Chapter 5

Math Review

WorkConservation of Energy

can never get more work out than you put in

trade-off between force and distance

Win = Wout

Fe × de = Fr × dr

EfficiencyEfficiency

measure of how completely work input is converted to work output

100%WW

Efficiencyin

out

always less than 100% due to friction

Force

Effort Force (Fe)force applied to the machine“what you do”

Resistance Force (Fr)force applied by the machine“what the machine does”

Mechanical AdvantageMechanical Advantage (MA)

number of times a machine increases the effort force

e

r

FF

MA

MA > 1 : force is increasedMA < 1 : distance is increasedMA = 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:

Fe = ?

Fr = 2000 N

MA = 10

WORK:Fe = Fr ÷ MA

Fe = (2000 N) ÷ (10)

Fe = 200 N

MA

Fr

Fe

LeverLever

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

Resistancearm

Fulcrum

Lever Ideal Mechanical Advantage (IMA)

frictionless machine

r

e

LL

IMA Effort arm length

Resistancearm length

Le must be greater than Lr 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:

Lr = 20 cm

Le = 140 cm

IMA = ?

WORK:IMA = Le ÷ Lr

IMA = (140 cm) ÷ (20 cm)IMA = 7

IMA

Le

Lr

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:

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 mIMA

Le

Lr

0.3m

?

150N

15N

Wheel and AxleWheel and Axle

two wheels of different sizes that rotate together

a pair of “rotatinglevers”

Wheel

Axle

Wheel and Axle Ideal Mechanical Advantage

(IMA) effort force is usu. applied to wheel axle moves less

distance but with greater force

a

w

rr

IMA 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:

rw = 20 cm

ra = 5 cm

IMA = ?

WORK:IMA = rw ÷ ra

IMA = (20 cm) ÷ (5 cm)IMA = 4

IMA

rw

ra

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 = 6rw = ?

ra = 4 cm

WORK:rw = IMA · ra

rw = (6)(4 cm)

rw = 24 cm

IMA

rw

ra

rarw

Inclined Plane Inclined Plane

Slanted surface used to raise objects

hl

hl

IMA

Problems What is the mechanical advantage

of a ramp that is 3 m long and 1.2 m high?

GIVEN:

IMA=?l = 3 mh = 1.2 m

WORK:IMA = l ÷ hIMA = (3 m)÷(1.2 m)IMA =

IMA

l

h

2.5