The final exam will
include:
1. Dimensional
Analysis
2. 6 Simple
Machines
3.
Robot Drive Trains and Gear Ratios
Simple
Machines:
These devices were all in common use for centuries
before Leonardo's time. Each one makes work easier to do by providing
some trade-off between the force applied and the distance over which
the force is applied.
Wheel
and Axle
In this machine a wheel or spoke is locked to a
central axle so that when one is turned the other must turn. A
longer motion at the edge of the wheel is converted to a shorter
more powerful motion at the axle. In reverse, a short powerful force
at the axle will move the wheel's edge a greater
distance.
Pulley
System
A single pulley simply reverses the direction of a
force. When two or more pulleys are connected together, they permit
a heavy load to be lifted with less force. The trade-off is that the
end of the rope must move a greater distance than the load.
Wedge
A
wedge converts motion in one direction into a splitting motion that
acts at right angles to the blade. Nearly all cutting machines use
the wedge. A lifting machine may use a wedge to get under a load.
Screw
A
screw is a central core with a thread or groove wrapped around it to
form a helix. While turning, a screw converts a rotary motion into a
forward or backward motion.
Lever
A
lever is a stiff rod that rotates around a pivot point. Downward
motion at one end results in upward motion at the other end.
Depending on where the pivot point is located, a lever can multiply
either the force applied or the distance over which the force is
applied.
Other
Elements of Machines:
Though
these devices were also used in machines in Leonardo's time, he
experimented with them and made changes to improve how they worked.
He also combined them in many exciting new ways to createmachines
and inventions that had never been seen before.
Gears
Gears
are toothed or pegged wheels meshed together to transmit motion and
force. In any pair of gears the larger one will rotate more slowly
than the smaller one, but will rotate with greater force. Each gear
in a series reverses the direction of rotation of the previous
gear.
Bevel
Gears
Gears that mesh at an angle change the direction of
rotation.
Worm
Gear
A worm gear is a combination of a gear meshed with the
threads of a screw. This combination changes the direction of
turning motion by ninety degrees. Worm gears also decrease the speed
of turning from screw to gear and increase its force.
Rack
and Pinion
A single gear, the pinion, meshes with a sliding
toothed rack. This combination converts rotary motion to back and
forth motion. Windshield wipers in cars are powered by a rack and
pinion mechanism. A small pinion at the base of the wiper meshes
with a sliding rack below.
Cam
A
cam is a wheel with shaped bumps on it. Cams are often connected to
rods, levers, or springs. In the gravity trip hammer shown here, the
bumps on the turning cam push down on the end of the lever making it
raise the hammer again and again.
Crank
and Rod
The crank is a wheel with a pivoting arm attached
near its edge. The arm is attached by a hinge to a rod. When the
crank turns, the rod is pushed back and forth. Alternatively, if the
rod is pushed back and forth at the right speed, the crank will
turn. The crank and rod shown here are part of giant steam engine.
Chains
and Belts
A chain or belt connects two separated wheels so
that one turns, the other will turn in the same direction.
Ratchet
A
ratchet is a device that allows a wheel to turn in only one
direction. The ratchet wheel has specially shaped teeth. A bar on a
pivot called the "pawl" is fixed above the ratchet wheel.
The pawl slides over the teeth of the ratchet in one direction, but
blocks the motion of the teeth if the wheel turns in the other
direction.
Which
elements of machines do you see in each gadget? Click on only the
ones you see.
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Hand Powered Drill
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Wing-Handle Corkscrew
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Scissor Style Jack
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A simple
machine is
any device that only requires the application of a single force to
work. Simple machines all convert a smaller amount of force
exerted over a larger distance to a greater amount of force exerted
over a shorter distance, or vice versa.
The
ratio of the output force to the input force is the called mechanical
advantage. The mechanical advantage of a lever is equal to the ratio
of its two arms. For example, the mechanical advantage of an inclined
plane (with the force acting parallel to the plane) is the cosecant
of the angle of inclination.
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Note:
Real machines are also affected by factors such as friction and
elasticity, so the actual mechanical advantage of a simple machine
will usually differ from its theoretical value.
Simple
machines are often used in combination as components of more complex
machines; for example the Archimedes screw, which is a pump, is an
example of a complex machine where the screw is a helical inclined
plane.
The
Lever
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The
principle of the lever tells us that the above is in static
equilibrium, with all forces balancing, if:
F1
X D1 = F2 X D2.
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In
physics, a lever (from
Old French levier,
the agent noun to lever "to
raise", c. f. levant)
is a rigid object that is used with an appropriate fulcrum or
pivot point to multiply the mechanical force that can be applied to
another object. This is also termed mechanical advantage, and is
one example of the principle of moments.
Definition
1: The fulcrum is the pivotal
point where the lever rotates.
Definition
2: The mechanical
advantage (MA) is
the factor by which a machinemultiplies
the force put into it.
Definition
3: The effort arm represents where force is input. The effort arm is
always larger than the resistance arm.
How
do we Calculate Mechanical Advantage of a Lever-
For
the Lever:
MA = length of effort arm ÷ length
of resistance arm.
About
the Lever
Force
and levers
The
force applied (at end points of the lever) is proportional to the
ratio of the length of the lever arm measured between the fulcrum
and application point of the force applied at each end of the lever.
Mathematically, this is expressed by M = Fd.
What
are the three classes of levers
There
are three classes of levers representing variations in the location
of the fulcrum and the input and output forces.
First-class
levers
Examples:
Seesaw
(also known as a teeter-totter)
Crowbar
Pliers
(double lever)
Scissors
(double lever)
Second-class
levers
Examples:
Wheelbarrow
Nutcracker
(double lever)
The
handle of a pair of nail clippers
An
oar
Third-class
levers
Examples:
Human
arm
Tongs
(double lever) (where hinged at one end, the style with a central
pivot is first-class)
Catapult
Any
number of tools, such as a hoe or scythe
The
main body of a pair of nail clippers, in which the handle exerts
the incoming force
Screw
Rotating
screw and fixed trough
A
screw is a specialized application of the wedge or inclined
plane. It contains a wedge, wound around an interior cylinder or
shaft, that either fits into a corresponding plane in a nut, or forms
a corresponding plane in the wood or metal as it is inserted. The
technical analysis (see also statics, dynamics) to determine the
pitch, thread shape or cross section, coefficient of friction (static
and dynamic), and holding power of the screw is very similar to that
performed to predict wedge behavior. Wedges are discussed in the
article on simple machines.
Critical
applications of screws and bolts will specify a torque that must be
applied when tightening. The main concept is to stretch the bolt, and
compress the parts being held together, creating a spring like
assembly. The stretch introduced to the bolt is called apre-load.
When external forces try to separate the parts, the bolt sees no
strain unless the pre-load force is exceeded (this takes some effort
to imagine).
As
long as the pre-load is never exceeded, the bolt or nut will never
come loose (assuming the full strength of the bolt is used). If the
full strength of the bolt is not used (eg. a steel bolt into aluminum
threads) then a thread locking adhesive may be used.
If
the pre-load is exceeded during normal use the joint will eventually
fail. The pre-load is calculated as a percentage of the bolt's yield
tensile strength, or the strength of the threads it goes into,
whichever is less.
Wedge
A
wedge is a simple machine used to separate two objects, or portions
of objects, through the application of force. A wedge is made up of
two inclined planes. These planes meet and form a sharp edge. This
edge can split things apart. Wedges are used as either separating or
holding devices. There are two major differences between inclined
planes and wedges. First, in use, an inclined plane remains
stationary while the wedge moves. Second, the effort force is applied
parallel to the slope of an inclined plane, while the effort force is
applied to the vertical edge (height) of the wedge. Force
multiplication varies inversely with the size of the wedge angle; a
sharp wedge ( small inclined angle ) yields a large force.
Wedges
are used as either separating or holding devices. A wedge can either
be composed of one or two inclined planes. A double wedge can be
thought of as two inclined planes joined together with their sloping
surfaces outward.
Examples
of wedges are: knives, axes, forks and nails
Inclined
plane
An Inclined
plane or a ramp is one of the basic
machines. It reduces the force necessary to move a load a certain
distanc'e up by providing a path for the load to move at a low angle
to the ground. This lessens the needed force but increases the
distance involved, so that the amount of work stays the same.
Examples
are ramps,
sloping roads, chisels, hatchets, plows, air hammers, carpenter's
planes and wedges.
The most canonical example of an inclined plane is a sloped surface;
for example a roadway to bridge a height difference. The inclined
plane is used to reduce the force necessary
to overcome the force of gravity when elevating or lowering a heavy
object. The ramp makes it easier to move a physical body vertically
by extending the distance traveled horizontally (run) to achieve the
desired elevation change (rise).
In
civil engineering the slope or ratio of rise/run is often referred
to as a grade or gradient. Others may also call it tilt.
Ramps
are used as an alternative for a stairway for wheelchairs, buggies
and shopping carts. Ramps may zigzag. There are also moving ramps.
By
changing the angle of the ramp one can usefully vary the force
necessary to raise or lower a load. For example:
A
wagon trail on a steep hill will often traverse back and forth to
reduce the gradient experienced by a team pulling a heavily loaded
wagon. This same techique is used today in modern freeways which
travel through steep mountain passes. Some steep passes have
separate truck routes that reduce the grade by winding along a
separate path to rejoin the main route after a particularly steep
section is past while smaller automobiles take the straighter
steeper route with a resulting savings in time.
It
is important in the history of science, engineering and technology
for a variety of reasons:
The
ramp or inclined plane was useful in building early stone edifices,
in roads and aqueducts, and military assault of fortified positions.
Experiments
with inclined planes helped early physicists such as Galileo Galilei
quantify the behavior of nature with respect to gravity, mass,
acceleration, etc.
Detailed
understanding of inclined planes and their use helped lead to the
understanding of how vector quantities such as forces can be
usefully decomposed and manipulated mathematically. This concept of
superposition and decomposition is critical in many modern fields of
science, engineering, and technology.
Other
simple machines based on the inclined plane include the blade, in
which two inclined planes placed back to back allow the two parts of
the cut object to move apart using less force than would be needed
to pull them apart in opposite directions.
MECHANICAL
ADVANTAGE OF THE INCLINED PLANE
If
an object is put on an inclined plane it will move if the force of
friction is smaller than the combined force of gravity and normal
force. If the angle of the inclined plane is 90 degrees (rectangle)
the object will free fall.
Wheel
and Axle
The
wheel and axle consists of a handwheel (a disc or lever arm with a
handle) which turns an axle around which a chord is wound. A heavy
weight attached to the chord can be lifted more easily because
ofmechanical advantage.
The
mechanical advantage of a wheel and axle is the ratio of the radius
of the wheel to the radius of the axle. If the radius of the wheel is
four times greater than the radius of the axle, every time you turn
the wheel once, your force will be multiplied four times.
Examples
of wheel and axles are:
Bicycles,
Ferris wheels, gears, wrenches, door-knobs and steering wheels.
Pulley
A pulley is
a wheel with a groove along its edge, for holding a rope or cable.
Pulleys are usually used in sets designed to reduce the amount of
force needed to lift a load. However, the same amount of work is
necessary for the load to reach the same height as it would without
the pulleys. The magnitude of the force is reduced, but it must act
through a longer distance. Pulleys are usually considered one of
the simple machines.
Types
of Pulleys
movable
pulley
A movable
pulley has a free axle, and is used to transform forces -
when stationary the total force on the axle balances the total force
provided by the tension in the rope (which is constant in magnitude
in each segment). As illustrated below, if one end of a rope is
attached to a fixed object, pulling on the other end will apply a
doubled force to any object attached to the axle.
compound
pulley
Block
and Tackle
A compound
pulley is a system of movable pulleys. The mechanical
advantage can be increased by using a block and tackle, where there
are several pulleys on each axle. Plutarch reported that Archimedes
moved an entire warship, laden with men, using compound pulleys and
his own strength.
Simple
Machines Quiz 1
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1.
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A screw
and a wedge are both similar to what other kind of simple machine?
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a.
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a lever
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b.
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a pulley
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c.
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a wheel
and axle
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d.
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an
inclined plane
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2.
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What is
the basic principle of using a lever?
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a.
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The
resistance force is greater than the effort force.
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b.
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The
amount of work the lever performs is more than the work you
perform.
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c.
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You
exert a small force over a long distance.
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d.
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All of
the above are correct.
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3.
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What is
the mechanical advantage of a lever if the effort arm is 2.0 m
long and the resistance arm is 0.5 m long?
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a.
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impossible
to calculate with the information provided
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b.
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1
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c.
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4
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d.
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0.25
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4.
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How are
work, power, and time related?
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a.
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Work is
power over time.
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b.
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Time is
power over work.
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c.
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Power is
work over time.
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d.
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Work is
time over power.
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5.
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How can
you increase the efficiency of a machine?
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a.
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by
reducing losses due to friction
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b.
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by
increasing the work output of the machine without increasing input
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c.
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by
decreasing the work input to the machine without increasing output
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d.
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All of
the above.
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6.
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What is
the mechanical advantage of a lever if the effort force is 2 N and
the resistance force is 8 N?
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7.
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How does
increasing the number of threads on a screw change the mechanical
advantage of a screw?
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a.
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The
mechanical advantage of the screw increases.
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b.
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The
mechanical advantage of the screw is unaffected.
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c.
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The
mechanical advantage of the screw decreases.
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d.
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It is
impossible to calculate the mechanical advantage of a screw.
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8.
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What is
the science of making machines suitable for the people who use
them?
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a.
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biotechnology
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b.
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ergonomics
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c.
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tectonics
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d.
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economics
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9.
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What is
the term used for the force applied to a machine?
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a.
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resistance
force
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b.
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machine
force
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c.
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applied
force
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d.
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effort
force
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10.
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Suppose
you need to place a 5-kg block of ice onto the shelf of a freezer
2.0 m high. The ice will soon start melting, so you must place the
ice in the freezer in 10 s. How much power will you need?
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a.
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1 W
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b.
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25 W
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c.
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2.5 W
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d.
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10 W
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11.
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What
does the mechanical advantage of a machine tell you?
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a.
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how much
the effort force is multiplied
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b.
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how
useful a machine may be
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c.
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how much
bigger resistance force is compared to effort force
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d.
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All of
the above are correct.
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12.
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Which of
the following is an example of a compound machine?
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a.
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a
screwdriver
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b.
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an axe
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c.
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a
doorknob
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d.
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a can
opener
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13.
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What is
the basic advantage of single fixed pulley?
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a.
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The
force exerted is reduced compared to the resistance force.
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b.
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The
pulley changes the direction of the effort force.
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c.
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The
pulley changes the distance over which the effort force is
applied.
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d.
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The
pulley decreases the amount of work needed to move an object.
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14.
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Suppose
you must lift a 100-kg box to a table 1 m high. How does the use
of an inclined plane affect the amount of work you must perform to
lift the box?
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a.
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You can
perform less work with a longer inclined plane.
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b.
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An
inclined plane does not change the amount of work you must
perform.
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c.
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An
inclined plane increases the amount of work you must perform.
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d.
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You can
perform less work with a shorter incline plane.
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15.
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When
using a wheel and axle, what happens as the size of the wheel
increases?
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a.
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You need
more effort force to turn the axle.
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b.
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You need
less effort force to turn the axle.
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c.
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The
distance the axle moves as you turn the wheel increases.
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d.
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The
amount of work you must do is less.
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Simple Machines Quiz 2
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Figure 1
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6.
A simple machine that is actually a kind of inclined plane is a
a)
pulley
b) wedge
c) gear
d) balance
7.
A pulley system has 2 sections of ropes that lift the load. The
mechanical advantage of the system is
a) 0
b)
1
c) 2
d) 3
8.
The efficiency of a simple machine
a) is always greater
than 100%
b) is equal to 100%
c) is always 50%
d)
is always less than 100%
9.
A knife is an example of a (an)
a)
inclined plane
b) ramp
c) wedge
d) pulley
10. Increasing
the slant of an inclined plane decreases its
a) work
output
b) effort force
b) power
d)
mechanical advantage
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In the example below, the
DRIVER has 60 teeth and because it is the largest we say that it
revolves once. The DRIVEN gear has 30 teeth. Simply divide 60
teeth by 30 teeth to work out the number of revolutions of the
driven gear.
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Work out the Velocity Ratio (Gear
Ratio);
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Work out the Velocity Ratio (Gear Ratio);
Work out the Velocity Ratio (Gear Ratio);