Chapters 10/11 Work, 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…

v

1 2 

v

0 2  2

ad

v

1 2 

v

0 2  2

ad

add V 2

to

both

sides

v

1 2 

v

0 2  2

ad

v

1 2 

v

0 2  2

ad

Substitute F/m for a

v

1 2 

v

0 2  2

fd m

v

1 2 

v

0 2  2

fd m

Multiply by ½ m 1 2

mv

1 2  1 2

mv

0 2 

fd

1 2

mv

2 

fd

Let’s look at each side of this equation, one side at a time….

• Left side contains terms that describe energy of a system

K E

 1 2

mv

2 …where the change in velocity is due to work being done .

Kinetic Energy

Kinetic energy is the energy of motion. By definition kinetic energy is given by:

K

E

= ½ m v

2 The equation shows that

. . .

• the more mass a body has • or the faster it’s moving

. . .

the more kinetic energy it has.

Derive the unit for Energy, the Joule!!!!

K is proportional to v 2 , so doubling the speed quadruples kinetic energy, and tripling the speed makes it nine times greater.

The formula for kinetic energy, K E = 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

SI Kinetic Energy Units

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.

= Joule

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 K E 1 2

mv

2



fd

Work-Energy Theorem Looking at both sides of the equation…..



K E



fd

Simply says that

by doing work on a system, you increase the kinetic energy

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 an object if the force causes the object to move in the direction of the force.

Objects that are at rest may have many forces acting on them, but no work is done if there is no movement.

Work

The 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

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….

Big Heavy Mass 50 N 10 m

Negative Work

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 BHM f k = 20 N 7 m

When zero work is done

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.

N BHM mg 7 m

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 Work

The 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, W net = 200 J - 80 J + 0 + 0 = 120 J Note that (F net ) (distance) = (30 N) (4 m) = 120 J. Therefore, W net = F net d N BHM f k = 20 N mg F A = 50 N 4 m

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 not in the exact same direction , then

work = Fd(cos

q

)

, where

q

is the angle between the force direction and displacement direction.

F =40 N

35 

d = 3.0 m The work done in moving the block 3.0 m to the right by the 40 N force at an angle of 35 to the horizontal is ...

W = Fd(cos

q

) = (40N)(3.0 m)(cos 35) = 98 J

B. Energy of Position

•

Called Potential Energy

Potential Energy

energy of position or condition

U

g

= m g h

The equation shows that

. . .

• the more mass a body has • or the stronger the gravitational field it’s in • or the higher up it is

. . .

the more gravitational potential energy it has.

SI Potential Energy Units

From the equation U g = m g h the units of gravitational = potential energy must be: m

·

g · h = kg · (m/s 2 ) ·m = (kg · m/s 2 ) ·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

– Nope—it

exchanges

• E=mc 2 energy?

mass for energy

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 K.E. = 0; P. E. =

mgh

h P.E. = 0; K.E. =

mgh

K.E. = 0; P. E. =

mgh

height reference

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 PE = mgh KE = 0 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.

h

Law of Conservation of Energy PE = mgh KE = 0 If the ball rolls off the shelf, the potential energy becomes kinetic energy h PE = 0 KE = ½ mv 2

Law of Conservation of Energy Since the energy at the top MUST equal the energy at the bottom… PE top + KE top = PE bottom + KE bottom 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?

Where is the ball the fastest? Why?

3.0 kg ball Energy at A?

Energy at B?

Energy at C?

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,

by definition, is the rate of doing work .

P = W / t

Unit=????

Power

• US Customary units are generally hp (horsepower) – Need a conversion factor 1

hp

 550

ft lb

 746

W s

– 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

Simple Machines

Ordinary machines are typically complicated combinations of simple machines. There are six types of simple machines:

Simple Machine Example / description

•

Lever

•

Incline Plane

•

Wedge

•

Screw

•

Pulley

•

Wheel & Axle 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!

continued on next slide

150 lb 300 lb 1.3 m 30º 1.3 m

Silly String

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 Advantage Mechanical 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, F in and the force required without the machine is the force that, in effect, we’re getting out of the machine, F out which is often the weight of an object being lifted. M.A. = F out F in Note: a mechanical advantage has no units and is typically > 1.

Ideal vs. Actual Mechanical Advantage

When 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 Advantage Efficiency 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.

is the distance over which F friction is present. To prove this, first remember that W out (the work we get out of the machine) is the same as F in

×

d when there is no friction, where d in is applied. Also, W in is the F in

×

d when A.M.A.

F out / F in w/ friction F in w/ no friction

= =

I.M.A.

F out / F in w/ no friction F in w/ friction

=

d F in w/ no friction d F in w/ friction

=

W out W in

=

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. a

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

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”

I.M.A. for a Lever A lever magnifies an input force (so long as d F > d o ). Here’s why: In equilibrium, the net torque on the lever is zero. So, the action reaction pair to F out

d F

(the force on the lever due to the rock) must balance the torque produced by the applied force, F in . This means F in ·d F = F out ·d o d F Therefore, I.M.A. = F out F in = d o

d o = distance from object to fulcrum = distance from applied force to fulcrum

d F d 0 F out F in

fulcrum

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

Johnstown, PA inclined Plane Archimede’s Screw

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 q is small, so is mg sin q (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 screw Determined 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???????

F out mg m F in

M.A. for a Single Pulley #1

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

. The AMA of this pulley would be less than one, depending on how much friction is present.

Pulley systems, with multiple pulleys, can have large mechanical advantages, depending on how they’re connected.

F mg m F in F

M.A. for a Single Pulley #2

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.

F in F F F mg m

M.A: Pulley System #1

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.

I.M.A: Pulley System #2 1. Number of pulleys:

3, but this matters not

2. Number of supporting ropes:

3, and this does matter

3. I.M.A. =

3, since there are 3 supporting ropes

4. Force required to lift box if no friction:

20 N

5. If 2 m of rope is pulled, box goes up:

0.667 m

F 6. Potential energy of box 0.667 m up:

40 J

F 7 a. Work done by input force to lift box 0.667 m up with no friction:

20 N · 2 m = 40 J

7 b. Work done lifting box 0.667 m straight up without pulleys:

60 N · 0.667 m = 40 J

If the input force needed with friction is 26 N, 60 N 9. A.M.A. =

(60 N) / (26 N) = 2.308 < I.M.A.

10. Work done by input force now is:

26 N · 2 m = 52 J

60 N F F in

Efficiency Note that in the last problem: Work done using Work done lifting

=

Potential energy pulleys (no friction)

=

straight up at high point

little force × big distance big force × little distance mgh

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 distance,” but the “ little force

”

“ big 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: mgh. Efficiency is defined as: W out = work done with no friction (often mgh) eff = W in work actually done by input force In 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.

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.

I.M.A. = r in / r out 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.

Human Body The center of mass of the forearm w/ hand is as a Machine tendon shown. Their combined weight is 4 lb.

F bicep bicep humerus 40 lb dumbbell radius ligament c.m.

4 lb 4 cm Because the biceps attach so close to the elbow, 40 lb 14 cm the force it exerts must be great in order to match the torques of the forearm’s weight and dumbbell: F bicep F bicep = 314 lb !

30 cm (4 cm) = (4 lb)(14 cm) + (40 lb)(30 cm)

continued on next slide

Human Body as a Machine F bicep 4 cm 14 cm 4 lb 30 cm (cont.) 40 lb Let’s calculate the mechanical advantage of this human lever: F out / F in = (40 lb) / (314 lb) = 0.127

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!