1. Chapter 5 Math Review

2. Work • Conservation of Energy • can never get more work out than you put in • trade-off between force and distance Win = Wout Fe × de = Fr × dr

3. Efficiency • Efficiency • measure of how completely work input is converted to work output • always less than 100% due to friction

4. Force • Effort Force (Fe) • force applied to the machine • “what you do” • Resistance Force (Fr) • force applied by the machine • “what the machine does”

5. Mechanical Advantage • Mechanical Advantage (MA) • number of times a machine increases the effort force • MA > 1 : force is increased • MA < 1 : distance is increased • MA = 1 : only direction is changed

6. Fr Fe MA Mechanical Advantage • Find the effort force needed to lift a 2000 N rock using a jack with a mechanical advantage of 10. GIVEN: Fe = ? Fr = 2000 N MA = 10 WORK: Fe = Fr ÷ MA Fe = (2000 N) ÷ (10) Fe = 200 N

7. Resistance arm Effort arm Fulcrum Engraving from Mechanics Magazine, London, 1824 “Give me a place to stand and I will move the Earth.” – Archimedes Lever • Lever • a bar that is free to pivot about a fixed point, or fulcrum

8. Effort arm length Resistance arm length Lever • Ideal Mechanical Advantage (IMA) • frictionless machine • Le must be greater than Lrin order to multiply the force.

9. 20cm Le 160cm Lr IMA Problems • You use a 160 cm plank to lift a large rock. If the rock is 20 cm from the fulcrum, what is the plank’s IMA? GIVEN: Lr = 20 cm Le = 140 cm IMA = ? WORK: IMA = Le ÷ Lr IMA = (140 cm) ÷ (20 cm) IMA = 7

10. 15N ? 0.3m 150N Le Lr IMA Problems • You need to lift a 150 N box using only 15 N of force. How long does the lever need to be if the resistance arm is 0.3m? GIVEN: Fr = 150 N Fe = 15 N Lr = 0.3 m Le = ? MA = 10 WORK: Le = IMA · Lr Le = (10)(0.3) Le = 3 m Total length = Le + Lr Total length = 3.3 m

11. Wheel and Axle • Wheel and Axle • two wheels of different sizes that rotate together • a pair of “rotating levers” Wheel Axle

12. effort radius resistance radius Wheel and Axle • Ideal Mechanical Advantage (IMA) • effort force is usu. applied to wheel • axle moves less distance but with greater force

13. rw 5 cm 20 cm ra IMA Problems • A crank on a pasta maker has a radius of 20 cm. The turning shaft has a radius of 5 cm. What is the IMA of this wheel and axle? GIVEN: rw = 20 cm ra = 5 cm IMA = ? WORK: IMA = rw ÷ ra IMA = (20 cm) ÷ (5 cm) IMA = 4

14. rw ra IMA Problems • A steering wheel requires a mechanical advantage of 6. What radius does the wheel need to have if the steering column has a radius of 4 cm? GIVEN: IMA = 6 rw = ? ra = 4 cm WORK: rw = IMA · ra rw = (6)(4 cm) rw = 24 cm rw ra

15. h l Inclined Plane • Inclined Plane • Slanted surface used to raise objects

16. l h IMA Problems • What is the mechanical advantage of a ramp that is 3 m long and 1.2 m high? GIVEN: IMA=? l = 3 m h = 1.2 m WORK: IMA = l ÷ h IMA = (3 m)÷(1.2 m) IMA = 2.5