Welcome back to Physics 211

1. Welcome back to Physics 211 Today’s agenda: Mechanical energy Conservation of energy

2. Reminder … • Exam 2 in class on Thursday • Seating will be posted • Closed book • Material: forces, Newton’s laws, work, power, kinetic and potential energy (not oscillations) • Practice tests, formula summary online

3. Work, kinetic energy, and the work-kinetic energy theorem

4. Work done on an object by the earth W(on object by earth) = – m gDh, where Dh = hfinal – hinitial is the change in height. The work done on the object by the earth only depends on the change in height from the initial to the final point.

5. Defining gravitational potential energy The change in gravitational potential energy of the object-earth system is just another name for the negative value of the work done on an object by the earth.

6. Changes in gravitational potential energy • If the object moves upward: (Note that Dh is a positive quantity.) • If the object moves downward all signs are reversed.

7. Total energy for object moving under gravity • Notice: definition of gravitational potential energy allows us to write down a conservation law for the total energy D (K+Ug)=0 => E=Ug+K=constant

8. Demo – hot wheels • Give alternative description of motion of car on track • Car moves so the sum of its potential and kinetic energies remains constant • (Equal to initial potential energy)

9. Potential energy U(x) • Energy object possesses by virtue of its position x • Equal to (-) work done on object moving it to x • Implies existence of a (conservative) force which acts on object

10. Conservative forces • Those forces which do work on a object in moving it from one position to another in such a way that the work done is path independent. • eg. gravity, electric forces, spring forces • Not friction, air resistance

11. Conservation of mechanical energy The (total) mechanical energy of an object or system is said to be conserved if the sum of the kinetic and potential energies never changes. This is the case whenever work is done only by conservative forces (such as the weight force or spring force).

12. Pendulum demo • Energy (K+U) should be constant • If pendulum released with zero speed will return to same point (height) with zero speed (ignoring air drag, friction etc)

13. Eg. springs … • Force F=-kx (Hooke’s law). • How do we calculate work done in stretching spring ? • Need general definition of work for variable forces …

14. General work (1D) Consider spring at extension x Extra work stretching spring to x+Dx DW=F Dx Total work W= F(x)dx=area under F vs x x Area=area of large triangle – area of small triangle F x

15. Work on spring Stretch from xI to xF W= -kxdx = -1/2kxF2+1/2kxI2 Thus change in elastic potential energy = work done stretching spring 1/2k(xF2-xI2) -- stored energy in spring

16. Energy in simple-harmonic motion K=1/2mv2 U=1/2kx2 Use x=acos wt, v=dx/dt, w2=k/m

17. Conservation of total energy The total energy of an object or system is said to be conserved if the sum of all energies (including those outside of mechanics that have not yet been discussed) never changes. This is believed to be always the case