WORK

1. WORK • The quantity obtained when force is multiplied by displacement • Connected to concept of energy • Some has energy when it has the capacity to due work, that is, apply force over some distance • Conversely, work changes the energy status of a system

2. Work  W = F x • Scalar • Unit: N m = joule, J • The force & a component of displacement must coincide • Positive Work = Force is applied in same direction as motion

4. A component of force must be applied in direction of displacement • F ┴ x = no work done • Pushing on wall = no work • Can be positive -  energy status, negative -  energy status, or zero

6. Multiple forces acting on object, the total work = work done by each force separately  Wtotal = W1 + W2 + … = Σ W • Can also be found by determining vector sum of forces: Ftotal  Wtotal = Ftotal x

7. KINETIC ENERGY & THE WORK – KINETIC ENERGY THEOREM • Positive work  energy status • Causes objects to move faster • Negative work = slower

8. Consider an apple falling through air • As apple falls through distance y, a kinematic describes its motion

9. The quantity measuring the change in motion = kinetic energy, K  K = ½ m v2 Unit: kg (m/s)2 = J • Scalar quantity, but never negative

10. Connection between work & energy = work-energy theorem  Wnet = Δ K • V is not linear with respect to K • Doubling v ≠ 2 K • V is squared  4 K

11. CONSERVATIVE & NONCONSERVATIVE FORCES • The work done is stored in the form of energy that can be released later • Consider gravity • Lifting a box at constant speed does work  W = F x = w y = mgy • Releasing the box, gravity does the same work equal to the same amount of K

12. Contrast to friction • Work is done sliding a box around on the floor  W = μ m g x • But, when released, the box does not move  no K • A nonconservative force – work is converted into other energy forms

13. Conservative forces are path independent • Only depends on initial & final positions • Route taken does not effect results • Any closed path, total work = zero

16. POTENTIAL ENERGY • Work done lifting an object higher position changes energy status • W did not  K, it is stored energy by virtue of its position • It gained potential energy, U • It has potential to do work • Gravitational Potential Energy, Ug  Ug = m g y Unit: J

17. The location of zero potential is problem dependent • Usually a convenient location • Often the lowest point of problem

18. POTENTIAL ENERGY OF SPRINGS • Springs are another conservative force • Springs can store energy like gravity can • Experiments show a linear relationship between force applied and amount of stretch  F α x • Constant that creates an equality = k – spring constant F = - k x Hooke’s Law

19. Negative indicates F is a restoring force  F is in opposite direction as x • Unit for k = N / m • How stiff the spring is • Large k = stiff spring

20. Since force is not constant, we cannot use W = F x • Consider the graphical representation of F vs. x • If we were to stretch spring from x = 0 to x, the area under curve represents work

21. Area = ½ (base)(height) = ½ (x)(kx)  W = ½ k x2 • Work done stretching/compressing = the energy stored  Us = ½ k x2

22. CONSERVATION OF MECHANICAL ENERGY • Mechanical energy, E = U + K • Regardless of what happens in systems with conservative forces, E is always conserved  E = constant • Energy can be converted in form, but the sum remains constant

23. Makes many problems a simple matter of bookkeeping  Initial energy = final energy • For conservative forces only

24. WORK DONE BY NONCONSERVATIVE FORCES • Nonconservative forces change the amount of mechanical energy in a system • Typically a decrease due to conversion of thermal energy K0 + U0 ≠ K + U Einitial > Efinal

25. To balance equation, an additional factor must be added to right • Call it Wnc K0 + U0 = K + U + Wnc