Work Work is the bridge between Force and Energy.

The General Work Equation W = F Dx cos  F: force (N) Dx : displacement (m) : angle between force and displacement (often 0o)

Units of Work SI System: Joule (N m) British System: foot-pound (not used in Phys B) cgs System: erg (dyne-cm) Atomic Level: electron-Volt (eV)

More on the Joule SI unit for force = N SI unit for length = m SI unit for work is = N•m or Joule(I J = 1N•m) 1 Joule of work is done when 1 N acts on a body moving it a distance of 1 meter

Counterintuitive Results There is no work done by a force if it causes no displacement. Forces perpendicular to displacement can do no work. The normal force and gravity do no work when an object is slid on a flat floor, for instance.

Work and Energy By doing positive work on an object, a force or collection of forces increases its mechanical energy in some way. The two forms of mechanical energy are called potential and kinetic energy.

Kinetic Energy Energy due to motion K = ½ m v2 K: Kinetic Energy in Joules. m: mass in kg v: speed in m/s

The Work-Energy Theorem Wnet = DKE When net work due to all forces acting on an object is positive, the kinetic energy of the object will increase (positive acceleration). When net work due to all forces is negative, the kinetic energy of the object will decrease (deceleration). When there is no net work due to all forces acting on an object, the kinetic energy is unchanged (constant speed).

Work and graphs F(x) The area under the curve of a graph of force vs displacement gives the work done by the force in performing the displacement. xa xb x

Power The rate of which work is done. When we run upstairs, t is small so P is big. When we walk upstairs, t is large so P is small.

Power in Equation Form P = W/t work/time P = F V (force )(velocity)

Unit of Power SI unit for Power is the Watt. 1 Watt = 1 Joule/s Named after the Scottish engineer James Watt (1776-1819) who perfected the steam engine.

How We Buy Energy… 1 kWh = 1000J/s • 3600s = 3.6 x 106J 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 106J

Hooke’s Law: Springs stretching Fapp = kx 100 -100 -200 200 F(N) 1 2 3 4 5 x (m) m m x F Wapp = ½ kx2

Springs:compressing Fapp = kx m m x F Wapp = ½ kx2 100 -100 -200 200 Fapp = kx 100 -100 -200 200 F(N) -4 -3 -2 -1 x (m) m m x F Wapp = ½ kx2

Work and Energy Work done on a system by a force or forces causes a change in energy of the system. Positive work on a system increases energy in some way. Negative work decreases energy in some way.

More about force types Conservation of Mechanical Energy holds Conservative forces: Work in moving an object is path independent. Work in moving an object along a closed path is zero. Work done against conservative forces increases potential energy; work done by them decreases it. Ex: gravity, 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 Conservation of Energy holds only for ALL Forces Conservation of Mechanical Energy holds only for Conservative Forces

Potential energy Energy an object possesses by virtue of its position or configuration. Represented by the letter U. Examples: Gravitational Potential Energy Spring Potential Energy

Potential gravitational energy Ug = mgh

Another form of Potential Energy Springs also can possess potential energy (Us). Us = ½ kx2 Us: spring potential energy (J). k: force constant of a spring (N/m). x: the amount the spring has been stretched or compressed from its equilibrium position (m).

Where is Spring Potential Energy Zero? Us is zero when a spring is in its preferred, or equilibrium, position. This means the spring is neither compressed nor extended.

The Work-Energy Theorem again Wnet =  U Work is done when the potential energy of an object changes For example, if you apply a force to a spring and store energy in the spring, you have done work on the spring. This work is against the way the spring wants to be (relaxed), so in this example, we say this work is negative.

Law of Conservation of Energy In any isolated system, the total energy remains constant. E = constant

Law of Conservation of Energy Energy can neither be created nor destroyed, but can only be transformed from one type of energy to another.

Law of Conservation of Mechanical Energy E = U + K = C E = U + K = 0 for gravity Ug = mghf - mghi K = ½ mvf2 - ½ mvi2

Law of Conservation of Mechanical Energy E = U + K = C E = U + K = 0 for springs Ug = ½ kxf2 - ½ kxi2 K = ½ mvf2 - ½ mvi2

Pendulums and Energy Conservation Energy goes back and forth between K and U. At highest point, all energy is U. As it drops, U goes to K. At the bottom , energy is all K.

Pendulum Energy ½mvmax2 = mgh K1 + U1 = K2 + U2 For minimum and maximum points of swing h K1 + U1 = K2 + U2 For any points 1 and 2.

Springs and Energy Conservation Transforms energy back and forth between K and U. When fully stretched or extended, all energy is U. When passing through equilibrium, all its energy is K. At other points in its cycle, the energy is a mixture of U and K.

Spring Energy K1 + U1 = K2 + U2 = E m -x m ½kxmax2 = ½mvmax2 m x All U All U All K K1 + U1 = K2 + U2 = E For any two points 1 and 2 m -x m ½kxmax2 = ½mvmax2 For maximum and minimum displacements from equilibrium m x

Law of Conservation of Energy (with dissipative forces) E = U + K + Eint= C U + K +  Eint = 0 Eint is thermal energy. Mechanical energy may be converted to and from heat.