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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