Conservation of Energy

Path Dependence What happens to work as a rollercoaster goes down hill then up again? What if the roller coaster took a less steep path?

Reversible Process If an object is acted on by a force has its path reversed the work done is the opposite sign. This represents a reversible process. y2 F = -mg h y1

Closed Path If the work done by a force doesn’t depend on the path it is a conservative force. Conservative forces do no work on a closed path. From 1 to 2, the path A or B doesn’t matter From 1 to 2 and back to 1, the path A then the reverse path B gives no work

Nonconservative Force Not all forces are conservative. In particular, friction and drag are not conservative. d Negative work is done by friction to get here F = -mFN -d Negative work is also done returning the box F = mFN

Net Work The work-energy principle is DK = Wnet. The work can be divided into parts due to conservative and non-conservative forces. Kinetic energy DK = Wcon + Wnon d Ff Fg

Kinetic and Potential Energy Potential energy is the negative of the work done by conservative forces. Potential energy DU = -Wcon The kinetic energy is related to the potential energy. Kinetic energy DK = -DU + Wnon The energy of velocity and position make up the mechanical energy. Mechanical energy Emech = K + U

Conservation of Energy Certain problems assume only conservative forces. No friction, no air resistance The change in energy, DE = DK + DU = 0 If the change is zero then the total is constant. Total energy, E = K + U = constant Energy is not created or destroyed – it is conserved.

No Absolute Potential energy reflects the work that may be done. The point U = 0 is arbitrary At the top of a table of height h: U = mg(y+h) The same experiment is shifted by a constant potential mgh: U = mgy + mgh = mgy + C y2 y y1 h

Solving Problems There are some general techniques to solve energy conservation problems. Identify all the potential and kinetic energy at the beginning Identify all the potential and kinetic energy at the end Set the initial and final energy equal to one another Nonconservative forces reduce the final energy

Using Friction and Energy The hill is 2.5 km long with a drop of 800 m. The skier is 75 kg. The speed at the finish is 120 km/h. How much energy was dissipated by friction? q

Friction and Height Find the total change in kinetic energy. Find the total change in potential energy. The difference is due to friction (air and sliding). DK = ½ mv2 - 0 = ½(75 kg)(130 m/s)2 = 5.4 x 105 J DU = mgh = (75 kg)(9.8 m/s2)(-800 m) = -5.9 x 105 J Wnon = DK + DU = -0.5 x 105 J

Universal Gravitational Work Gravity on the surface of the Earth is a local consequence of universal gravitation. How much work can an object falling from very far from the Earth do when it hits the surface? r RE

Universal Gravitational Potential The work doesn’t depend on the path. Universal gravity is a conservative force The potential is set with U = 0 at an infinite distance. Gravity acts at all ranges Gravity is weakest far from the source