 18, 11
In part (a) of the figure, an air track cart attached to a spring
rests on the track at the position xequilibrium and the spring is
relaxed. In (b), the cart is pulled to the position xstart and
released. It then oscillates about xequilibrium.
Which graph correctly represents the potential energy of the
spring as a function of the position of the cart?
Tuesday, October 18, 11
In part (a) of the figure, an air track cart attached to a spring
rests on the track at the position xequilibrium and the spring is
relaxed. In (b), the cart is pulled to the position xstart and
released. It then oscillates about xequilibrium.
Which graph correctly represents the potential energy of the
spring as a function of the position of the cart?
Answer: 3.The cart starts at xstart with no kinetic energy, and so the spring’s
✔
potential energy is a maximum. Once released, the cart accelerates to the
right and its kinetic energy increases as the potential energy of the spring
is converted into kinetic energy of the cart. As the cart passes the equilibrium
position, its kinetic energy is a maximum and so the spring’s potential
energy is a minimum. Once to the right of xequilibrium, the cart starts to
compress the spring and it slows down as its kinetic energy is converted
back to potential energy of the recompressed spring. At the rightmost point
it reaches, the cart reverses its direction of travel. At that instant, it has no
kinetic energy and the spring again has maximum potential energy.
Tuesday, October 18, 11
Tuesday, October 18, 11
• We’ve studied how Newton’s
Second Law allows us to
calculate an acceleration from a
force but what if the force
changes during its application?
We must be able to account for
things like an archer’s bow.
• We must look at action–
reaction pairs that are not
immediately obvious (like the
shotgun expelling the pellets
with expanding gas but having
the expanding gas do work on
the shotgun at the same
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Tuesday, October 18, 11
The Law of Conservation of Energy
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The Work-Energy Equation
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Work
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Work Done by Force at an Angle to Displacement
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Use the parallel component
if the force acts at an angle
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How can it be such a great
“workout” with no work?
When positive and negative work cancel, the net
work is zero even though muscles are exercising.
Tuesday, October 18, 11
The work-energy theorem
Work done on an object can change its motion and energy.
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We can compare the kinetic
energy of different bodies
Changes in the
energy of a moving
body under the
influence of an
applied force change
differently depending
on the direction of
application.
Tuesday, October 18, 11
The Basic Equation
Kf + Uf + ΔEth = Ki + Ui
A few things to note:
• Work can be positive (work in) or negative (work out)
• We are, for now, ignoring heat.
• Thermal energy is…special. When energy changes to
thermal energy, this change is irreversible.
Tuesday, October 18, 11
Stepwise
solution of
work done
by several
forces
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Stepwise
solution of
work done
by several
forces
Tuesday, October 18, 11
Energy Equations
Tuesday, October 18, 11
Checking Understanding
Each of the boxes, with masses noted, is pulled for
10 m across a level, frictionless floor by the noted force. Which box
experiences the largest change in kinetic energy?
Tuesday, October 18, 11
Answer
Each of the boxes, with masses noted, is pulled for
10 m across a level, frictionless floor by the noted force. Which box
experiences the largest change in kinetic energy?
D.
Tuesday, October 18, 11
Checking Understanding
Each of the boxes, with masses noted, is pulled for
10 m across a level, frictionless floor by the noted force. Which box
experiences the smallest change in kinetic energy?
Slide 10-33
Tuesday, October 18, 11
Answer
Each of the boxes, with masses noted, is pulled for
10 m across a level, frictionless floor by the noted force. Which box
experiences the smallest change in kinetic energy?
C.
Tuesday, October 18, 11
Example Problem
A 200 g block on a frictionless surface is pushed against a spring
with spring constant 500 N/m, compressing the spring by 2.0 cm.
When the block is released, at what speed does it shoot away from
the spring?
Tuesday, October 18, 11
Example Problem
A 200 g block on a frictionless surface is pushed against a spring
with spring constant 500 N/m, compressing the spring by 2.0 cm.
When the block is released, at what speed does it shoot away from
the spring?
Use :
F = ma
(Fsp )x = −kΔx
v = v + 2ax (x − x0 )
2
x
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2
0x
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Work and energy with varying forces
Perhaps the best
example is driving a
car, alternating your
attention between the
gas and the brake.
The effect is a variable
positive or negative
force of various
magnitude along a
straight line.
Tuesday, October 18, 11
The stretch of a spring and
the force that caused it
The force applied to
an ideal spring will be
proportional to its
stretch.
The graph of force on
the y axis versus
stretch on the x axis
will yield a slope of k,
the spring constant.
Tuesday, October 18, 11
Stepping on a scale
Whether you like the result or not, stepping on a
scale is an excellent example of applied force and
the work being done to compress that spring.
Tuesday, October 18, 11
Motion with a varying force
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Motion with a varying force
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Motion on a curved path
If you watch a child on a swing set, you can also
consider the motion of a particle along a curved
path.
Tuesday, October 18, 11
Watt about power?
Once work is calculated, dividing
by the time that passed
determines power.
The pun is credit to James Watt.
(You will see that scientists of
that era often were privileged to
leave their names on the topic of
their efforts.)
Also note the popular culture
power unit of horsepower.
The energy you use may be noted
from the meter the electric
company probably installed to
measure your consumption of
energy in kilowatt-hours.
Tuesday, October 18, 11
Power
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Power
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Power
•
•
Same mass...
Both reach 60 mph...
Same final kinetic
energy, but
different times mean
different powers.
Tuesday, October 18, 11
An example you might do
if the elevator is out
It’s interesting how a lighter stair climber and heavier stair
climber can expend the same power by using different
amounts of time.
Tuesday, October 18, 11
Elastic Collisions
Tuesday, October 18, 11
Checking Understanding
Four toy cars accelerate from rest to their top speed in a certain
amount of time. The masses of the cars, the final speeds, and the
time to reach this speed are noted in the table. Which car has the
greatest power?
Tuesday, October 18, 11
Car
Mass (g)
Speed (m/s)
Time (s)
A
100
3
2
B
200
2
2
C
200
2
3
D
300
2
3
E
300
1
4
Answer
Four toy cars accelerate from rest to their top speed in a certain
amount of time. The masses of the cars, the final speeds, and the
time to reach this speed are noted in the table. Which car has the
greatest power?
Tuesday, October 18, 11
Car
Mass (g)
Speed (m/s)
Time (s)
A
100
3
2
B
200
2
2
C
200
2
3
D
300
2
3
E
300
1
4
Checking Understanding
Four toy cars accelerate from rest to their top speed in a certain
amount of time. The masses of the cars, the final speeds, and the
time to reach this speed are noted in the table. Which car has the
smallest power?
Tuesday, October 18, 11
Car
Mass (g)
Speed (m/s)
Time (s)
A
100
3
2
B
200
2
2
C
200
2
3
D
300
2
3
E
300
1
4
Answer
Four toy cars accelerate from rest to their top speed in a certain
amount of time. The masses of the cars, the final speeds, and the
time to reach this speed are noted in the table. Which car has the
smallest power?
Tuesday, October 18, 11
Car
Mass (g)
Speed (m/s)
Time (s)
A
100
3
2
B
200
2
2
C
200
2
3
D
300
2
3
E
300
1
4
Summary
Tuesday, October 18, 11
Summary
Tuesday, October 18, 11
Additional Questions
Each of the 1.0 kg boxes starts at rest and is then pushed for 2.0
m across a level, frictionless floor by a rope with the noted force.
Which box has the highest final speed?
Tuesday, October 18, 11
Answer
Each of the 1.0 kg boxes starts at rest and is then pushed for 2.0
m across a level, frictionless floor by a rope with the noted force.
Which box has the highest final speed?
E.
Slide 10-51
Tuesday, October 18, 11