Energy

In the everyday world energy is often the topic of discussion.  People talk about the energy used and the cost of energy. Fuel and electricity costs are talked about by many people.

Energy is present in many forms. It exists as mechanical energy, chemical energy, electromagnetic energy and as nuclear energy.

Energy can be changed from one form of energy into another form of energy as long as the principle of the conservation of energy is not violated.

The total amount of energy in a closed system must be a constant and the total energy in the universe is also a constant.

As one form of energy decreases some other form of energy must increase.

A dry cell, commonly but mistakenly called a battery, can convert chemical energy into electrical energy, which can be converted into mechanical energy.

The two main types of energy we will discuss is potential energy and kinetic energy.

We will start our study of energy by a brief study of a concept called work.

Work

The common conception of work is incorrect. Work is often considered the same as a job.  People go to work each day but they may not actually do any work.

Imagine that you job is to carry boxes of documents from a safe in the basement, up 10 flights of stairs, to an office where someone will look at these documents, and that after they have looked at the documents you carry the boxes back down to the safe and store them for the next day.  At the end of the day, even though you may have been busy all day long and are very tired, the total amount of your work will be zero.

Work is defined as being the product of the component of a force along the direction of displacement times the magnitude of the displacement.

W = FΔx 

If the force is applied at some angle to the direction of travel the equation becomes:

W = (FcosŲ)Δx

where W = work, F = force in the direction of the displacement,  Δx is the displacement , and  Ų is the angle between the line of force and the direction of the motion.

Older textbooks, and older instructors,  may use 's' as the symbol for displacement.

This is commonly stated as work equals force times distance.  Remember that the distance refers to the distance moved in the same direction as the force.  When the force is as an angle to the direction of motion a component of the force can be in the direction of the motion.

If you exert a lot of force on an immovable object and the object does not move then you have done no work because nothing was moved. 

Work is a scalar quantity.

Work can be positive or negative.

When a baseball bat hits a baseball work is done by the baseball bat on the baseball.  There is an interaction between the baseball bat and the baseball

Imagine that you pick up a 50 kilogram object off the floor, carry it to the other side of a large flat room, and place it back on the floor.  You have done no net work.  Your boss may pay you for doing this even though you have actually done no work for him, according to the definition of work.

Work Done by a Force that Varies.

The work done by a force that changes can be determined by a graphing the force in the y direction and the distance in the x direction.  The work done will be equal to the area under the curve. 

Work done in Rotary Motion.

W = FrŲ = TŲ where:

W = work
F = force that is applied tangentially to the circle
r = radius of the circle
Ų = angle in radians 
T = torque 

Work done when stretching or compressing a spring 

Compressing or stretching a spring requires a force.  This force acting over a distance causes work to be done on the spring. The more a spring is compressed or stretched the more force is needed and the more work is done.

In the next equation the (1/2) is introduced because the average force is 1/2 the difference of final force minus the initial force. When dealing with the compression or expansion of a spring the formula for work is as follows:  

W = (1/2) (Force) (Distance the spring is stretched or compressed) or

W = (1/2)FΔx where:

W = work
F = force that stretches  or compresses the spring in newtons 
Δx  is the distance the spring is stretched or compressed in meters

If the force needed is not known but the spring constant, 'k', is known use the following formula.

The force to stretch or compress a spring is found using the following equation:

F = kΔx where

F = force needed to stretch or compress the spring in newtons 
k = spring constant measured in N/m
Δx = distance the  force stretches or compresses the spring in meters

W = (1/2)(spring constant)( Distance stretched or compressed)2 or simply 

W = (1/2)(k)(Δx)2 

Units of Work

In the SI system of measurement work is measured in newton-meters (n-m) while in the US Customary system of measurement it is measured in foot-pounds (ft-lb).  Other units such as inch-pounds are sometimes used.

The newton-meter is also called a joule which uses the symbol of the upper case 'J'.

Joule

The joule, named after James Prescott Joule, is the SI unit of work and of energy. The joule is the same a a newton-meter

Kinetic Energy

Kinetic Energy is the energy that an object has because of its motion. The symbol used for kinetic energy is 'KE'. Kinetic energy is also a scalar quantity with the same unit as work, which is the newton-meter, also known as the joule.

KE = (1/2)mv2 where:

KE = Kinetic Energy
m = mass of the object
v = velocity of the object

Work and Kinetic Energy 

Wnet = KEfinal -KEinitial = ΔKE 

The work done by a net force on an object is equal to the change in the kinetic energy of the object.

Potential Energy

Potential energy is the mechanical energy associated with an object because of its position in a system.  Potential energy uses the symbol 'PE' as an abbreviation. Potential energy is, like work and kinetic energy, a scalar quantity.

PE = mgh = mgΔy where:

PE = Potential Energy
m = mass of the object
g = acceleration due to gravity
h = height of the object above the reference location.
Δy = displacement in the vertical direction

Potential energy caused by the objects relative position to the surface of Earth  is called gravitational potential energy.

Gravitational Potential Energy

Gravitational potential energy associated with the potential energy of an object based upon its relative position to the surface of the Earth.

Gravitational force can cause work to be done.

Wgravity = PEinitial - PEfinal

Even though the acceleration due to gravity changes slightly with altitude we normally consider 'g' to be a constant when working with objects close to the Earth.  If a really accurate value is needed the change in the value of 'g' can be taken into account. We will consider 'g' to remain a constant for our work.

When working problems involving gravitational potential energy you must select a reference point which is the elevation where you consider the potential energy to be cipher, or zero.

Reference Levels for Gravitational Potential Energy

Some elevation must be chosen as the point where the gravitational potential energy is zero. The surface of the Earth is often chosen as the zero point, but the surface of a floor or desk  or table can also be used.  If concerned about objects falling upon your head you might choose the top of your head to be the point where the gravitational potential energy is zero.