Equations Introduced and Used in this Topic:

• Work [latex](W) = \vec{F}_{\text{net}} \cdot \vec{d}[/latex] or [latex]\vec{F}_{\text{net}} \vec{d} \cos ø[/latex]
• [latex]E_p = m g h[/latex]
• [latex]\Delta E_p = m g \Delta h[/latex]
• [latex]\Delta h = h_f - h_i[/latex]
• [latex]\Delta v = v_f - v_i[/latex]
• [latex]W = \Delta[/latex] Mechanical Energy
• [latex]E_k = \frac{1}{2} m v^2[/latex]
• [latex]\Delta E_k = \frac{1}{2} m \Delta v^2[/latex]
• [latex]V[/latex] or [latex]V_g = g h[/latex]
• [latex]w = m g[/latex]

(All equations are esolved as either scalar or vector)

Where…

• [latex]W[/latex] is Work, measured in joules (J)
• [latex]E_p[/latex] or [latex]\Delta E_p[/latex]   is Gravitational Potential Energy or change in Gravitational Potential Energy, measured in joules (J)
• [latex]E_k[/latex] or [latex]\Delta E_k[/latex]  is Kinetic Energy or change in Kinetic Energy, measured in joules (J)
• [latex]E[/latex] or [latex]\Delta E[/latex] is Energy or change in Energy, measured in joules (J)
• [latex]V[/latex] or [latex]V_g[/latex] is the Gravitational Potential, measured in newtons per kilogram (N/kg)
• [latex]m[/latex] is the Mass of the object, measured in kilograms (kg)
• [latex]g[/latex] or [latex]a_g[/latex] is the Gravitational Field Strength, measured in metres per second squared (m/s²)
• [latex]v[/latex] is the Speed of the object, measured in metres per second (m/s)
• [latex]v[/latex] or [latex]\Delta v[/latex]  is the Speed or change in Speed of the object, measured in metres per second (m/s)
• [latex]v_f[/latex] & [latex]v_i[/latex]  is the final and initial Speed of the object, measured in metres per second (m/s)
• [latex]F_{\text{net}}[/latex]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, measure in degrees (°)
• [latex]d[/latex] is the displacement through which the Net Force acts on a body or a System, measured in metres (m)
• [latex]h[/latex] or [latex]\Delta h[/latex] is the height or change in Vertical Height, measured in metres (m)
• [latex]h_f[/latex] & [latex]h_i[/latex]  is the final and initial height of the object measured in metres (m)

Notes: 

The Dot Product “[latex]\cdot[/latex]“ 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 “[latex]\times[/latex]“ 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

The Dot Product for these two vectors is: 

• [latex]\vec{A} \cdot \vec{B} = \vec{A}\vec{B} \cos ø[/latex]
• [latex]\vec{A} \cdot \vec{B} = (20)(18) \cos 42^{\circ}[/latex]
• [latex]\vec{A} \cdot \vec{B} = 267.5[/latex] or 270

The Cross Product for these two vectors is:

• [latex]\vec{A} \times \vec{B} = \vec{A}\vec{B} \sin ø[/latex]
• [latex]\vec{A} \times \vec{B} = (20)(18) \sin 42^{\circ}[/latex]
• [latex]\vec{A} \times \vec{B} = 240.9[/latex] or 240

10.1 Kinetic and Potential Energy

Conceptual Development:

The concept of energy is generally credited to Aristotle’s (384 BCE-322 BCE) 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 BCE) 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 speed^[1] 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)^[2]. 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) did introduce the word energy in 1800 but it did not become a term of 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^[3] and work all helped to lay the foundations for a complete rewrite of the understanding of science away from 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:

The change in Potential Energy ([latex]\Delta E_p[/latex]) 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.

10.2 Work

Equations Introduced or Used for this Section:

• Work [latex](W) = \vec{F}_{\text{net}} \cdot \vec{d}[/latex] or [latex]\vec{F}_{\text{net}} \vec{d} \cos ø[/latex]
• [latex]w = m g[/latex]

Conceptual Development:

The concept of work is largely accepted as having been introduced by Gaspard-Gustave de Coriolis (1792 – 1843) from his thoughts that work resulted in a transfer of energy by lifting some weight through 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:

[latex]W = \Delta \text{ Energy}[/latex] (or [latex]W = \Delta E_p[/latex])

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 through a height, one is able to define the measure of work using algebra. Specifically: [latex]W = F_{\text{net}} d[/latex]  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 [latex]W = F d[/latex] is corrected by the addition of the cosine function of the angle between the force vector and the displacement vector.  Specifically:

[latex]W = F_{\text{net}} d \cos ø[/latex] or [latex]W = F_{\text{net}} \cdot d[/latex]

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

10.3 Work & Mechanical Energy

Equations Introduced or Used for this Section:

• Work [latex](W) = \vec{F}_{\text{net}} \cdot \vec{d}[/latex] or [latex]\vec{F}_{\text{net}} \vec{d} \cos ø[/latex]
• [latex]E_p = m g h[/latex]
• [latex]\Delta E_p = m g \Delta h[/latex]
• [latex]\Delta h = h_f - h_i[/latex]
• [latex]\Delta v = v_f - v_i[/latex]
• [latex]W = \Delta[/latex] Mechanical Energy
• [latex]E_k = \frac{1}{2} m v^2[/latex]
• [latex]\Delta E_k = \frac{1}{2} m \Delta v^2[/latex]
• [latex]V[/latex] or [latex]V_g = g h[/latex]
• [latex]w = m g[/latex]

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 is at 90° to direction as 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 [latex]W = \Delta \text{ Mechanical Energy}[/latex].  Using this restriction, we will only be considering changes to potential energy ([latex]\Delta E_p = m g h[/latex]) and kinetic energy ([latex]\Delta E_k = \frac{1}{2}m v^2[/latex]).  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 some change in the objects displacement, specifically.

Work [latex](W) = \vec{F}_{\text{net}} \cdot \vec{d}[/latex] or [latex]\vec{F}_{\text{net}} \vec{d} \cos ø[/latex]

Net conservative forces^[10] 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 found by the resulting change in an object’s kinetic energy.  Specifically:

[latex]W = \Delta E_k[/latex] or [latex]E_{k f} - E_{k i}[/latex]

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

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

It is more complicated in defining 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 [latex]\Delta E_p = 0 \text{ J}[/latex] even though a lot of energy was expended in raising and lowering the object. Changes in potential energy are path dependent.

For the [latex]W = \Delta E_k[/latex] to be correct in usage for these problems, the force acting must be a conservative force, where the only changes in energy are either 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 [latex]W = \Delta E_k[/latex] and the angle between the force and displacement ø = 0°.

First:

[latex]W = \Delta[/latex] Energy where [latex]W = \vec{F}_{\text{net}} \cdot \vec{d}[/latex] & [latex]\vec{F}_{\text{net}} = m a (d)[/latex]

Yielding:

[latex]\Delta \text{ Energy } = m a d[/latex]

Using the kinematic equation...

• [latex]2 a d = v_f^2 - v_i^2[/latex]
• [latex]a d = \dfrac{(v_f^2 - v_i^2)}{2}[/latex]

Substituting into the first equation [latex]\Delta \text{ Energy } = m a d[/latex], we get

[latex]\Delta E = \dfrac{(v_f^2 - v_i^2)}{2}[/latex]

And is generally written as:

[latex]E_k = \frac{1}{2} m v^2[/latex]

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 one has 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 [latex]\Delta \text{ Energy } = W[/latex]  or ([latex]F_{\text{net}} d \cos ø[/latex] or [latex]magd[/latex]) and requires 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:

[latex]W = \Delta \text{ Energy}[/latex] where [latex]W = \vec{F}_{\text{net}} \cdot d[/latex] & [latex]\vec{F}_{\text{net}} = m a (d)[/latex]

Yielding:

[latex]\Delta \text{ Energy } = m a d[/latex]

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

This results in:

[latex]\Delta \text{ Energy } = m g \Delta h[/latex]

And is generally written as:

[latex]E_p = m g h[/latex]

Exercise Answers

10.1 Kinetic and Potential Energy

1. 2500 J
2. [latex]2.9 \times 10^4[/latex] J
3. [latex]4.56 \times 10^{-17}[/latex] J
4. 1. 38 Ns North
   2. 660 J
5. 2800 J
6. − 19 J

10.2 Work

1. 1. 16 J
   2. 6.6 J
   3. 45 J
   4. [latex]3.08 \times 10^6[/latex] J
   5. 420 J

2. 8.0 m South
3. Zero joules, (0 J)
4. 58000 J
5. 320 J

10.3 Work & Mechanical Energy

1. 1. [latex]1.225 \times 10^3[/latex] J
   2. [latex]1.225 \times 10^4[/latex] J
   3. [latex]1.35 \times 10^4[/latex] J
2. [latex]9.8 \times 10^5[/latex] J
3. 19600 J
4. 37.5 N/kg
5. 5 000 J
6. − 540000 J
7. − 2500 N
8. − 1100 N
9. − 2500 N
10. − 13.7 J
11. − 510 J
12. 17000 N
13. 800 N
14. 1. − 1800 N
    2. − 90 J
15. 1. [latex]4.0 \times 10^5[/latex] J
    2. − 4000 N
    3. − 4000 N
16. 140 m

10.4.1 Bennu on JPL’s Sentry: A Potential Major Natural Disaster

1. [latex]E_k = 5.37 \times 10^{18}[/latex] J
2. 1300 mega tons of TNT

Media Attributions

• "Niagara Falls" by martajozsa is licensed under a Pixabay licence.
• "Bennuasteroid" from NASA is in the public domain.