1. Work and energy 1. Work Wf = |fk| |Δx| cos(180°) = -|fk| |Δx| < 0
Definition:
Work done by forces that oppose the direction of motion will be negative.
Units:
[W] = N*m = J
Example: A block slides down a rough inclined surface. The forces acting on the block are depicted below. The work done by the frictional force is:
A. Positive B. Negative C. Zero f
Wf = |fk| |Δx| cos(180°) = -|fk| |Δx| < 0
Work done by the normal force:
WN = |N| |Δx| cos(90°) = 0
Work done by weight:
Wmg = |mg| |Δx| cos(θ ) > 0

2. W=K2 - K1 2. Work kinetic energy principle Definition:
Example: An 80-g arrow is fired from a bow whose string exerts an average force of 100 N on the arrow over a distance of 49 cm. What is the speed of the arrow as it leaves the bow?
m = 80 g
F = 100 N
d = 49 cm
v1= 0
v2 - ?

3. Same force, same distance Same work
Example: Two blocks (m1=2m2) are pushed by identical forces, each starting at rest at the same start line. Which object has the greater kinetic energy when it reaches the same finish line? 
Box1              
Box 2
They both have the same kinetic energy
Same force, same distance
Same work
Same change in kinetic energy
Example: A ball is dropped and hits the ground 50 m below. If the initial speed is 0 and we ignore air resistance, what is the speed of the ball as it hits the ground?
We can use kinematics or… the WKE theorem
Work done by gravity: mgh

4. 8. Conservation of energy
3. Potential energy
a) Gravitational potential energy:
b) Elastic potential energy (spring):
8. Conservation of energy

5. Example: A box of unknown mass and initial speed v0 = 10 m/s moves up a frictionless incline. How high does the box go before it begins sliding down? m
Only gravity does work (the normal is perpendicular to the motion), so mechanical energy is conserved.
We can apply the same thing to any “incline”! h
Turn-around point: where K = 0
E K U
v = 0

6. Example: A roller coaster starts out at the top of a hill of height h.
How fast is it going when it reaches the bottom? h
Example: An object of unknown mass is projected with an initial speed, v0 = 10 m/s at an unknown angle above the horizontal. If air resistance could be neglected, what would be the speed of the object at height, h = 3.3 m above the starting point?

7. Example: Pendulum (Conservation of energy)
Only weight of the pendulum is doing work; weight is a conservative force, so mechanical energy is conserved: θ0 θ0 L m
The angle on the other side is also θ0!

8. 4. Energy in the simple harmonic motionU E x –A A K t U t E t
Total mechanical energy is constant through
oscillation: conservation of energy!

9. x(t) t b – damping constant 5. Damped Harmonic Motion Damping force is
proportional to velocity:
b – damping constant
(Shows how fast oscillations decay)
Optional math:

10. 6. Resonance