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

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 - ?

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

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

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

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?

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!

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

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:

6. Resonance