Chapter 5
Math Review
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
Efficiency
 Efficiency
 measure of how completely
work input is converted to
work output
Wout
Efficiency 
 100%
Win
 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 Advantage
 Mechanical Advantage (MA)
 number of times a machine
increases the effort force
Fr
MA 
Fe
 MA
 MA
 MA
> 1 : force is increased
< 1 : distance is increased
= 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
Fr
MA Fe
WORK:
Fe = Fr ÷ MA
Fe = (2000 N) ÷ (10)
Fe = 200 N
Lever
 Lever
 a bar that is free to pivot
about a fixed point, or
fulcrum
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
 Ideal Mechanical Advantage
(IMA)
 frictionless machine
Le
IMA 
Lr
 Le
Effort arm length
Resistance
arm length
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 = ?
Le
IMA
Lr
WORK:
IMA = Le ÷ Lr
IMA = (140 cm) ÷ (20
cm)
20cm
IMA = 7
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
15N
Le = (10)(0.3)
?
Le = 3 m
Total length = Le + Lr
Total length = 3.3 m
0.3m
150N
Le
IMA
Lr
Wheel and Axle
 Wheel and Axle
 two wheels of different sizes
that rotate together
 a pair of
Wheel
“rotating
levers”
Axle
Wheel and Axle
 Ideal Mechanical Advantage
(IMA)


effort force is usu.
applied to wheel
axle moves less
distance but with
greater force
rw
IMA 
ra
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 = ?
rw
IMA
ra
WORK:
IMA = rw ÷ ra
IMA = (20 cm) ÷ (5 cm)
IMA = 4
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?
WORK:
GIVEN:
IMA = 6
rw = ?
ra = 4 cm
rw
IMA
ra
rw = IMA · ra
rw = (6)(4 cm)
rw = 24 cm
ra
rw
Inclined Plane
 Inclined Plane

Slanted surface
used to raise
objects
l
IMA 
h
l
h
Problems
 What is the mechanical advantage of a
ramp that is 3 m long and 1.2 m high?
WORK:
GIVEN:
IMA=?
l=3m
h = 1.2 m
IMA = l ÷ h
IMA = (3 m)÷(1.2 m)
IMA = 2.5
l
IMA
h