1. 1. Work [W] = N*m = J Units: Work done by forces that oppose the direction of motion will be negative. Work and energy A. PositiveB. NegativeC. Zero 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: W f = |f k | |Δx| cos(180°) = -|f k | |Δx| < 0 f Work done by the normal force: W N = |N| |Δx| cos(90°) = 0 Work done by weight: W mg = mg|Δx| cos(θ ) > 0 Definition: 1

2. 2. Work kinetic energy principle Definition: W=K 2 - K 1 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 v 1 = 0 v 2 - ? 2

3. Example: Two blocks (m 1 =2m 2 ) 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? Same force, same distance Same work Same change in kinetic energy 1.Box1 2.Box 2 3.They both have the same 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 3

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

5. Example: A box of unknown mass and initial speed v 0 = 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 5

6. h 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? Example: An object of unknown mass is projected with an initial speed, v 0 = 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? 6

7. Only weight of the pendulum is doing work; weight is a conservative force, so mechanical energy is conserved: L m θ0 θ0 The angle on the other side is also θ 0 ! θ0 θ0 Example: Pendulum (Conservation of energy) 7

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

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

10. 6. Resonance 10