Chapter 5 Math Review

Work  Conservation of Energy can never get more work out than you put in trade-off between force and distance W in = W out F e × d e = F r × d r

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

Force  Effort Force (F e ) force applied to the machine “what you do”  Resistance Force (F r ) force applied by the machine “what the machine does”

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

Mechanical Advantage  Find the effort force needed to lift a 2000 N rock using a jack with a mechanical advantage of 10. GIVEN: F e = ? F r = 2000 N MA = 10 WORK : F e = F r ÷ MA F e = (2000 N) ÷ (10) F e = 200 N MA FrFr FeFe

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

Lever  Ideal Mechanical Advantage (IMA) frictionless machine Effort arm length Resistance arm length L e must be greater than L r in order to multiply the force.

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: L r = 20 cm L e = 140 cm IMA = ? WORK : IMA = L e ÷ L r IMA = (140 cm) ÷ (20 cm) IMA = 7 IMA LeLe LrLr 20cm 160cm

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: F r = 150 N F e = 15 N L r = 0.3 m L e = ? MA = 10 WORK : L e = IMA · L r L e = (10)(0.3) L e = 3 m Total length = L e + L r Total length = 3.3 m IMA LeLe LrLr 0.3m ? 150N 15N

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

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

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: r w = 20 cm r a = 5 cm IMA = ? WORK : IMA = r w ÷ r a IMA = (20 cm) ÷ (5 cm) IMA = 4 IMA rwrw rara 5 cm 20 cm

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 r w = ? r a = 4 cm WORK : r w = IMA · r a r w = (6)(4 cm) r w = 24 cm IMA rwrw rara rara rwrw

Inclined Plane  Inclined Plane Slanted surface used to raise objects h l

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 = IMA l h 2.5