10.1. Concepts: Work, Energy, and Power
Work
Introduction
When a force applied to an object so that the displacement of the object is in the same direction as the force, it is said that work is being done on the object.
Do you do work when:
- You carry a platter of food across the kitchen?
- Push a lawnmower on your front yard?
- Lift a bucket of water from the ground to the kitchen sink?
Items b) and c) describe situations in which a force is being applied and produces a displacement in the same direction as the force; therefore, these are situations in which you do work.
a) describes a situation in which the applied force is vertical (you are holding the platter against gravity), but the platter is being moved horizontally. Forces perpendicular to the displacement do not do work on the object, so this is a situation in which you do not do work.
Concepts
Consider a box being pulled on a level floor as represented in the diagram below. The pulling force makes an angle [latex]𝜃[/latex] with the horizontal.
The components of the force on the x and y axes can be calculated as:
[latex]\small F_x = F \hspace{.2cm} cos𝜃[/latex] (1)
[latex]\small F_y = F \hspace{.2cm} sin𝜃[/latex] (2)
Since only the x component produces work (it is the component that has the same direction as the displacement), then work can be calculated as:
[latex]\small W = F[/latex] x [latex] \small d[/latex] x [latex] \small cos𝜃 \hspace{.4cm}[/latex](3)
In formula (3):
[latex]F[/latex] – represents the applied force in [latex][\text N ][/latex]
[latex]d[/latex] – represents the displacement in [latex][\text m][/latex]
[latex]𝜃[/latex] – represents the angle between the force and the displacement
[latex]W[/latex] – represents the work measured in [latex][\text J][/latex]
Work is a scalar quantity, but it can be positive or negative, as follows:
- When work done on an object increases the energy of the object, the work is positive. Example: a child kicking a ball; a wagon being pulled forward so that its speed increases, etc.
- When work done on an object decreases the energy of the object, the work is negative.
- Example: Pushing against a rolling car to stop it, etc.
Examining the formula of work for different situations
a) If 𝜃=0°, then cos 0° = 1, and the formula for work becomes:
[latex]\small W = F[/latex] x [latex] \small d[/latex]
It follows that maximum work is done when the force is parallel and in the same direction as the displacement.
b) If 𝜃=90°, then cos 90 = 0, and work becomes:
[latex]\small W = 0[/latex]
This is consistent with the fact that forces perpendicular to the displacement do not do work.
c) If 𝜃=180°, then cos 180 °= −1, and the formula for work becomes:
[latex]\small W = - F[/latex] x [latex] \small d[/latex]
This shows that if the force is opposite to the direction of the displacement, work is negative (since the effect of the work done will decrease the energy of the object).
Net work
When more than one force does work on an object, a net work can be calculated.
Net work can be calculated in two ways:
- Net work = sum of all works done on the object
- Net work = Net force multiplied by the displacement
Mathematically speaking:
- [latex]\small W_{net} = \small \sum W[/latex] (4)
- [latex]\small W_{net} = F_{net}[/latex] x [latex]d[/latex] (5)
Mechanical Energy
Introduction
Work and energy are closely related. When we do work to move an object, we change the object’s energy.
Energy can be thought as the ability to do work. Energy can take a variety of different forms, and one form of energy can transform to another.
Mechanical energy comes in two forms: kinetic energy and potential energy.
Kinetic energy is the energy of motion, while gravitational potential energy is stored energy as a result of the object’s position above Earth’s surface.
Example
A car speeding on a highway has kinetic energy, while a skier at the top of a hill has gravitational potential energy.
Relationship between work and energy
Example
If we apply force to lift a rock from the ground, we increase the rock’s potential energy, PE. If we drop the rock, the force of gravity increases the rock’s kinetic energy as the rock moves downward until it hits the ground. In both situations, a force does work on the rock, changing the rock’s energy.
Concepts
When work is done on an object and results in the object increasing its speed, the work is said to increase the object’s kinetic energy. Therefore, the change in the kinetic energy of an object equals the work done on the object.
Mathematically speaking:
[latex]\small {W = ∆KE} \hspace{.2cm}[/latex](the work – energy theorem) (1)
The general formula for kinetic energy is:
[latex]\small {KE} \hspace{.2cm} = \hspace{.2cm} \frac {1}{2}mv^2[/latex] (2)
where:
[latex]m[/latex] – represents the mass of the object, in [latex][\text kg][/latex]
[latex]v[/latex] – represents the speed, in [latex][\text m/s][/latex]
[latex]KE[/latex] – represents kinetic energy expressed in [latex][\text J][/latex]
When work is done to lift an object vertically against the force of gravity, then work done on the object changes the potential energy of the object, so:
[latex]\small {W = ∆PE} \hspace{.2cm}[/latex](the work – energy theorem) (3)
The general formula for potential energy is:
[latex]\small {PE} \hspace{.2cm} = \hspace{.2cm} mvh[/latex] (4)
where:
[latex]m[/latex] – represents the mass of the object, in [latex][\text kg][/latex]
[latex]g[/latex] – acceleration due to gravity, in [latex][\text m/s][/latex]
[latex]h[/latex] – represents the height of the object, above or below the reference point, in[latex][\text m][/latex]
Conservation of mechanical energy
Energy cannot be created or destroyed, but only transferred from one form to another.
In the absence of friction, mechanical energy is conserved, meaning the initial mechanical energy equals the final mechanical energy. As mentioned before, mechanical energy consists of two components: kinetic energy and gravitational potential energy.
Therefore:
[latex]\small {ME_1} \hspace{.2cm} = \small {ME_2} \hspace{.4cm}[/latex](5)
When expanding the mechanical energy formula by using the expressions of the kinetic energy and the potential energy, we get:
[latex]\small {KE_1} \hspace{.2cm} + \hspace{.2cm} \small {PE_1} = \small {KE_2} \hspace{.2cm} + \hspace{.2cm} \small {PE_2} \hspace{.4cm}[/latex](6)
[latex]\small {\frac{1}{2}mv_2^1} \hspace{.2cm} + \hspace{.2cm} \small {mgh_1} = \small {\frac{1}{2}mv_2^2} \hspace{.2cm} + \hspace{.2cm} \small {mgh_2} \hspace{.4cm}[/latex](7)
Expression (7) is the most used formula when solving questions related to the conservation of energy principle.
To summarize, there are two distinct principles that can be applied when solving energy related questions:
- The work-energy theorem that applies to solving questions with external forces that do work on an object, thus changing the total mechanical energy of the object. The change can be in the kinetic energy, potential energy, or both. If the work done is positive, the total mechanical energy will increase, and if the work is negative, the total mechanical energy will decrease.
[latex]\small {W = ∆E} \hspace{.2cm}[/latex] (8)
- The conservation of mechanical energy principle that applies to solving questions with internal forces acting on an object (such as gravity), when friction and air resistance are considered negligible. The total mechanical energy of the object remains constant. In such cases, the object’s energy changes form.For example, as an object goes from a higher elevation o a lower one, some of the potential energy of that object is transformed into kinetic energy. Yet, the sum of the kinetic and potential energies remains constant (Figure 10.2a).
[latex]\small {ME_1} \hspace{.2cm} = \small {ME_2} \hspace{.4cm}[/latex](9)
Power
Introduction
Power is the rate at which the energy is being transferred.
Therefore, power is directly proportional to the energy (or work), and inversely proportional to the time in which this energy is transferred.
For instance, runners in a race might achieve the same change in kinetic energy (doing the same amount of work) when they accelerate from rest to a maximum speed, but the athlete developing the greatest amount of power is the one that does this in the shortest amount of time.
Concepts
Power is defined as work done in a certain amount of time, therefore:
[latex]\small P \hspace{.2cm} = \hspace{.2cm} \large {\frac {W}{t}} \hspace{.2cm} = \hspace{.2cm} {\frac {∆E}{t}}[/latex] (1)
If the object is moving at a constant speed, and the force acts in the same direction as the displacement, then:
[latex]\small W = F[/latex] x [latex] \small d[/latex] (2)
Substituting equation (2) in equation (1), we get a new formula for power:
[latex]\small P = \frac{FXd}{t} = v[/latex] x [latex] t[/latex] (3)
In the formulas above:
[latex]P[/latex] – represents the power in [latex][\text W][/latex]. 1 Watt is equal to 1Joule/1second
[latex]W[/latex] – represents the work done in [latex][\text J][/latex]
[latex]t[/latex] – represents time in [latex][\text s][/latex]
[latex]F[/latex] – represents force doing the work in [latex][\text N][/latex]
[latex]d[/latex] – represents displacement in [latex][\text m][/latex]
[latex]v[/latex] – represents speed in [latex][\text {m/s}][/latex]
Another unit for power is horsepower (hp).
The conversion formula is:
[latex]\small 1 \text{ hp} \approx 746 \text{ W}[/latex]
1 horsepower represents the power developed by a horse lifting a mass of 550 lb in 1 second at a height of 1 foot (Figure 10.2b).
Image Attributions
- Figure 10.1 adapted from:
- Figure 10.2a adapted from:
- Figure 10.2b adapted from:
