Chapters 10/11Work, Power, Energy, Simple Machines
10.1 Energy and Work
• Some objects, because of their– Composition– Position– movement
Possess the ability to cause change, or to do Work.
Anything that has energy has the ability to do work.
In this chapter, we focus on Mechanical Energy only….
Old Man on the Mountain (before and after)
A. Energy of Things in Motion
• Called Kinetic Energy… here’s the derivation…starting with an acceleration equation…
advv 220
21
sidesto both V- add 2
advv 220
21
advv 220
21
advv 220
21 Substitute F/m for a
m
fdvv 22
021
m
fdvv 22
021
Multiply by ½ m
fdmvmv 20
21 2
1
2
1
fdmv 2
2
1
Let’s look at each side of this equation, one side at a time….
• Left side contains terms that describe energy of a system
2
2
1mvKE
…where the change in velocity is due to work being done.
Kinetic EnergyKinetic energy is the energy of motion. By definition kinetic energy is given by:
KE = ½ m v 2
The equation shows that . . .
. . . the more kinetic energy it has.
• the more mass a body has
• or the faster it’s moving
K is proportional to v2, so doubling the speed quadruples kinetic energy, and tripling the speed makes it nine times greater.
Derive the unit for Energy, the Joule!!!!
SI Kinetic Energy UnitsThe formula for kinetic energy,
KE = ½ m v 2
shows that its units are:
kg · (m/s)2
= kg · m 2 / s 2
= (kg · m / s 2 ) m
= N · m
= Joule
So the SI unit for kinetic energy is the Joule, just as it is for work. The Joule is the SI unit for all types of energy.
Sample Calculations….
• What is the kinetic energy of a 75.0 kg warthog sliding down a muddy hill at 35.0 m/s?
• What is the kinetic energy of a 50.0 kg anvil after free-falling for 3.0 seconds?
Mechanical Work
Right Side of out earlier equation implies that a force, applied through a distance, causes changes in KE
fdmv 2
2
1
Work-Energy Theorem
fdKE Simply says that by doing work on a system, you increase the kinetic energy
Looking at both sides of the equation…..
Work is done when…..• Work done against a force,
including friction, or gravity – (no net work is done however)
• Work done to change speed (momentum)– (net work is done)
Work is only done by a force on anWork is only done by a force on anobject if the object if the force causes the objectforce causes the object
to move in the direction of the forceto move in the direction of the force..
Objects that are at rest mayObjects that are at rest mayhave many forces acting on them,have many forces acting on them,
but no work is donebut no work is doneif there is no movement.if there is no movement.
WorkThe simplest definition for the amount of work a force does on an object
is magnitude of the force times the distance over which it’s applied:
W = F d
This formula applies when:
• the force is constant
• the force is in the same direction as the displacement of the object
F
d
Work Example
Big Heavy Mass50 N
10 m
A 50 N horizontal force is applied to a 15 kg crate of BHM over a distance of 10 m. The amount of work this force does is
W = 50 N · 10 m = 500 N · m = 500 J
In this problem, work is done to change the kinetic energy of the box….
Negative Work
BHM
7 mfk = 20 N
A force that acts opposite to the direction of motion of an object does negative work. Suppose the BHM skids across the floor until friction brings it to a stop. The displacement is to the right, but the force of friction is to the left. Therefore, the amount of work friction does is -140 J.
v
When zero work is done
BHM
7 m
N
mg
As the crate slides horizontally, the normal force and weight do no work at all, because they are perpendicular to the displacement. If the BHM were moving vertically, such as in an elevator, then each force would be doing work. Moving up in an elevator, the normal force would do positive work, and the weight would do negative work.
Another case when zero work is done is when the displacement is zero. Think about a weight lifter holding a 200 lb barbell over her head. Even though the force applied is 200 lb, and work was done in getting over her head, no work is done just holding it over her head.
Work done in lifting an object
• If you lift an object at constant velocity, there is no net force acting on the object….therefore there is no net work done on the object.
• However, there is work done, but not on the object, but against gravity
Net WorkThe net work done on an object is the sum of all the work done on it by the individual forces acting on it. Net Work is a scalar, so we can simply add work up. The applied force does +200 J of work; friction does -80 J of work; and the normal force and weight do zero work.
So, Wnet = 200 J - 80 J + 0 + 0 = 120 J
BHMFA = 50 N
4 mfk = 20 N
N
mg
Note that (Fnet ) (distance) = (30 N) (4 m) = 120 J.
Therefore, Wnet = Fnet d
Net Work done????• Is work done in…
–Lifting a bowling ball???
–Carrying a bowling ball across the room???
–Sliding a bowling ball along a table top???
If the force and displacement are notin the exact same direction, then
work = Fd(cos),where is the angle between the forcedirection and displacement direction.
F =40 N
d = 3.0 m
The work done in moving the block 3.0 mThe work done in moving the block 3.0 mto the right by the 40 N force at an angleto the right by the 40 N force at an angle
of 35 to the horizontal is ...of 35 to the horizontal is ...
35
W = Fd(cos W = Fd(cos ) = (40N)(3.0 m)(cos 35) = 98 J) = (40N)(3.0 m)(cos 35) = 98 J
B. Energy of Position
• Called Potential Energy
Ug = m g hThe equation shows that . . .
. . . the more gravitational potential energy it has.
• the more mass a body has
• or the stronger the gravitational field it’s in• or the higher up it is
Potential EnergyPotential Energyenergy of position or conditionenergy of position or condition
SI Potential Energy Units
From the equation Ug = m g h
the units of gravitational potential energy must be:= m · g · h= kg · (m/s2) · m
= (kg · m/s2) · m = N · m
= J What a surprise!!!!!
This shows the SI unit for potential energy is still the Joule, as it is for work and all other types of energy.
Reference point for U is arbitrary
Example: A 190 kg mountain goat is perched precariously atop a 220 m mountain ledge. How much gravitational potential energy does it have?
Ug = mgh = (190kg) (9.8m/s2) (220m) = 410 000J
This is how much energy the goat has with respect to the ground below. It would be different if we had chosen a different reference point.
Conservation and Exchange of Energy
Law of Conservation of Energy
In Conservation of Energy, the total
mechanical energy remains constant In any isolated system of objects interacting
only through conservative forces, the total
mechanical energy of the system remains
constant.
Law of Conservation of Energy
Energy may neither be created, nor destroyed, but is transformed from one form to another.
Example:kinetic energy of flowing water is converted into electrical energy using magnets.
Energy is Conserved• Conservation of Energy is different from Energy
Conservation, the latter being about using energy wisely
• Don’t we create energy at a power plant?– That would be cool…but, no, we simply transform
energy at our power plants, from one form to another• (fossil fuel energy or nuclear energy or potential energy of
water to electrical energy)
• Doesn’t the sun create energy?– Nope—it exchanges mass for energy
• E=mc2
Energy Exchange
• Though the total energy of a system is constant, the form of the energy can change
• A simple example is that of a simple pendulum, in which a continual exchange goes on between kinetic and potential energy
pivot
height reference
hK.E. = 0; P. E. = mgh K.E. = 0; P. E. = mgh
P.E. = 0; K.E. = mgh
Perpetual Motion• Why won’t the pendulum swing forever?• It’s hard to design a system free of energy paths• The pendulum slows down by several mechanisms
– Friction at the contact point: requires force to oppose; force acts through distance work is done
– Air resistance: must push through air with a force (through a distance) work is done
– Gets some air swirling: puts kinetic energy into air (not really fair to separate these last two)
• Perpetual motion means no loss of energy– solar system orbits come very close
Law of Conservation of Energy
The law says that energy must be conserved.
On top of the shelf, the ball has PE.
Since it is not moving, it has NO kinetic energy.
PE = mghKE = 0
h
Law of Conservation of Energy
If the ball rolls off the shelf, the potential energy becomes kinetic energy
PE = mghKE = 0
h
PE = 0KE = ½ mv2
Law of Conservation of Energy
Since the energy at the top MUST equal the energy at the bottom…
PEtop + KEtop = PEbottom + KEbottom
Notice that the MASS can cancel!
Example 1
A large chunk of ice with mass 15.0 kg falls from a roof 8.00
m above the ground. a) Find the KE of the ice when it reaches
the ground.
b) What is the velocity of the ice when it reaches the ground?
Energy at A?
Energy at B?
Energy at C?
Where is the ball the fastest? Why?
3.0 kg ball Calculate the energy values for A-K
Bouncing Ball• Superball has gravitational potential energy• Drop the ball and this becomes kinetic
energy• Ball hits ground and compresses (force times
distance), storing energy in the spring• Ball releases this mechanically stored energy
and it goes back into kinetic form (bounces up)
• Inefficiencies in “spring” end up heating the ball and the floor, and stirring the air a bit
• In the end, all is heat
Power,Power,by definition, isby definition, is
the the raterate of doing of doing workwork..
PP == WW // ttUnit=????
Power
• US Customary units are generally hp (horsepower)
– Need a conversion factor
– Can define units of work or energy in terms of units of power:
• kilowatt hours (kWh) are often used in electric bills
• This is a unit of energy, not power
Ws
lbfthp 7465501
Simple Machines
Ordinary machines are typically complicated combinations of simple machines. There are six types of simple machines:
• Lever
• Incline Plane
• Wedge
• Screw
• Pulley
• Wheel & Axle
Simple Machine Example / description
crowbar
ramp
chisel, knife
drill bit, screw (combo of a wedge & incline plane)
wheel spins on its axle
door knob, tricycle wheel (wheel & axle spin together)
Simple Machines: Force & Work
A machine is an apparatus that changes the magnitude or direction of a force.
Machines often make jobs easier for us by reducing the amount of force we must apply.
However, simple machines do not reduce the amount of work we do! The force we apply might be smaller, but we must apply that force over a greater distance.
Force / Distance Tradeoff Suppose a 300 lb crate of silly string has to be loaded onto a 1.3 m high silly string delivery truck. Too heavy to lift, a silly string truck loader uses a handy-dandy, frictionless, ramp, which is at a 30º incline. With the ramp the worker only needs to apply a 150 lb force (since sin 30º = ½). A little trig gives us the length of the ramp: 2.6 m. With the ramp, the worker applies half the force over twice the distance. Without the ramp, he would apply twice the force over half the distance, in comparison to the ramp. In either case the work done is the same!
Silly String
30º
1.3 m
150 lb
Silly String
300 lb
1.3 m
continued on next slide
Force / Distance Tradeoff (cont.)So why does the silly string truck loader bother with the ramp if he does as much work with it as without it? In fact, if the ramp were not frictionless, he would have done even more work with the ramp than without it.
answer: Even though the work is the same or more, he simply could not lift a 300 lb box straight up on his own. The simple machine allowed him to apply a lesser force over a greater distance. This is the “force / distance tradeoff.”
A simple machine allows a job to be done with a smaller force, but the distance over which the force
is applied is greater. In a frictionless case, the product of force and distance (work) is the same
with or without the machine.
Mechanical AdvantageMechanical advantage is the ratio of the amount of force that must be applied to do a job with a machine to the force that would be required without the machine. The force with the machine is the input force, Fin and the force required without the machine is the force that, in effect, we’re getting out of the machine, Fout which is often the weight of an object being lifted.
M.A. = Fout
Fin
Note: a mechanical advantage has no units and is typically > 1.
Ideal vs. Actual Mechanical AdvantageWhen friction is present, as it always is to some extent, the actual mechanical advantage of a machine is diminished from the ideal, frictionless case.
Ideal mechanical advantage = I.M.A. = the mechanical advantage of a machine in the absence of friction. Determined by comparing physical attributes of the machine.
Actual mechanical advantage = A.M.A. = the mechanical advantage of a machine in the presence of friction. Determined by comparing the output force with the input force
I.M.A. > A.M.A, but if friction is negligible we don’t distinguish between the two and just call it M.A.
Efficiency & Mechanical AdvantageEfficiency always comes out to be less than one. If eff > 1, then we would get more work out of the machine than we put into it, which would violate the conservation of energy. Another way to calculate efficiency is by the formula:
eff = A.M.A.I.M.A.
To prove this, first remember that Wout (the work we get out of the machine) is the same as Fin × d when there is no friction, where d
is the distance over which Fin is applied. Also, Win is the Fin × d when friction is present.
A.M.A.
I.M.A.
Fout / Fin w/ friction
Fout / Fin w/ no friction
Fin w/ no friction
Fin w/ friction
d Fin w/ no friction
d Fin w/ friction
Wout
Win
= =
= = = eff In the last pulley problem, I.M.A. = 3, A.M.A. = 2.308.
Check the formula: eff = 2.308 / 3 = 76.9%, which is the same answer we got by applying the definition of efficiency on the last slide.
Levers
• a lever (from French lever, "to raise", c. f. alevant) 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
Mechanical Advantage?
Crowbar (or pry bar)
Archimedes said “Give me a lever long enough, and a place to stand and I can move the earth”
do
I.M.A. for a Lever
d0
dF
fulcrum
Fin
Fout
A lever magnifies an input force (so long as dF > do). Here’s why: In equilibrium, the net torque on the lever is zero. So, the action-reaction pair to Fout (the force on the lever due to the rock) must balance the torque produced by the applied force, Fin. This means
Fin· dF = Fout· do I.M.A. = Fout
Fin
dF=do = distance from object to fulcrumdF = distance from applied force to fulcrum
Therefore,
Inclined Plane• A more common word for an inclined plane
is a ramp.• It is a surface that is set at an angle. • The smaller the angle of a ramp the less
effort is needed, but it will take a longer distance to gain the same height.
• A screw is an inclined plane and a curved ramp
• A wedge is a modification of an inclined plane it is made of two inclined planes
I.M.A. for an Incline Plane
I.M.A. = d / h A more gradual the incline will have a greater mechanical advantage. This is because when is small, so is mg sin (parallel component of weight)(equal to the force necessary to push box up hill).
d is very big, though, which means, with the ramp, we apply a small force over a large distance, rather than a large force over a small distance without it. In either case we do the same amount of work (ignoring friction).
IMA of a screwDetermined by the “pitch” of the threads
Pulleys
• A pulley is a grooved wheel, called a sheave, and a block.
• Used with a rope or chain to change direction or magnitude of a force.
• IMA = # of support strands
What’s Wrong with this Cartoon???????
M.A. for a Single Pulley #1
m
Fin
Fout
mg
With a single pulley the IMA is only 1.
The only purpose of this pulley is that it allows you to lift something up by applying a force down.
Pulley systems, with multiple pulleys, can have large mechanical advantages, depending on how they’re connected.
The AMA of this pulley would be less than one, depending on how much friction is present.
M.A. for a Single Pulley #2
m
Fin
mg
With a single pulley used in this way the I.M.A. is 2.
The reason for this is that there are two supporting ropes.
The tradeoff is that you must pull out twice as much rope as the increase in height, e.g., to lift the box 10 feet, you must pull 20 feet of rope.
FF
M.A: Pulley System #1
m
Fin
mg
In this type of 2-pulley system the I.M.A. = 3
A 300 lb object could be lifted with a 100 lb force if there is no friction.
The reason for this is that there are three supporting ropes
The tradeoff is that you must pull out three times as much rope as the increase in height, e.g., to lift the box 4 feet, you must pull 12 feet of rope.
F
F
F
I.M.A: Pulley System #2
60 N
Fin60 N
F
FF
1. Number of pulleys:
2. Number of supporting ropes:
3. I.M.A. =
4. Force required to lift box if no friction:
5. If 2 m of rope is pulled, box goes up:
6. Potential energy of box 0.667 m up:
7 a. Work done by input force to lift box 0.667 m up with no friction:7 b. Work done lifting box 0.667 m straight up without pulleys: If the input force needed with friction is 26 N,
9. A.M.A. =
10. Work done by input force now is:
3, but this matters not
3, and this does matter
3, since there are 3 supporting ropes
20 N
0.667 m
40 J
20 N · 2 m = 40 J
60 N · 0.667 m = 40 J
(60 N) / (26 N) = 2.308 < I.M.A.
26 N · 2 m = 52 J
EfficiencyNote that in the last problem:
Work done using pulleys (no friction)
Work done lifting straight up
little force × big distance
big force × little distance
Potential energy at high point
m g h
= =
All three of the above quantities came out to be 40 J. When we had to contend with friction, though, the rope still had to be pulled a “big distance,” but the “little force” was a little bigger. This meant the work done was greater: 52 J. The more efficient a machine is, the closer the actual work comes to the ideal case in lifting: m g h. Efficiency is defined as:
eff = Wout
WinIn the last example eff = (40 J) / (52 J) = 0.769, or 76.9%. This means about 77% of the energy expended actually went into lifting the box. The other 13% was wasted as heat, thanks to friction.
work actually done by input forcework done with no friction (often m g h)
=
Wheel & Axle The axle and wheel move together here, as in a doorknob. Not all wheel and axles are actually simple machines……..a wheel on a little red wagon does NOT act as a simple machine.
With a wheel and axle a small force can produce great turning ability. (Imagine trying to turn a doorknob without the knob.) Note that this simple machine is almost exactly like the lever. Using a bigger wheel and smaller axle is just like moving the fulcrum of a lever closer to object being lifted.
I.M.A. = rin / rout
Human Body as a Machine
bicep
tendon
ligament
radius
humerus
4 cm
30 cm
The center of mass of the forearm w/ hand is shown. Their combined weight is 4 lb.
40 lb dumbbell
c.m.
14 cm
Fbicep
40 lb
4 lb
Because the biceps attach so close to the elbow, the force it exerts must be great in order to match the torques of the forearm’s weight and dumbbell:
Fbicep= 314 lb !Fbicep(4 cm) = (4 lb) (14 cm) + (40 lb) (30 cm)
continued on next slide
Human Body as a Machine (cont.)
Fbicep
40 lb
4 lb
Let’s calculate the mechanical advantage of this human lever:
Fout / Fin = (40 lb) / (314 lb) = 0.127
4 cm
30 cm
14 cm
Note that since the force the biceps exert is less than the dumbbell’s weight, the mechanical advantage is less than one. This may seem pretty rotten. It wouldn’t be so poor if the biceps didn’t attach so close to the elbow. If our biceps attached at the wrist, we would be super duper strong, but we wouldn’t be very agile!
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