Work, Energy and Power

Georgia Perimeter College

Objectives

  1. To be able to calculate a value for work.

  2. To recognize when work is done, and not done, in a physical science sense.

  3. To identify a system that is conservative.

  4. To be able to calculate the potential and kinetic energies of a body in a conservative system.

  5. To be able to explain what makes a system NOT conservative.

  6. To be able to explain and calculate the different powers of two objects undergoing the same work at different time intervals.

  7. To explain why energy and work have the same units.

This section addresses, in whole or in part, the following Georgia GPS standard(s):
  • S8P3.  Students will investigate relationship between force, mass, and the motion of objects.
  • S8P2.  Students will be familiar with the forms and transformations of energy.
    a.  Explain energy transformation in terms of the Law of Conservation of Energy.
    b.  Explain the relationship between potential and kinetic energy.


 

This section addresses, in whole or in part, the following Benchmarks for Science Literacy:
  • Energy cannot be created or destroyed, but only changed from one form into another.
  • Tools are used to do things better or more easily and to do some things that could not otherwise be done at all. In technology, tools are used to observe, measure, and make things.
  • Throughout all of history, people everywhere have invented and used tools. Most tools of today are different from those of the past but many are modifications of very ancient tools.
  • Changes in speed or direction of motion are caused by forces. The greater the force is, the greater the change in motion will be. The more massive an object is, the less effect a given force will have.  

  • An unbalanced force acting on an object changes its speed or direction of motion, or both. If the force acts toward a single center, the object's path may curve into an orbit around the center.
  • The change in motion of an object is proportional to the applied force and inversely proportional to the mass.

This section addresses, in whole or in part, the following National Science Education Standards:
  • The total energy of the universe is constant. Energy can be transferred by collisions in chemical and nuclear reactions, by light waves and other radiations, and in many other ways. However, it can never be destroyed. As these transfers occur, the matter involved becomes steadily less ordered.
  • All energy can be considered to be either kinetic energy, which is the energy of motion; potential energy, which depends on relative position; or energy contained by a field, such as electromagnetic waves.
  • People have always had problems and invented tools and techniques (ways of doing something) to solve problems. Trying to determine the effects of solutions helps people avoid some new problems.
  • Tools help scientists make better observations, measurements, and equipment for investigations. They help scientists see, measure, and do things that they could not otherwise see, measure, and do.
     

 

Work, Energy and Power

 

Enough of all this theoretical folderol; let’s get some work done!

 

But what is work, in a physical science sense?  It has nothing to do with biochemistry or ‘work’ you do with your muscles.  It has to do with the exchange of energy between two or more objects or systems.  The energy is exchanged via traveling through some distance simultaneously with the use of, or interaction with, forces.

 

Work is done when the motion status of an object is changed.  Recall from the Forces section that when motion status is changed, there has to be an acceleration, which means the object has to have been affected by a force.  So Work, in a physics sense, requires the application of some kind of force.  The object has been pushed or pulled and this must be in the direction of motion to be work.  Forces perpendicular to the motion (and vice versa) do no work. 

 

Vocabulary

Potential Energy (PE)

Kinetic Energy (KE)

Total Energy (TE)

Conservative system

Nonconservative system

Work

Energy

Watts

Joules

Power

Horsepower

 

What is Work?

 

Long before George Bush appropriated his middle initial for his logo on bumper stickers, scientists used “W” to stand for Work (and other things like Weight but we won’t go there). 

 

 

You already know that force is written as “F”. 

 

You should also remember that acceleration under Newton’s Laws takes place ONLY during the time a force is being applied, thereafter inertia takes over again.  Well, if work also involves the change in motion status, then Work only takes place during the distance that the force has been applied.  So our relationship is work equals the product of force multiplied by the distance the force is applied during the interaction with the force. 

 


W = F * D
 

 

Again, this is different from biological work.  You can push a car all you want but if it doesn’t move, no physical work has been done, even if you burned a few dozen calories in the effort.  But if you succeeded in moving the car, then the force F you pushed on the car times the distance D it moved while you were actively pushing it, gives you the amount of work you did in a physical science sense.  Once you stopped pushing the car, even though the car continues to coast to a stop some extra distance d ahead, no more work is being done ON the car because it no longer has your force F affecting it.

 

The metric units of work are joules (abbreviated J).

 

Since W = F * D, then joules = N * meters, or also Nm or Nm. 

 

1 Joule is also 1 kg*m2/s2 from the relationship F = m*a (Newton’s Second Law).

 


One joule of is work done when:
one Newton of force is applied for one meter
or
one Newton of force moves something one meter
 

 

A sample of work values….lifting one common apple, weighing 1 N, from your belt to the top of your head (about 1 meter), takes 1 J of work.  But we aware that while you may push the apple from your belt forward for some distance and do work, gravitational work is NOT done, because your push was perpendicular to gravity. 

 

If a pulley lifts 10 N up 0.4 m, how much work is done by the pulley on the weight?  4 J.

Let’s try a simple quantitative measurement.  To climb stairs you have to work against gravity.  The force of gravity on you is your weight so you have to lift your weight over each step, a distance of D.  Let’s see how much work and power it takes to climb a flight of stairs.

 

Find your weight in Newtons.  If you know it in pounds, convert it to N by multiplying  your weight by 4.45 N/pound.  If you have a kilogram scale, multiply your mass by 9.8 m/sec2 to get N.  This is the F for your equation.

 

Find the distance up you climb.  This is not the linear the distance you walk from the bottom step to the top step.  It is the sum total of the step heights.  This is your D.

 

The Work is simply the product of F times D.  In my case, a 3.5 meter stair height and a 890 N force yields a work effort of 3115 J (or 3.115 kilojoules or 3.115 kJ). 

 

Power

 

Notice that there is no time in any of the equations or concepts above.  It doesn’t matter how quickly or slowly you raise the apple, nor pushed the car.  Did you shove the car hard and fast, or just lean on it for an hour?  Didn’t matter, the work done was the same.  But clearly the amount of work per unit of time is very different between shoving and leaning on the car!  To discuss how much work you do in an amount of time, we use the concept of Power.

 

Power is simply the amount of work per unit of time, or P = W/t. 

 


Power = Work/Time
 

 

In the metric system, Power P has the unit of Watt, which equal the number of joules per second.  (We also use the letter W for Watts so be sure you know if you are discussing power in Watts or Work!!)  So if you used 1 J of work to raise the apple in 1 second to your head, you did a Watt of power.  If you took 5 seconds (lazy!) you expended your workload at the power rate of only .20 Watts. The Watt, by the way, is named for Scottish engineer James Watt, who co-invented the modern steam engine.

 

Another unit of power is the horsepower

 

 

A horsepower referred to the average power output of a typical farm workhorse.  18th and 19th century farmers, scientists and industrialists were much more familiar with horse power ("How many horses to move this thing, Joe?") than we are.  Hp is now found in motors and cars and not elsewhere much.  1 Hp equals 746 Watts of power. 

Return to our walking up the stairs problem….  Now you have to time how long it takes to walk up the stairs.  Let’s say it takes 5 seconds.  Then my power is 3115 J / 10 sec, or 312 W.  If I rush it and take the steps two at a time and get up there in 5 seconds, my power is 623 W.  How does your power and work rates compare?


 

It takes energy to do work, too

 

Two simple ways to look at the relationship between energy and work -

 

1)      Energy is the ability to do work.  No source of energy, no work done. 

 

2)      When work is done, energy is transferred between systems, or simply transformed from one type of energy to another type.

 

Because energy is present when work is done, it shares the same units of measure, joules. 

 

Do 6 joules of work, 6 joules of energy get shifted around. 

 

Perhaps they came from you and got transferred to the object that moved. 

Perhaps they came from elsewhere. 

Perhaps they came from a stored source but got mutated into energy of motion.  

 

In fact, energy in a system (object and environment) or object comes in two types, potential and kinetic

 


 

Potential energy (PE)

 

Potential energy is energy stored. 

 

For simple mechanical things, the energy can be stored two ways, both related to position. 

 

1.  If the amount of potential energy is a function of stretching or compressing (i.e. how much distance stretched, or compressed), then this is named elastic potential energy

Springs, rubber bands, archery bows, bungee cords, and mousetraps are all examples of places where energy can be stored as elastic PE.  So are strings on guitars and violins.

2.  If If the amount of potential energy has to do with height above the earth, it is gravitational potential energy

Gravitational PE equals the amount of work needed to raise an object up into the sky or air from the ground. 

 

Since work is F*D, and F of gravity = m*g, and the d is the height h off the ground,

gravitational work = m*g*h and gravitational PE equals that value.

 

There are other forms of PE though they don’t have much say in mechanical physics. 

 

These include PE in chemicals.  Energy is stored in molecular bonds which is released in chemical reactions and becomes available to move things, for example. 

Biochemical reactions release PE stored in molecules used by living things. 

Nuclear reactions release stored energy from atomic nuclei; this is what powers the sun and generates its heat and light.  Ditto for earthly nuclear power plant reactions that make heat and electricity.

Find your mass in kilograms and multiply it by 9.8 to get your weight in Newtons

Then calculate how much potential energy from gravity you will have if you are at some height, say one story height of a building, 4 meters.


 

Kinetic Energy

 

The flip side of stored energy is energy in use, or kinetic energy

 

Since energy released does work, which causes some kind of motion, the opposite form to potential energy is kinetic energy, the energy of motion. 

 

Kinetic energy (KE) depends on mass but even more on velocity. 

 

The relationship is KE = ˝ * m * v2

 

A car over the speed limit has a lot more kinetic energy than a slower car, so the damage in a collision at high speeds is far worse.  That kinetic energy when the car stops has to go somewhere, and it goes into damaging the car and occupants.

 

 

A car damaged in a collision by kinetic energy

 

Take that PE value from the previous problem, and convert it to kinetic energy.  How fast will you be going when you reach the ground?  Is this a safe speed?  How does it compare to the ‘miles per hour’ you might have to drive (convert from meters per second to miles per hour).


 

Total Energy
and the Law of Conservation of Energy

 

The total energy (TE) in a system is the sum of the KE and the PE. 

 

As long as no energy gets lost out of the system, the total energy is a constant. 

 

Thus, if some of the total energy gets converted to more kinetic energy, the amount of potential energy has to go down. 

 

If the object then slows down, the lost KE gets put back into potential energy. 

 

This principle is called the Law of Conservation of Energy.

 

Because of this law we can state several truisms. 

1.  Energy doesn’t appear out of nowhere. 

2.  It doesn’t disappear either. 

3.  Energy can not be created or destroyed. 

4.  The total energy MUST remain a constant value.

 

(Note:  Students may confuse this kind of energy conservation with ecological energy conservation, recycling materials, turning off lamps, and so on.  While related, they are not the same thing.)

 


 

Conservative systems

 

We say a system is "conservative" not as a statement about its political leanings, but in that it obeys the Law of Conservation of Energy

 

Energy is transformed during processes but never lost. 

 

There are three common forms of conservative systems, all of which involve some kind of positional potential energies.

 

1.  Gravitational systems

 

This is so common you probably will say “I should have seen that!” 

 

You are high on a ladder painting a wall.  You accidentally drop your paint brush.  It falls to the ground reaching that point with quite some speed.  Where did it get its energy?

 

You have worked against gravity to get that brush up to your height above the ground.  It has no KE but a lot of PE.  The PE is equal in amount calculated above, the mass of the brush times the gravitational acceleration rate times the height h.  But when it fell, the PE was gradually released and converted to KE.

More dangerous than a paint brush, try this out to see how potential energy and kinetic energy are related:  http://www.visionlearning.com/library/module_viewer.php?mid=46

The same thing happens on roller coasters only now you get KE and PE shifting back and forth. 

You gain KE and speed when going down the slope, maximized at the bottom of the track valley, and slow down as you go up as the KE gets converted to PE.

 

 

 

Roller coaster with potential energy

 

 

 

Rollercoaster losing potential energy and gaining kinetic energy

 

 

 

Rollercoaster gaining potential energy

 

Another variant is the downhill skier.  Here’s a web animation that shows you the KE and PE as Jean-Paul races down the Alps:  http://www.physicsclassroom.com/mmedia/energy/se.html

 

A pendulum is a variation of the roller coaster and paint brushes.  Unlike them, it is periodic, in that the variation goes through cycles and would keep going forever if it could.  The PE of the pendulum is maximized when it swings the most it can from being straight up and down, and the KE is maximized when the pendulum mass (the bob) is hanging straight down. 

 

A student/child’s playground swing is just a pendulum for human ‘bob’s.

 

 

 

2.  It’s Springtime!  - Springs 'N Things

 

The same "periodic thing" happens with a spring with a mass on the end of it. 

 

The PE of a spring is some "spring constant k" that relates to how tight the spring naturally is, and the distance x that the spring is compressed or stretched from its equilibrium position. 

 

So, like gravity, the PE of a spring is related linearly to a distance, it is just is usually horizontal, not vertical. 

 

The KE is still ˝ mv2 for the mass at the end of the spring. 

 

The total energy is the sum of any kinetic energy + spring potential energy

 

You CAN have a spring vertical as well but then we have to have an equation with all the terms, the KE, the spring PE and the gravitational PE of the spring mass.  It still works but we won’t go into that kind of complication.

 

If you move the mass so the spring is compressed as much as possible, the spring has all the energy, all PE.  But when it gets released and moves the mass, some of the PE gets converted into moving the mass, which then has KE. 

 

Back at the equilibrium position, neither stretched nor compressed, the energy is all kinetic. 

 

The mass then starts to stretch the spring out, increasing its PE and reducing the mass’s KE until once again the mass stops moving.  Then everything reverses again.

 

 

 

 

A spring trap.  

 

Example:  The spring trap.  There is a horizontal spring across the top of the trap.  When the lever at top left is pulled to the left, the trap door opens, and the spring is stretched, increasing its potential energy.  When an animal enters the trap and touches the trigger, the trap is sprung, and the spring's potential energy is converted into kinetic energy, and the trap door slams shut.

 

 

3.  The Real World

 

But what do you know happens to a roller coaster if it could go without another pull up the big first hill?  Or the spring as its mass glides back and forth over a table top?  Or the pendulum swinging in the air?  The child on the swing who stops pumping? 

 

Yes, they all slow down and stop.  Why? 

 

Because we live in a world with friction, not an ideal friction-free place. 

 

So energy is lost each time we/bob/child and swing/roller coaster rubs against air/wood/metal track molecules. 

 

This friction transfers energy, as heat, to the interfering material, and the total energy lowers. 

 

But often, for short swings, for short times, we can treat a system as an ideal one, and compute speeds of masses in kinetic motion, and heights of roller coasters and maximum swings or stretched distances of springs.

 

This means that the "work in" (energy input) SHOULD equal the "work out" (energy output), but because of the real world situation, it never does. 

 

To some degree, the amount lost can be used to determine the efficiency (or inefficiency) of any machine, and all machines are indeed not 100% efficient. 

 

Friction, the creation of sound energy (squeaks), an accidentally sparking or metal on metal, all these reduce machine efficiency from the ideal amount.

 

That’s why pumping the legs on a swing, getting a battery supplied push on a pendulum swing, another pull by the roller coaster cable under the track, an extra shove on the spring, and lubricants help keep things going. 

 

But then the system isn’t really conservative any more.

 

Some rubber lengths used in flooring, when turned over, make good flexible roller coaster tracks.  You can also use foam pipe insulation, cut in half to make two U-shaped tracks that work well with a marble.  Connect several together in such a way that the ball comes to a stop at the end of your track.  You almost certainly will have to adjust heights and lengths to do this, and your kids will love the lab! 

 

Also, you can determine efficiency through frictional loss by making a U track and seeing how much height is lost if you start the rolling from one end of the track and see how much less it goes on the other end.

 

 


 

Machines


A machine is any mechanical or electrical device that uses energy to perform or help perform a human task.

 

Mechanical machines are popular because they save us work.  More precisely, they multiply our ability to do work.  They increase the force we need, add some energy, do work we couldn’t do before.  I’m not talking about machines like computers or televisions or toasters.  These are not mechanical machines.  But they HAVE some mechanical machines in them.  We shall see later if you can recognize them.

 

There are two kinds of mechanical machines, simple and compound

 

As you might guess, a simple machine is a single mechanical device.

 

A compound machine uses more than one simple machine in it. 

 

Machines usually multiply force. The amount by which a machine can multiply a force is called the machine's mechanical advantage.  Whether a machine is useful depends on whether it gives us more mechanical ‘strength’ (mechanical advantage, or M.A.) than doing the work ourselves.  Can we raise a heavier load up in the air better with a machine?  If so, then it is useful and has an advantage of some value times more than if we did it ourselves.

Mechanical advantage - the ratio of force output to force input.  Since both forces are in Newtons, the units cancel  (unitless).

The work a machine does is called work output.  The energy or work put in the machine is called work input.

The work output of a machine can never exceed the work input.

Efficiency = percentage of input work that is converted to usable output of work. 

Efficiency = work output/work input

Machines help us by taking the input work (energy/forces) we provide and get some more force or energy out than we supplied.  Uh, doesn’t that violate the law of conservation of energy?  No, because the extra force comes at a price.  Remember that work is a function of force AND distance.  If we get more force out, it comes at a sacrifice of some distance during which the force will be applied.  In actual fact, the work in and the work out of the machine are equal.  But the forces in and out, and distances applied in and out, will not be.  One increases, the other decreases to compensate.  We have to decide if we want more force out or greater distance to apply a smaller force.

 

 

 

                  

   


Review Questions

  1. Given a force of 25 N pulling a toy cart for 10 meters, calculate a value for work.


  2. Name two situations in which work is not done, in a physical science sense.


  3. Identify a system that is conservative.


  4. A roller coaster has heights of 100 m, 50 m, and 25m.  Calculate the potential energies available at all three heights, and how much kinetic energies a moving body would have at each point (provided that it IS moving!).


  5. What makes a system NOT conservative?


  6. You have to push a wheelchair-bound friend up a ramp that is 2m high.  He and the chair weigh 1000N.  You could lift him; how much energy would you need?  Now, if the ramp is 10m long, what is the mechanical advantage of the ramp?  Use this to determine how much force you now need to push him up the ramp.


  7. In the above example, it took you 10 seconds to push him up the ramp.  By himself, your friend took 30 seconds.  If it took 100 W of work to do it, what would be your amounts of power?


  8. Explain why energy and work have the same units.


Define

Potential Energy (PE)

Kinetic Energy (KE)

Total Energy (TE)

Conservative system

Nonconservative system

Work

Energy

Watts

Joules

Power

Horsepower

Thrills of the Hills WebQuest  http://campus.pc.edu/students/ctacke00/Middle%20School/index1.htm

You are part of a roller coaster design team. The “Thrills of the Hills Amusement Park” is looking to add a new coaster and begin its construction by the spring of 2004. Your job is to work together as a team of four and design a fun and safe roller coaster, present your coaster design to the Board of Directors at “Thrills of the Hills,” and try to persuade them to choose your design. Do you have what it takes to design the newest attraction at “Thrills of the Hills Amusement Park?”  


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Content provided by Larry Krumenaker, Georgia Perimeter College

Photos courtesy of Pamela J. W. Gore

Screw images courtesy of NASA,  http://www.grc.nasa.gov/WWW/K-12/Summer_Training/KaeAvenueES/The_Screw.html

Page created by Pamela J.W. Gore
Georgia Perimeter College,
Clarkston, GA

Page created March 6-8, 2007
Modified May 19, 2007
Modified May 27, 2007