WORK, POWER & ENERGY

Work When a force acts upon an object

to cause a displacement of the object, it is said that work was done upon the object.

There are three key ingredients to work - force, displacement, and cause.

Work In order for a force to qualify as

having done work on an object, there must be a displacement and the force must cause the displacement.

Examples (In everyday life) a horse pulling a plow through the

field a father pushing a grocery cart

down the aisle of a grocery store a freshman lifting a backpack full of

books upon her shoulder a weightlifter lifting a barbell above

his head

Examples (In everyday life) an Olympian launching the shot-

put, etc. In each case described here there

is a force exerted upon an object to cause that object to be displaced.

The Mathematics of Work Mathematically, work can be

expressed by the following equation

where F is the force d is the displacement, the angle (theta) is defined as the

angle between the force and the displacement vector.

Angle: Scenario A A force acts rightward upon an

object as it is displaced rightward. In such an instance, the force

vector and the displacement vector are in the same direction.

Thus, the angle between F and d is 0 degrees.

Angle: Scenario B A force acts leftward upon an object

which is displaced rightward. In such an instance, the force vector

and the displacement vector are in the opposite direction.

Thus, the angle between F and d is 180 degrees.

Angle: Scenario C A force acts upward on an object

as it is displaced rightward. In such an instance, the force

vector and the displacement vector are at right angles to each other.

Thus, the angle between F and d is 90 degrees.

To Do Work, Forces Must Cause Displacements

Scenario C involves a situation similar to the waiter who carried a tray full of meals above his head by one arm straight across the room at constant speed.

Explanation It was mentioned earlier that the

waiter does not do work upon the tray as he carries it across the room.

The force supplied by the waiter on the tray is an upward force and the displacement of the tray is a horizontal displacement.

Explanation As such, the angle between the force

and the displacement is 90 degrees. If the work done by the waiter on the

tray were to be calculated, then the results would be 0.

Regardless of the magnitude of the force and displacement, F*d*cosine 90 degrees is 0 (since the cosine of 90 degrees is 0).

Explanation A vertical force can never cause a

horizontal displacement; thus, a vertical force does not do work on a horizontally displaced object!!

It can be accurately noted that the waiter's hand did push forward on the tray for a brief period of time to accelerate it from rest to a final walking speed.

Explanation But once up to speed, the tray will stay

in its straight-line motion at a constant speed without a forward force.

And if the only force exerted upon the tray during the constant speed stage of its motion is upward, then no work is done upon the tray.

Again, a vertical force does not do work on a horizontally displaced object.

Situational Analysis

The Negative Work On occasion a force acts upon a moving

object to hinder a displacement. Examples might include a car skidding to

a stop on a roadway surface or a baseball runner sliding to a stop on the infield dirt.

In such instances, the force acts in the direction opposite the objects motion in order to slow it down.

The force doesn't cause the displacement but rather hinders it.

The Negative Work These situations involve what is

commonly called negative work. The negative of negative work

refers to the numerical value which results when values of F, d and theta are substituted into the work equation.

The Negative Work Since the force vector is directly

opposite the displacement vector, theta is 180 degrees.

The cosine(180 degrees) is -1 and so a negative value results for the amount of work done upon the object.

Units of Work the standard metric unit is the

Joule (abbreviated J)

The Joule is the unit of work.1 Joule = 1 Newton * 1 meter

1 J = 1 N * m

SUMMARY Work is done when a force acts upon an

object to cause a displacement. Three quantities must be known in

order to calculate the amount of work. Those three quantities are force,

displacement and the angle between the force and the displacement.

Sample Problem

Answers Diagram A Answer: 500 J It is said (or shown or implied) that the force and

the displacement are both rightward. Since F and d are in the same direction,the angle is 0 degrees.

Diagram B Answer:433 J It is said that the displacement is rightward. It is

shown that the force is 30 degrees above the horizontal. Thus, the angle between F and d is 30 degrees.

Diagram C Answer: 735 J

Power The quantity work has to do with a

force causing a displacement. Work has nothing to do with the

amount of time that this force acts to cause the displacement.

Sometimes, the work is done very quickly and other times the work is done rather slowly.

Power For example, a rock climber

takes an abnormally long time to elevate her body up a few meters along the side of a cliff.

On the other hand, a trail hiker (who selects the easier path up the mountain) might elevate her body a few meters in a short amount of time.

Power The two people might do the same

amount of work, yet the hiker does the work in considerably less time than the rock climber.

The quantity which has to do with the rate at which a certain amount of work is done is known as the power.

The hiker has a greater power rating than the rock climber.

Power Power is the rate at which work is

done. It is the work/time ratio. Mathematically, it is computed

using the following equation.

Units of Power The standard metric unit of power is the Watt. As is implied by the equation for power, a unit

of power is equivalent to a unit of work divided by a unit of time.

Thus, a Watt is equivalent to a Joule/second. For historical reasons, the horsepower is

occasionally used to describe the power delivered by a machine.

One horsepower is equivalent to approximately 750 Watts.

MACHINE Machine is a device used to

multiply forces or simply to change the direction of forces.

Most machines are designed and built to do work on objects.

All machines are typically described by a power rating.

MACHINE The power rating indicates the rate

at which that machine can do work upon other objects.

Thus, the power of a machine is the work/time ratio for that particular machine.

Example: lever and pulley

MACHINE No machine can put out more energy

than is put into. No machine can create energy. A machine can only transfer energy from

one place to another or transform it from one form to another.

A machine which is strong enough to apply a big force to cause a displacement in a small mount of time (i.e., a big velocity) is a powerful machine.

MACHINE Theoretically, all machines are

considered ideal. Meaning, all the work input was

transferred to work output. (100% efficient)

But, it never happens. Some energy is transformed to

other forms which makes machines warmer.

Situational Analysis A tired squirrel (mass of

approximately 1 kg) does push-ups by applying a force to elevate its center-of-mass by 5 cm in order to do a mere 0.50 Joule of work. If the tired squirrel does all this work in 2 seconds, then determine its power.

Situational Analysis (Answer) The tired squirrel does 0.50 Joule

of work in 2.0 seconds. The power rating of this squirrel is found by

P = W / t = (0.50 J) / (2.0 s) = 0.25 Watts

ENERGY Whenever work is done on an object, it

gains energy. If 100 J of work is done, the object gains 100J of energy.

The objects doing the work exchange energy in one form to do work on another object to give it energy.

The energy acquired by the objects upon which work is done is known as mechanical energy.

ENERGY

The energy may be stored by the objects in a variety of forms:

Kinetic Energy Potential Energy Elastic Potential Energy Internal Energy

KINETIC ENERGY - the energy an object has because it is moving. For example, a moving hammerhead has kinetic energy while one at rest has none.

POTENTIAL ENERGY - the energy an object has because of its vertical separation from the earth.

ELASTIC POTENTIAL ENERGY – the energy stored in a stretched or compressed elastic material such as spring.

INTERNAL ENERGY – the atomic and molecular energy of matter consisting of (1) the kinetic energy of the atoms and molecules due to their random motion (called thermal energy) and (2) the energy atoms and molecules have as a result of their bonds and interactions with each other.

Potential Energy An object can store energy as the

result of its position. For example, the heavy ball of a

demolition machine is storing energy when it is held at an elevated position.

This stored energy of position is referred to as potential energy.

Similarly, a drawn bow is able to store energy as the result of its position.

Potential Energy When assuming its usual position (i.e.,

when not drawn), there is no energy stored in the bow.

Yet when its position is altered from its usual equilibrium position, the bow is able to store energy by virtue of its position.

This stored energy of position is referred to as potential energy.

Potential energy is the stored energy of position possessed by an object.

Potential Energy

Gravitational Potential Energy Gravitational potential energy is

the energy stored in an object as the result of its vertical position or height.

The energy is stored as the result of the gravitational attraction of the Earth for the object.

Gravitational Potential Energy There is a direct relation between

gravitational potential energy and the mass of an object; more massive objects have greater gravitational potential energy.

Gravitational Potential Energy There is also a direct relation

between gravitational potential energy and the height of an object; the higher that an object is elevated, the greater the gravitational potential energy.

Gravitational Potential Energy

PEgrav = mass * g * height

PEgrav = m * g * h

Elastic Potential Energy Elastic potential energy is the

energy stored in elastic materials as the result of their stretching or compressing.

Elastic potential energy can be stored in rubber bands, bungee chords, trampolines, springs, an arrow drawn into a bow, etc.

Elastic Potential Energy The amount of elastic potential

energy stored in such a device is related to the amount of stretch of the device - the more stretch, the more stored energy.

Elastic Potential Energy Springs are a special instance of a

device which can store elastic potential energy due to either compression or stretching.

A force is required to compress a spring; the more compression there is, the more force which is required to compress it further.

Elastic Potential Energy

Situational Analysis A cart is loaded with a brick and

pulled at constant speed along an inclined plane to the height of a seat-top. If the mass of the loaded cart is 3.0 kg and the height of the seat top is 0.45 meters, then what is the potential energy of the loaded cart at the height of the seat-top?

Situational Analysis (Answer)

PE = m*g*h PE = (3 kg ) * (9.8 m/s/s) * (0.45 m)

PE = 13.2 J

Kinetic Energy Kinetic energy is the energy of motion. An object which has motion - whether it be

vertical or horizontal motion - has kinetic energy.

There are many forms of kinetic energy – vibrational (the energy due to vibrational motion) rotational (the energy due to rotational motion) translational (the energy due to motion from one

location to another).

Kinetic Energy To keep matters simple, we will

focus upon translational kinetic energy.

The amount of kinetic energy which an object has depends upon two variables: the mass (m) of the object the speed (v) of the object.

Kinetic Energy

Situational Analysis Determine the kinetic energy of a

625-kg roller coaster car that is moving with a speed of 18.3 m/s.

Situational Analysis (Answer)

KE = 0.5*m*v2 KE = (0.5) * (625 kg) * (18.3 m/s)2

KE = 1.05 x105 Joules

Sources of Energy Sunlight

A daily energy supply Photosynthesis Photocells (Solar Cells)

Water Hydroelectric energy

Wind windmills

Sources of Energy Ancient Energy Supply

Oil Coal Natural gas Fossil fuel

Other Energy Supplies Nuclear Geothermal

Forms of Energy Potential Energy

the energy that a body possesses because of its position.

Kinetic Energy the energy that a body possesses by virtue

of its motion. Heat Energy

the measure of the internal energy of substance that is due to its temperature.

Forms of Energy Radiant Energy

energy associated with ordinary light, x-rays, radio waves or infrared rays.

Chemical Energy “stored energy” the energy possessed by a

substance that allows it to be changed into a new substance.

Atomic or Nuclear Energy energy associated with the manner in which

atoms are constructed or formed.

Law of Conservation of Energy Energy can neither be created nor

destroyed. It can only be transformed from one form to another, but the total amount of energy never changes.