Energy and Work Energy is defined as the ability of a body or system of bodies to perform Work.

Energy and 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.

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 Work  Work is the dot product of force and displacement.  Work is a scalar resulting from the interaction of two vectors.

Calculating Work  W = F s = F s cos   W =  F(x) dx  W =  F ds

Vectors and Work F s W = F s W = F s cos 0 o W = F s Maximum positive work

Vectors and Work s W = F s W = F s cos  Only the component of force aligned with displacement does work. Work is less. F 

Vectors and Work Fs  W = F s W = F s cos 180 o W = - F s Maximum negative work.

Gravity often does negative work. mgmg 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.

Units of Work  SI System:  Joule (N m)  British System:  foot-pound  cgs System:  erg (dyne-cm)  Atomic Level:  electron-Volt (eV)

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) x xaxa xbxb W =  F(x) dx xaxa xbxb

Net Work Net work (W net ) is the sum of the work done on an object by all forces acting upon the object.

The Work-Energy Theorem  W net =  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.

Kinetic Energy  A form of mechanical energy  Energy due to motion  K = ½ m v 2 –K: Kinetic Energy in Joules. –m: mass in kg –v: speed in m/s

Mechanical Energy: Potential energy  Energy 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 (U g )  For objects near the earth’s surface, the gravitational pull of the earth is constant, so  W g = mg  x –The force necessary to lift an object at constant velocity is equal to the weight, so we can say   U g = -W g = mgh

Power  Power 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  The rate at which work is done.  P ave = W / t  P = dW/dt  P = F v = Fvcosθ

Units of Power  Watt = J/s  ft lb / s  horsepower 550 ft lb / s 746 Watts

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 F D = bv 2. 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 W  1 h = 3600 s 1 kWh = 1000J/s 3600s = 3.6 x 10 6 J

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

Ideal Spring  Obeys Hooke’s Law.  F s (x) = -kx –F s is restoring force exerted BY the spring.  W s =  F s (x)dx = -k  xdx –W s is the work done BY the spring.  Us = ½ k x 2

Spring Problem Three 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? XY Z

System Boundary

Isolated System E = U + K + E int = 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 = C  E =  U +  K = 0 for gravity  U g = mgh f - mgh i  K = ½ mv f 2 - ½ mv i 2

Law of Conservation of Mechanical Energy E = U + K = C  E =  U +  K = 0 for springs  U g = ½ kx 2  K = ½ mv f 2 - ½ mv i 2

Law of Conservation of Energy E = U + K + E int = C  E =  U +  K +  E int = 0 E int 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 ½mv max 2 = mgh For minimum and maximum points of swing ½mv mgh 1 = ½mv mgh 2 For any points two points in the pendulum’s swing

Spring Energy mm -x m x 0 ½ kx 2 = ½ mv max 2 For maximum and minimum displacements from equilibrium ½ kx ½ mv 1 2 = ½ kx ½ mv 2 2 For any two points in a spring’s oscillation

Spring and Pendulum Energy Profile E x Total Energy U K

Equilibrium  The 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 Equilibrium  Examples: –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/dx When 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/dx When 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 Equilibrium  Examples: –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 positive, dU/dx is negative, so F is positive and pushes system further from equilibrium.

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/dx F = -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 Equilibrium  Examples: –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

Potential Energy and Force F(r) = -dU(r)/dr F x = -  U/  x F y = -  U/  y F z = -  U/  z

Equilibrium equations  Stable –  U/  x = 0,  2 U/  x 2 > 0  Unstable –  U/  x = 0,  2 U/  x 2 < 0  Neutral –  U/  x = 0,  2 U/  x 2 = 0

Atomic bonds and Equilibrium x U Lowest energy inter-atomic separation Atoms too close Atoms too far apart

Where is Gravitational Potential Energy Zero?  U g 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 U g is zero. Usually this is the lowest point an object can reach in a given situation  Then, anything above this point is a positive U g.