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12/22/2011 1
Energy & Work
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Work Done by a Constant ForceWork Done by a Constant ForceThe definition of work, when the force is
parallel to the displacement:
SI work unit: newton-meter (N·m) = joule, J
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Typical Work
If the force is at an angle to the displacement:
Only the horizontal component of the force does any work (horizontal displacement).
Work for Force at an Angle
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Energy is transferred from person to spring as the person stretches the spring. This is “work”.
W F x= ∆
cosxW F x F xθ= ∆ = ∆
Work = 0
SI Units for work:1 joule = 1 J = 1 N·m
1 electron-volt = 1 eV = 1.602 x 10-19 J
The work done may be positive, zero, or negative, depending on the angle between the force and the displacement:
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Perpendicular Force and Work
A car is traveling on a curved highway. The force due to friction fspoints toward the center of the circular path.
How much work does the frictional force do on the car?
Zero!General Result: A force that is everywhere perpendicular to the motion does no work.
After algebraic manipulations of the equations of motion, we find:
Therefore, we define the kinetic energy:
2 2 2 22 2f i f iv v a x mv mv F x= + ∆ ⇒ = + ∆
Kinetic Energy & The Work-Energy Theorem
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12/22/2011 9
Work/KE TheoremW = F dx
x1
x2
∫ F = ma = mdvdt
= m1
2(v2
2 − v12 ) =
1
2mv2
2 −1
2mv1
2 = ∆KE
= mdv
dtdx
x1
x2
∫
= m v dvv1
v2
∫= m vdv
dxdx
v1
v2
∫
dv
dt=
dx
dt
dv
dx= v
dv
dx chain rule
Example:If you pull the sled (mass 80 kg) with a force of 180 N at 40° above the horizontal. The sled moves ∆x = 5.0 m, starting from rest. Assume that there is no friction.(a) Find the work you do.
(b) Find the final speed of your sled.
total you cos
(180 N)(cos40 )(5.0 m) 689 JxW W F x F xθ= = ∆ = ∆
= ° =
1 1 12 2 2total 2 2 2f i fW mv mv mv= − =
2 total2f
Wvm
=
total2 2(689 J) 4.15 m/s(80 kg)f
Wvm
= = =
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Energy
• What do we mean by an isolated system ?
• What do we mean by a conservative force ?
• If a force acting on an object act for a period of time then we have an Impulse à change (transfer) of momentum
If only “conservative” forces are present, the total energy (sum of potential, U, and kinetic energies, K) of a system is conserved.
Et = EK + EP = constant
Et = EK + EP
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Conservation of Energy
If only conservative forces are present, the total kinetic plus potential energy of a system is conserved i.e. the total “mechanical energy” is conserved. .
E = EK + EP constant!!!
Both EK and EP can change, but E = EK + EP remainsconstant.
But, if non-conservative forces act, then energy can be dissipated in other forms (thermal, sound)
E = EK + EP∆E = ∆EK + ∆EP
= W + ∆EP= W + (-W) = 0
⇒ using ∆EK = W⇒ using ∆EP = -W
What speed will the skateboarder reach at bottom of the hill if there is no friction and the skeateboarder starts at rest?
Assume we can treat the skateboarder as “point” Zero of gravitational potential energy is at bottom of the hill
R=5 m
..m = 25 kg
Example Skateboard
..
Use E = K + U = constantEbefore = Eafter
0 + mgR = ½ mv2 + 02gR = v2à v= (2gR)½v = (2 x 10 x 5)½ = 10 m/s
R=5 m
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Force vs. Energy for a Hooke’s Law spring
F = - k x (Hooke’s Law) F = ma = m dv/dt
= m (dv/dx dx/dt)= m dv/dx v= mv dv/dx
So – k x dx = mv dv
m
∫∫ =−f
i
f
i
v
v
x
xdv mvdx kx
f
i
f
i
vkx vxx mv |2
21 |2
21 =−
2212
212
212
21 ifif mvmvkxkx −− =+
2212
212
212
21 ffii mvkxmvkx +=+
Changes in EK with a constant F
In one-D, from F = ma = m dv/dt = m dv/dx * dx/dt to net work.
∫∫ =xf
xi
f
i
v
vxx
x
xx dvmvdxF
F is constant∫∫ =
xf
xi
f
i
v
vxx
x
xx dvmvdxF
Kxixfxifx EmvmvxFxxF ∆=−=∆=− 2212
21)(
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Net Work: 1-D Example (constant force)
(Net) Work is F ∆x = = 10 10 x x 5 5 N m = N m = 50 50 JJ 1 Nm is defined to be 1 Joule and this is a unit of energy Work reflects energy transfer
∆xx
A force FF = 10 N pushes a box across a frictionlessfloor for a distance ∆x x = 5 m.
FF θ = 0°
Start Finish
Work: “2-D” Example (constant force)
(Net) Work is Fx ∆x = F cos(-45°) = = 50 50 x x 00..71 71 Nm = Nm = 35 35 JJWork reflects energy transfer
∆xx
FF
A force FF = 10 N pushes a box across a frictionless floor for a distance ∆x x = 5 m and ∆y y = 0 m
θ = -45°
Start Finish
FFxx
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A falling object
What is the speed of an object after falling a distance H, assuming it starts at rest?
Wg = F F . ∆r r = mg ∆rr cos(0) = mgH
Wg = mgH
Work / Kinetic Energy Theorem:
Wg= mgH = 1/2mv2
∆rrmg g
H
j j
v0 = 0
v v = 2gH
Work & Power:
Two cars go up a hill, a Corvette and a ordinary Malibu. Both cars have the same mass.
Assuming identical friction, both engines do the same amount of work to get up the hill.
Are the cars essentially the same ?NO. The Corvette can get up the hill quickerIt has a more powerful engine.
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v Power is the rate at which work is done.v Average Power is,
v Instantaneous Power is,v If force constant, W= F ∆x = F (v0 t + ½ at2)v and P = dW/dt = F (v0 + at)
tWP∆
=
dtdWP =
v Power is the rate at which work is done.
tWP∆
=dt
dWP =
InstantaneousPower:
AveragePower:
A person of mass 80.0 kg walks up to 3rd floor (12.0m). If he/she climbs in 20.0 sec what is the average power used.
Pavg = F h / t = mgh / t = 80.0 x 9.80 x 12.0 / 20.0 WP = 470. W
Example :
Units (SI) areWatts (W):
1 W = 1 J / 1s
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12/22/2011 Dr. Mohamed Al- Fadhali 23
- Elasticity of Materials Materials Properties
The definition of the stresses is
stressesdAdFStress =
Ø dF is the element of force suffered by material on dA area. Ø Stress has dimensions of pressure (force/area), and we often measure it in
pascals, (1 N/m2 = 1 pascal = 1 Pa).Ø Stress can be classified into two different types.: One is called normal s
tress or stretching stress, the other is called shearing stress. Ø The normal (stretching) stress is perpendicular to the surface exerted by a
force. It is expressed by
dAdF
=σ
Ø Shearing stress is parallel to the acting area, expressed by
it is equal to F/A if the force is uniform on the area.
dAdF
=τ
12/22/2011 Dr. Mohamed Al- Fadhali 24
StrainThere are three kinds of strains, which are stretching, volume and shearing strains.The definition of the three strains is given below respectively.
•Stretching (tensile) strain is defined by0LL∆
=ε
where ∆L = L0–L denotes the length change and L0 is the original length of that object.
•Volume strain, expressed by ξ , is defined by0VV∆
−=ζ
where ∆V = V0–V, V0 is the volume before being depressed and V is the volume under stain. The minus sign means that the bulk of object is always depressed and becomes smaller.
•Shearing strain, denoted by γ, is defined as ϕγ tan=∆=hx
where ∆x is the length change on the direction of acting force, h is the height of the object and ϕ is the related angle deviated from the vertical line
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