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Energy and Work. Energy - the ability of a body or system of bodies to perform work . A body is given energy when a force does work on it. But What is Work?. A force does work on a body (and changes its energy) when it causes a displacement . - PowerPoint PPT Presentation
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Energy and Work
Energy - the ability of a body or system of bodies to perform work.
A body is given energy when a force does work on it.
But What is Work?
A force does work on a body (and changes its energy) when it causes a displacement.
If a force causes no displacement, it does no work.
Riddle Me This
If a man holds a 50 kg box at arms length for 2 hours as he stands still, how
much work does he do on the box?
Nad
a ZipZilch
NONEZERO
Counterintuitive Results There is no work done by a force
if it causes no displacement. Forces perpendicular to
displacement, such as the normal force, can do no work.
Likewise, centripetal forces never do work.
Calculating WorkWork is the dot product of
force and displacement.
Work is a scalar resulting from the interaction of two vectors.
Vector Multiplication
There are three ways to multiply vectors:
•Scalar Multiplication
•Dot Product
•Cross Product
Scalar Multiplication
•Magnitude of vector changes.•Direction of vector does not change.
amF
a = 10 m·s-1
F = 50 N
If m = 5 kg
Dot Product
BAW
Note that the dot
product of two vectors gives a scalar .
. and between angle theis BA
cosABBA
θ
A
B
Dot Product
Geometrically, the dot product is the projection of one vector on a second
vector multiplied by the magnitude of the second vector.
θ
A
B
cosA
Which does more work?
θ
F1
F2
Two forces are acting on the box shown causing it to move across the floor. Which force does more work?
Vectors and Work
s
W = F • sW = F s cos Only the component of force aligned with displacement does work. Work is less.
F
Gravity often does negative work.
mg
F
When the load goes up, gravity does negative work and the crane does positive work.
When the load goes down, gravity does positive work and the crane does negative work.
Positive, zero, or negative work?
A box is being moved with a velocity v by a force P (parallel to v) along a level floor. The normal force is FN, the frictional force is fk, and the weight of the box is mg.
Decide which forces do positive, zero, or negative work.
Work and variable force
The area under the curve of a graph of force vs displacement gives the work done by the force.
F(x)
xxa xb
W = F(x) dxxa
xb
Net Work
Net work (Wnet) is the sum of the work done on an object by all forces acting
upon the object.
The Work-Energy Theorem
• Consider a force applied to an object (ΣF ≠ 0).
• Newton’s second law tells us that this net force will produce an acceleration.
• Since the object is accelerating, its displacement will change, hence the net force does work.
WsF as
vv if 2
)( 22
2212
21
if mvmvW
The Work-Energy Theorem
maF massF masW
asvv if 222
2
)( 22if vv
mW
Kinetic Energy
A form of mechanical energyEnergy due to motionK = ½ m v2
– K: Kinetic Energy in Joules.
– m: mass in kg
– v: speed in m/s
The Work-Energy Theorem
Wnet = KE– When net work due to all forces acting upon
an object is positive, the kinetic energy of the object will increase.
– When net work due to all forces acting upon an object is negative, the kinetic energy of the object will decrease.
– When there is no net work acting upon an object, the kinetic energy of the object will be unchanged.
PowerPower is the rate of which work is
done.No matter how fast we get up the
stairs, our work is the same.When we run upstairs, power demands
on our body are high.When we walk upstairs, power
demands on our body are lower.
Power Problem
Develop an expression for the power output of an airplane cruising at constant speed v in level flight. Assume that the aerodynamic drag force is given by FD = bv2. By what factor must the power be increased to increase airspeed by 25%?
How We Buy Energy…The kilowatt-hour is a commonly used unit
by the electrical power company.Power companies charge you by the
kilowatt-hour (kWh), but this not power, it is really energy consumed.
1 kW = 1000 W1 h = 3600 s
1 kWh = 1000J/s • 3600s = 3.6 x 106J
More about force types
Conservative forces:– Work in moving an object is path independent.– Work in moving an object along a closed path is zero.– Work may be related to a change in potential energy
or used in the work-energy theorem.– Ex: gravity, electrostatic, magnetostatic, springs
Non-conservative forces:– Work is path dependent.– Work along a closed path is NOT zero.– Work may be related to a change in total energy
(including thermal energy).– Ex: friction, drag, magnetodynamic
Mechanical Energy:Potential energyEnergy an object possesses by virtue
of its position or configuration.Represented by the letter U.Examples:
– Gravitational potential energy.– Electrical potential energy.– Spring potential energy.
Gravitational Potential Energy (Ug)
For objects near the earth’s surface, the gravitational pull of the earth is constant, so
Wg = mgx – The force necessary to lift an object at
constant velocity is equal to the weight, so we can say
Ug = -Wg = mgh
Where is Gravitational Potential Energy Zero?Ug has been defined to be zero when
objects are infinitely far away, and becoming negative as objects get closer.
This literal definition is impractical in most problems.
It is customary to assign a point at which Ug is zero. Usually this is the lowest point an object can reach in a given situation
Then, anything above this point is a positive Ug.
Ideal SpringObeys Hooke’s Law.
Fs(x) = -kx
– Fs is restoring force exerted BY the spring.
Ws = Fs(x)dx = -k xdx
– Ws is the work done BY the spring.
Us = ½ k x2
Spring ProblemThree identical springs (X, Y, and Z) are hung as shown. When a 5.0-kg mass is hung on X, the mass descends 4.0 cm from its initial point. When a 7.0-kg mass is hung on Z, how far does the mass descend?
X Y
Z
Isolated System
E = U + K + Eint
= Constant
No mass can enter or leave!No energy can enter or leave!Energy is constant, or conserved!
Boundary allows no exchange with environment.
Law of Conservation of Energy
Energy can neither be created nor destroyed, but can only be transformed from one type of energy to another.
Applies to isolated systems.
Law of Conservation of Mechanical Energy
E = U + K = CE = U + K = 0
for gravityUg = mghf - mghi
K = ½ mvf2 - ½ mvi
2
Law of Conservation of Mechanical Energy
E = U + K = CE = U + K = 0
for springsUg = ½ kx2
K = ½ mvf2 - ½ mvi
2
Law of Conservation of Energy
E = U + K + Eint= CE = U + K + Eint = 0
Eint is thermal energy.Mechanical energy may be converted to and from heat.
James Prescott Joule
Father of Conservation of Energy.Studied electrical motors.Derived the mechanical equivalent of heat.Measured heat of water as it fell.Measured cooling of expanding gases.
h
Pendulum Energy
½mvmax2 =
mghFor minimum and maximum points of swing
½mv12 + mgh1 = ½mv2
2 + mgh2 For any points two points in the pendulum’s swing
Spring Energy
m
m -x
mx
0½ kx2 = ½ mvmax
2
For maximum and minimum displacements from equilibrium½ kx1
2 + ½ mv12
= ½ kx22 + ½
mv22
For any two points in a spring’s oscillation
EquilibriumThe net force on a system is zero
when the system is at equilibrium.
There are three types of equilibrium which describe what happens to the forces on a system when it is displaced slightly from the equilibrium position.
Force and Potential Energy
In order to discuss the relationships between displacements and forces, we need to know a couple of equations.
W = F(x)dx = - dU = -U
dU = - F(x)dx
F(x) = -dU(x)/dx
Stable Equilibrium
U
x
compressed extended
dU/dx
F = -dU/dx
When x is negative, dU/dx is negative, so F is positive and pushes system back to equilibrium.
Stable EquilibriumExamples:
– A spring at equilibrium position.
When the system is displaced from equilibrium, forces return it to the equilibrium position.
Often referred to as a potential energy well or valley.
Stable Equilibrium
U
x
compressed extended
dU/dxWhen x is positive, dU/dx is positive, so F is negative and pushes system back to equilibrium.
F = -dU/dx
Stable Equilibrium
U
x
compressed extended
dU/dxWhen x is zero, dU/dx is zero, so F is zero and there are no forces on the system pushing it anywhere.
F = -dU/dx
Unstable EquilibriumExamples:
– A cone on its tip.
When the system is displaced from equilibrium, forces push it farther from equilibrium position.
A potential energy peak or mountain.
Unstable Equilibrium
U
x
dU/dx
F = -dU/dx
When x is negative, dU/dx is positive, so F is negative and pushes system further from equilibrium.
Unstable Equilibrium
U
x
dU/dxF = -dU/dx
When x is positive, dU/dx is negative, so F is positive and pushes system further from equilibrium.
Unstable Equilibrium
U
x
dU/dxF = -dU/dx
When x is zero, dU/dx is zero, so F is zero and no forces are trying to push the system anywhere.
Neutral EquilibriumExamples:
– A book on a desk.
When the system is displaced from equilibrium, it just stays there.
A potential energy plane.
Neutral Equilibrium
U
x
F = -dU/dx
When x changes, dU/dx is zero, so F is zero and no force develops to push the system toward or away from equilibrium.
dU/dx