chap. 6: work and kinetic energy - texas a&m...
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Chap. 6: Work and
Kinetic Energy
1. What is Work? Work as Energy Transfer Work done by a (constant/varying) force (scalar product & integrals)
2. Kinetic Energy A way to describe the status of the motion Work-Energy Theorem
1
Case Study: Constant Force
Work and Energy
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Work and Energy Physics Message? 4
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Work and Energy
1) Effort or labor to get something done. Solving problems is hard work.
2) A person’s job. What kind of work do you do?
3) A piece of music, a painting, or sculpture. A work of art.
4) A scalar quantity involving both a force and a distance. When you lift a barbell against Earth’s gravity, work is done. The
heavier the barbell or the higher we lift the barbell, the more work is done on the barbell.
What is “Work”?
(force) (distance)
(force)
(distance)
W
W
W
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PHYS218
Work and Energy
4) A scalar quantity involving both a force and a distance. A weight lifter who holds a barbell weighting 1000 N does no
work on the barbell. (∵)Zero distance.
Won barbel = 0 But he may get really tired doing so.
Work may be done on the muscles by stretching and contracting,
which is force times distance on a biological (small) scale.
Won muscles 0
8
Work and Energy
Work (W) Done by
a Constant Force (I)
A tractor is doing work on (or “energy transfer” to) a sled
of firewood as it exerts the force over a displacement.
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Engine
Energy Transfer
Work and Energy
[Hint] Both magnitude (F) and
directions (q ) must be taken
into account.
FN
Ff = 3500 N
FG = 14,700 N
F.B.D. on sled
d = 20 m
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Work (W) Done by
a Constant Force (I)
[Q] Is the force effectively pulling a sled of firewood?
(distance)
cos
(force)energy kinetic of difference
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f
2
1
2
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Work and Energy
Ff = 3500 N
FG = 14,700 N
FN
m
Fa
d = 20 m
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Work and Energy
(distance)
cos
(force)energy kinetic of difference
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[Q] What is this?
m
Fa
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[A] That is a sum of Wtension and Wfriction (= Wnet).
Work-Energy Theorem
Work and Energy
Ff = 3500 N
FG = 14,700 N
FN
[Q] Find vf if vi = 2.0 m/s using
“work-energy theorem”.
Problem-solving steps:
(1) F.B.D. for an object
(2) Wi = Fi d cosqi = F||d (if all forces are constant)
3 Wnet SWi
4 Wnet DK
Analyze the motion
d = 20 m
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NOTE: You have noticed WFN and WFG are zero.
“No Work”
What was a special condition to see this?
Work and Energy
Special Case: “No Work”
[Example] “Work” done on the
bag by the person..
Special case: W = 0 J
a) WP = FP d cos ( 90o )
b) Wg = m g d cos ( 90o )
Nothing to do with the motion
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Work and Energy
Work Energy Theorem
Wnet = Kf – Ki
You measure Force(s) and Distance. You measure Velocities.
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Work and Energy
e.g., Work by a Baseball Pitcher
A baseball pitcher is doing work on (or
“energy transfer” to) the ball as he
exerts the force over a displacement.
vi = 0
Ki = (1/2) m vi 2
vf = 44 m/s
Kf = (1/2) m vf 2
Initial Status Final Status 27
Work and Energy
A 50.0-kg crate is pulled 40.0 m by a constant force
exerted (FP = 100 N and q = 37.0o) by a person. Assume
a coefficient of friction force mk = 0.110. Determine the
work done by each force acting on the crate and its net
work. Find the final velocity of the crate if d = 40 m and
vi = 0 m/s.
Example 1
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Work and Energy
Example 1 Solution (cont’d)
Wnet = SWi
= 1302 [J] (> 0)
The body’s speed increases.
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Work and Energy
WP = FP d /cos(30o)
WN = FN d
FN = FG cos(30o) or FG
Wf = Ff d cos(30o)
WG = FG d cos(90o)
Example 2 Wrong Solution
(1) F.B.D.
(2) W by each force
FN
FG
FP or FP/cos(30o)
Ff
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Work and Energy
Example 2 Wrong Solution
(4) W-E Theorem to find v2 :
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Chap. 6: Work and
Kinetic Energy
1. What is Work? Work as Energy Transfer Work done by a (constant/varying) force (scalar product & integrals)
2. Kinetic Energy A way to describe the status of the motion Work-Energy Theorem
42
Case Study: Varying Force
Work and Energy
W DK = Kb - Ka = ½ m vb2 ½ m va
2
“Work-Energy Theorem”
Work
Distance, d
Work: Area in F-d Graph
[F = constant, q = constant]
Work and Energy
Work Done by Work Done by
a Constant Force (I)a Constant Force (I)
Work (W)
Both magnitude (F) and
directions (q ) must be
taken into account.
W [Joule] = ( F cos q ) d = F d
How effective is the force in moving a body?
a b F
cosq
An ideal scenario …
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Work and Energy
W DK = Kb - Ka = ½ m vb2 ½ m va
2
“Work-Energy Theorem”
Work
Distance, d Work and Energy
Work Done by Work Done by
a Constant Force (I)a Constant Force (I)
Work (W)
Both magnitude (F) and
directions (q ) must be
taken into account.
W [Joule] = ( F cos q ) d = F d
How effective is the force in moving a body?
Fco
sq
Work: Area in F-d Graph [F = varied, q = constant]
She is exhausted.
If we know how to calculate the area in F-d graph, we are okay! 44
Work and Energy
Fco
sq
lb
W = F|| dl
la
Motion ab is
collection of straight-
line motion with a
constant force
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Work Done by a Varying Force (direction and magnitude)
Work and Energy
Spring Force (Hooke’s Law) [q = constant, F = varied]
FS(x) = k x
FP FS
Natural Length x > 0
x < 0
Spring Force
(Restoring Force):
The spring exerts
its force in the
direction opposite
the displacement.
x 0
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Work and Energy
Work Done on Box by FS
FS(x) = – k x
Natural Length
FP FS
xf
WS = FS(x) d x = – ½ k xf2 + ½ k xi
2
xi
Final position
Initial position
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Work and Energy
x2
WP = FP(x) dx
x1
Work Done “to Stretch a Spring”
FP(x) = k x
W
Natural Length
FP FS
xf
WP = FP(x) d x = ½ k xf2 – ½ k xi
2
xi
Initial position
Final position
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Work and Energy
Problem Solving Steps [Varying Force]
ifnet
net
4
3
d 2)
object eachfor F.B.D. 1)
2
1
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WW)
lFW
i
i
l
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ii
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Work and Energy
Example 3
A person pulls on the spring, stretching it
3.0 cm, which requires a maximum force
of 75 N. How much work does the person
do ? If, instead, the
person compresses
the spring 3.0 cm,
how much work
does the person do ?
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Work and Energy
Example 4
A person pulls on the spring, stretching it
3.0 cm, which requires a maximum force
of 75 N. How much work does the spring
do ? If, instead, the
person compresses
the spring 3.0 cm,
how much work
does the spring do ?
54
Work and Energy
You are weighing 600 N on a bathroom scale containing a stiff
spring. In equilibrium the spring is 1.0 cm under your weight.
Find the spring constant and the work done by the spring.
Example 5
57
Work and Energy
Example 6
A 1.50-kg block is pushed against a spring
(k = 250 N/m), compressing it 0.200 m, and
released. What will be the speed of the
block when it separates from the spring at
x = 0? Assume mk =
0.300.
(i) F.B.D. first !
(ii) x < 0
FS = k x
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Work and Energy
Example 7
A 1.50-kg block is pushed against a spring,
compressing it 0.200 m, and released. The
spring force is given as FS(x) = kx + bx2 where
k = 250 N/m and b = 300 N/m2. What will be
the speed of the block when it separates from
the spring at x = 0?
Assume mk = 0.300.
(i) F.B.D. first !
(ii) x < 0
FS = k x+bx2
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Work and Energy
Example 8 Today’s cars have “8 km/h bumpers” that are designed to
elastically compressed and rebound without any physical
damage at speeds below 8 km/h. If the material of the bumper
permanently deforms after a compression of 1.5 cm, but remains
like a elastic spring up to that point, what must the effective
spring constant of the bumper material be, assuming the car has
a mass of 1150 kg and is tested by ramming into a solid wall?
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1) Set up a model (simplified
version of a physical system) to
explain the phenomena.
2) …
Work and Energy
Today’s cars have “8 km/h bumpers” that are designed to
elastically compressed and rebound without any physical
damage at speeds below 8 km/h. If the material of the bumper
permanently deforms after a compression of 1.5 cm, but remains
like a elastic spring up to that point, what must the effective
spring constant of the bumper material be, assuming the car has
a mass of 1150 kg and is tested by ramming into a solid wall?
Bumper Car
1) Set up a model (simplified
version of a physical system) to
explain the phenomena.
2) Identify …
67
Example 8
Work and Energy
Problem 4: (25 points)
A particle of mass M moves in one dimension under the influence of a force that is given
by:
F(x) = A x + B x2 –C x3
where the direction of motion is taken to be +x and A, B, and C are given constants.
a. (15 pts) The particle starts at position –x0 and moves to position +x0. Express the
work done on this particle by the force given above in terms of A, B, C, and x0.
b. (10 pts) Assume that when the particle is at –x0 it has zero velocity. Find the velocity
of the particle (in terms of A, B, C, x0, and M) when its reaches +x0.
70
Example 9
Work and Energy
Problem 4: (25 points)
A block is sliding on a smooth, flat, horizontal plane with speed v0. Initially, there is no
friction. At some position, which you should call x = 0, the plane starts to get rough, and
friction increases linearly with distance, increasing by 1 in a distance L. This means that the
coefficient of kinetic friction is a function of position, and we can write:
m(x) = x / L .
Thus we have m(0) = 0 and m(L) = 1. Calculate how far into the rough surface the block
slides before coming to rest. Note that L is the distance over which m changes by 1, not the
distance the block travels. The distance the block travels depends on v0, g, and L.
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Example 10