Chapter 10: Work and Mechanical Energy

Equations Introduced and Used for for this Topic (all equations are solved as either scalar or vector):

[latex]Work(W) =\overrightarrow{F}_{net} •\overrightarrow{d} or \overrightarrow{F}_{net}\overrightarrow{d} cos Ø[/latex]                      W = ∆ Mechanical Energy

E_p = mgh                                    ∆E_p = mg∆h                               E_k = ½mv² ∆E_k = ½m∆v²

Where

W  is Work, measured in Joules (J)

E_p or ∆E_p   is Gravitational Potential Energy or change in Gravitational Potential Energy, measured in Joules (J)

E_k or ∆E_k  is Kinetic Energy or change in Kinetic Energy, measured in Joules (J)

E or ∆E  is Energy or change in Energy, measured in Joules (J)

V or V_g   is the Gravitational Potential, measured in Newtons per Kilogram (N/kg)

m  is the Mass of the object, measured in Kilograms (kg)

g or ag   is the Gravitational Field Strength, measured in Metres per Second ^squared (m/s²)

v   is the Speed of the object, measured in Metres per Second (m/s)

v or ∆v  is the Speed or change in Speed of the object, measured in Metres per Second (m/s)

v_f & v_i  is the final and initial Speed of the object, measured in Metres per Second (m/s)

F_net   is the Net Force or the Vector Sum of the Forces acting on a body or a system, measured in Newtons (N)

ø   is the angle between the vectors of the Net Force and Displacement, measured in degrees (°)

d   is the displacement through which the Net Force acts on a body or a System, measured in Metres (m)

h or ∆h   is the height or change in Vertical Height, measured in Metres (m)

h_f & h_i  is the final and initial height of the object measured in Metres (m)

Notes:

The Dot Product “ • “ represents the cosine of the angle between two vectors. For Work it represents the angle between the direction of the Net Force and the displacement change.

The Cross Product “x “ represents the sine of the angle between two vectors. Equations using the cross product will be given later in your study of physics, such as the measure of Torque or the force on a charged particle moving in a magnetic field.

Work and Energy are measured in Joules (J) or Newton-Metres (Nm)

For Example:  Consider the following multiplication of these two vectors shown below.

[latex]\overrightarrow{A}•\overrightarrow{B}[/latex]= (20(18) cos 42º

[latex]\overrightarrow{A}•\overrightarrow{B}[/latex]= 267.5 or 270

The Cross Product for these two vectors is …

[latex]\overrightarrow{A}\times\overrightarrow{B}= \overrightarrow{A}\overrightarrow{B}[/latex]= sin Ø

10.1 Kinetic and Potential Energy^5, 6

Equations Introduced and Used in this Topic:

E_p = mgh     ∆E_p = mg∆h                                       E_k = ½ mv²        ∆E_k = ½ m∆v²

       ∆h = h_f – h_i      ∆v = v_f  – v_i                                             V or V_g = gh

Conceptual Development:

The concept of energy is generally credited to Aristotle’s (384 BC-322 BC) enérgeia (ἐνέργειά), which is a word having no direct translation into English except for “being at work”.  Aristotle’s concept of work grew from Empedocles’s (490-430 BC) thoughts on the basic elements of all matter: earth, water, air, fire and quintessence, with fire being characterized as the driver behind his thoughts and understanding leading to the concept of enérgeia.  Aristotle’s concepts remained in vogue for nearly two thousand years before The Age of Enlightenment in European science began to take prominence in creating a new paradigm to understand energy.

The first of these developments was in 1669 by Christian Huygens (1629 -1695), who recognized that in a specific collision between two hardwood spheres (close to an elastic collision) the product of mass times the square of the speed⁷ was conserved along with what became known as momentum. This discovery is credited as the origin of the modern concepts of mechanical energy, specifically kinetic energy.  Huygens’s name for his discovery was not energy but vis viva (living force)⁸. A couple of decades later, Gottfried Wilhelm von Leibniz (1646-1716) in 1686 became the first to quantify the concept of vis-viva as the product of mass times the square of the speed.

Thomas Young (1773-1829) introduced the word “energy” in 1800 but it did not enter common usage at that time. Instead the name “energy” was popularized around fifty years later by William Thomson, Lord Kelvin (1824 – 1907) and William J. M. Rankine (1820-1872) as the concept of force was integrated to act across other related phenomena.

When one looks back at the explosion of European physics that occurred in The Age of Enlightenment, and following Leibniz’s vis viva, the concepts linking force and energy, heat ⁹ and work all helped to lay the foundations for a complete overhaul of the understanding of science in its Greek origins some two thousand years prior.  Eventually these developments led to Albert Einstein’s (1879-1955) rewriting of the Newtonian mechanical universe and arguably the most famous equation in physics equating the relationship between matter and energy:

E = mc²

Modern use of the word “energy” generally causes much confusion about the concepts of energy in introductory physics classes. The next few sections in this chapter will outline a physical definition of the energy concepts (scalar measures) and how to quantify energy in modern SI metric measures.

Two energy forms, termed “kinetic energy” and “potential energy”, are both classified as forms of mechanical energy.

The first of these, Kinetic Energy¹⁰, is the measure of the energy held by a body in motion. The measure of an object’s kinetic energy is almost identical to that of the vis viva proposed by Leibniz (mv²) except that it is 1/2 the measure.  Energy is often denoted by the capital letter E with a subscript of k to denote that it is the kinetic energy form (E_k)11.

In Newtonian mechanics, the measure of a body’s kinetic energy is represented by the equation:

Ek = ½ mv2

where the mass of the moving object is given in kilograms (kg) and the speed of the object is given in metres per second (m/s).  The measure of energy is given the name Joules (J) in honor of James Prescott Joule (1818 – 1889) for his work in his discovery of the mechanical equivalent of heat, which later led to the law of conservation of energy and the development of the First Law of Thermodynamics.

Kinetic Energy can simply be remembered as the energy of a moving body.

The second of the mechanical energies is termed Gravitational Potential Energy (E_p) or its shortened version, Potential Energy^12, 13, which is the energy of any object where gravity can accelerate an object through a vertical displacement, meaning that the change in position of an object is central. Common examples of gravitational potential energy can be seen in roller coasters, water wheels, and in the dams used to create hydroelectric power.

In Newtonian mechanics the measure of a body’s potential energy is given by the equation:

∆ E_p = mg∆h     or    E_p = mgh

where the mass of the object is given in kilograms (kg), gravity is the strength of gravity for this position measured in metres per second squared (m/s²) and the change in position or height is measured in metres (m).  As in kinetic energy, this textbook uses the symbol E to designate energy and the subscript p to indicate that it is potential energy (E_p)¹⁴. The units of measure are the same as those used in kinetic energy, Joules (J).

Potential Energy can simply be remembered as the energy of a body that can move under the influence of gravity, accelerating it.

Gravitational Potential or Gravitational Potential Difference¹⁵  is a concept shared with electricity. You will run into two variations of it in both gravitational and electrical situations. In this first case we only deal with a constant strength gravitational field. Gravitational potential and Electrical potential both use the same symbol (V), named in honor of Alessandro Volta.

Quite simply, gravitational potential in a constant strength gravitational field represents the product of the strength of gravity and change in height.  Or: V = gh. If you wish you can use V_gto represent gravitational potential andV_el to represent Electrical potential but these are non-standard notations.

You should note that Gravitational Potential V (aka: gravitational potential difference) is close in concept to that of Gravitational Potential EnergyE_p, with the only difference being that to find the potential energy involved, we must use the mass that is being moved through the potential difference. In this way, in a constant gravitational field,   E_p  = mV or mgh, the Units for Gravitational Potential are Joules/kilogram (J/kg).

The change in Potential Energy (∆E_p) can be calculated using gravitational potential if you have the amount of mass that is moving through the potential difference.  You will be using the potential difference concept in much greater detail in the later topics concerning electricity.

The following examples illustrate the equations using the potential and kinetic energy equations.

Equations used: 

[latex]\text{Work (W)}=\overrightarrow{F}_{net}\bullet\overrightarrow{d}\text{or}\overrightarrow{F}_{net}\overrightarrow{d}\text{cos Ø}[/latex]                         w = mg

Conceptual Development:

The concept of work is largely accepted as having been introduced by Gaspard-Gustave de Coriolis (1792 – 1843), who believed that work resulted in a transfer of energy by lifting a weight over a distance.  These thoughts are believed to have originated from the early use of steam engines used to lift buckets of water out of flooded mines. As such, work was originally defined as the transfer of energy from one place to another or from one form to another. A simple definition of Coriolis’s concept looks like:

W = ∆ Energy (or W = ∆E_p)

where energy can be given or extracted from a system by doing work on it. The units of work are Joules (J).  Often the units Newton-metre (Nm) are used as a measure of work but recent movements by academic organizations are acting to discourage this unit due to possible confusion with the concept of Torque, that uses Nm as its measure.

When one looks at the concept of lifting a weight to a height, one is able to define the measure of work using algebra. Specifically: _{W = Fnet d}  where the net force must be in a direction to change the height of the object.

This means that the Force used must be in the same direction or in the opposite direction to the change in displacement of the object.  The angle between the Force vector and the displacement vector must be either 0° or 180°.  The equation of W = F d is corrected by the addition of the cosine function of the angle between the force vector and the displacement vector.  Specifically:

W = F_net d cos ø  or  W = F_net • d

Consider the work done in changing the potential energy of the following tubs of water:

10.3  Work & Mechanical Energy ^{17, 18, 19, 20}

Equations Used in this Section:

Work (W) =[latex]\overrightarrow{F}_{net}•\overrightarrow{d} or\overrightarrow{F}_{net}\overrightarrow{d}[/latex] cos Ø            W = ∆ Mechanical Energy

E_p = mgh                   ∆E_p = mg∆h                      E_k = ½mv²              ∆E_k = ½m∆v²

∆E_p = mg(hf – hi)                            ∆E_k = ½m∆ (v_f²– v_i²)

∆h = hf – hi                         ∆v = vf – vi                           V or Vg = gh

Summarizing the previous section:

• If the net force is in the same direction as the displacement, then the system gains energy.

• If the net force is in the opposite direction to the displacement, then energy is removed from the system.

• If the net force direction is at 90° to the change in displacement, then there is no change in the energy of the system.

In this section we investigate the relationship between work and the change in the mechanical energy of a system using W = ∆ Mechanical Energy.  Using this restriction, we will only be considering changes to potential energy (∆E_p = mgh) and kinetic energy (∆E_k = ½ mv²).  As before, the force used to change the energy of an object is the net force (and must be conservative) acting on the object which results in a change in the object’s displacement, specifically:

[latex][Work (W) =\overrightarrow{F}_{net}•\overrightarrow{d} or\overrightarrow{F}_{net}\overrightarrow{d}\text{cos Ø}][/latex]

Net conservative forces²¹ are used to define work that acts to change the energy of a body or a system.

The easiest work energy relationship to work with comes from the Work-Energy Theorem, where the work done can be defined by the resulting change in an object’s kinetic energy.  Specifically:

W = ∆ E_k  or  E_kf –  E_ki

In this case all one needs to do is to find the change in kinetic energy to calculate the work done.

Expanding one  [Work (W) =[latex]\overrightarrow{F}_{net}•\overrightarrow{d} or\overrightarrow{F}_{net}\overrightarrow{d}[/latex] cos Ø}] one gets to extend the quantification of force to be either accelerating or braking, and if it is acting at a vertical angle to the change in the object’s displacement.

It is more complicated to define work as a change in potential energy, since one is restricted to the change in vertical height to calculate this change.  For instance, if an object is raised and lowered several times but ends up in the initial position at the end of this event, then ∆E_p = 0 Jeven though a lot of energy was expended in raising and lowering the object. Changes in potential energy are path dependent.

For the W = ∆ E_k to be used correctly in these problems, the force acting must be a conservative force, where the only changes in energy are potential and kinetic. As you advance in your physics studies, you will be required to distinguish between conservative forces and non-conservative forces.

The derivation of the kinetic energy equation can be completed using an equation termed the work-energy theorem, specifically W = ∆ E_k and the angle between the force and displacement ø = 0°.

First:                  [latex]W =\Delta\text{Energy where W}\overrightarrow{F}_{net}•\overrightarrow{d}\&\overrightarrow{F}_{net}[/latex]= ma(d)

The derivation of potential energy has two versions: one for a constant gravitational field strength and the second (which is covered later) for a variable gravitational field where there is a change in gravity as one moves away (or towards) a planet, sun or moon.

The derivation of the potential energy equation can also be done using the work-energy theorem., specifically ∆ Energy = W  or (F_net d cos ø or ma_gd) and requires that the acceleration of gravity affecting the body is constant and the angle between the net force and the change in vertical displacement is ø = 0°.  This derivation is easily completed.

First:               W = ∆ Energy where W = F_net • d & F_net = ma(d)

Yielding:                              ∆ Energy = mad

Replace a with the constant acceleration due to gravity and d with the vertical change in height.

This results in                                ∆ Energy = mg∆h

And is generally written as    E_p = mgh