# data[].attributes.attachment_link.readable_text
In the first three units of The Physics Classroom, we utilized Newton's laws to analyze the motion of objects. Force and mass information were used to determine the acceleration of an object. Acceleration information was subsequently used to determine information about the velocity or displacement of an object after a given period of time. In this manner, Newton's laws serve as a useful model for analyzing motion and making predictions about the final state of an object's motion. In this unit, an entirely different model will be used to analyze the motion of objects. Motion will be approached from the perspective of work and energy. The effect that work has upon the energy of an object (or system of objects) will be investigated; the resulting velocity and/or height of the object can then be predicted from energy information. In order to understand this work-energy approach to the analysis of motion, it is important to first have a solid understanding of a few basic terms. Thus, Lesson 1 of this unit will focus on the definitions and meanings of such terms as work, mechanical energy, potential energy, kinetic energy, and power.

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. 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. There are several good examples of work that can be observed 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, 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.

Read the following five statements and determine whether or not they represent examples of work. Then click on the See Answer button to view the answer.

Statement

Answer with Explanation

A teacher applies a force to a wall and becomes exhausted.

A book falls off a table and free falls to the ground.

A waiter carries a tray full of meals above his head by one arm straight across the room at constant speed. (Careful! This is a very difficult question that will be discussed in more detail later.)

A rocket accelerates through space.

Work Equation

Mathematically, work can be expressed by the following equation.

W = F * d * cos Θ

where F is the force, d is the displacement, and the angle ( theta) is defined as the angle between the force and the displacement vector. Perhaps the most difficult aspect of the above equation is the angle "theta." The angle is not just any 'ole angle, but rather a very specific angle. The angle measure is defined as the angle between the force and the displacement. To gather an idea of it's meaning, consider the following three scenarios.

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.

Scenario B: A force acts leftward upon an object that 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.

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

Let's consider Scenario C above in more detail. 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. 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. 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). 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. 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.

The equation for work lists three variables - each variable is associated with one of the three key words mentioned in the definition of work (force, displacement, and cause). The angle theta in the equation is associated with the amount of force that causes a displacement. As mentioned in a previous unit, when a force is exerted on an object at an angle to the horizontal, only a part of the force contributes to (or causes) a horizontal displacement. Let's consider the force of a chain pulling upwards and rightwards upon Fido in order to drag Fido to the right. It is only the horizontal component of the tension force in the chain that causes Fido to be displaced to the right. The horizontal component is found by multiplying the force F by the cosine of the angle between F and d. In this sense, the cosine theta in the work equation relates to the cause factor - it selects the portion of the force that actually causes a displacement.

The Meaning of Theta

When determining the measure of the angle in the work equation, it is important to recognize that the angle has a precise definition - it is the angle between the force and the displacement vector. Be sure to avoid mindlessly using any 'ole angle in the equation. A common physics lab involves applying a force to displace a cart up a ramp to the top of a chair or box. A force is applied to a cart to displace it up the incline at constant speed. Several incline angles are typically used; yet, the force is always applied parallel to the incline. The displacement of the cart is also parallel to the incline. Since F and d are in the same direction, the angle theta in the work equation is 0 degrees. Nevertheless, most students experienced the strong temptation to measure the angle of incline and use it in the equation. Don't forget: the angle in the equation is not just any 'ole angle. It is defined as the angle between the force and the displacement vector.

The Meaning of 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. These situations involve what is commonly called negative work. The negative of negative work refers to the numerical value that results when values of F, d and theta are substituted into the work equation. 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. Negative work will become important (and more meaningful) in Lesson 2 as we begin to discuss the relationship between work and energy.

Units of Work

Whenever a new quantity is introduced in physics, the standard metric units associated with that quantity are discussed. In the case of work (and also energy), the standard metric unit is the Joule (abbreviated J). One Joule is equivalent to one Newton of force causing a displacement of one meter. In other words,

The Joule is the unit of work.

1 Joule = 1 Newton * 1 meter

1 J = 1 N * m

In fact, any unit of force times any unit of displacement is equivalent to a unit of work. Some nonstandard units for work are shown below. Notice that when analyzed, each set of units is equivalent to a force unit times a displacement unit.

Non-standard Units of Work:

foot*pound kg*(m/s²)*m kg*(m ²/s²)

In 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.

Investigate!
We do work every day. The work we do consumes Calories ... err, should we say Joules. But how much Joules (or Calories) would be consumed by various activities? Use the Daily Work widget to investigate the amount of work that would be done to run, walk or bike a for a given amount of time at a specified pace.

Click to continue the lesson on Work

We Would Like to Suggest ...

Sometimes it isn't enough to just read about it. You have to interact with it! And that's exactly what you do when you use one of The Physics Classroom's Interactives. We would like to suggest that you combine the reading of this page with the use of our It's All Uphill Interactive. You can find it in the Physics Interactives section of our website. The It's All Uphill Interactive allows a learner to explore the effect of an incline angle upon the force and the work done in pulling a cart up a hill at a constant speed.

# data[].attributes.attachment_link.description
When a force acts upon an object while it is moving, work is said to have been done upon the object by that force. Work can be positive work if the force is in the direction of the motion and negative work if it is directed against the motion of the object.

# data[].attributes.attachment_link.readable_text
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. 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.

Gravitational Potential Energy

The two examples above illustrate the two forms of potential energy to be discussed in this course - gravitational potential energy and elastic 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. The gravitational potential energy of the massive ball of a demolition machine is dependent on two variables - the mass of the ball and the height to which it is raised. There is a direct relation between gravitational potential energy and the mass of an object. More massive objects have greater 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. These relationships are expressed by the following equation:

PEgrav = mass * g * height

PEgrav = m ** g * h

In the above equation, m represents the mass of the object, h represents the height of the object and g represents the gravitational field strength (9.8 N/kg on Earth) - sometimes referred to as the acceleration of gravity.

To determine the gravitational potential energy of an object, a zero height position must first be arbitrarily assigned. Typically, the ground is considered to be a position of zero height. But this is merely an arbitrarily assigned position that most people agree upon. Since many of our labs are done on tabletops, it is often customary to assign the tabletop to be the zero height position. Again this is merely arbitrary. If the tabletop is the zero position, then the potential energy of an object is based upon its height relative to the tabletop. For example, a pendulum bob swinging to and from above the tabletop has a potential energy that can be measured based on its height above the tabletop. By measuring the mass of the bob and the height of the bob above the tabletop, the potential energy of the bob can be determined.

Since the gravitational potential energy of an object is directly proportional to its height above the zero position, a doubling of the height will result in a doubling of the gravitational potential energy. A tripling of the height will result in a tripling of the gravitational potential energy.

Use this principle to determine the blanks in the following diagram. Knowing that the potential energy at the top of the tall platform is 50 J, what is the potential energy at the other positions shown on the stair steps and the incline?

Elastic Potential Energy

The second form of potential energy that we will discuss is 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. 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.

Springs are a special instance of a device that 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 that is required to compress it further. For certain springs, the amount of force is directly proportional to the amount of stretch or compression (x); the constant of proportionality is known as the spring constant (k).

Fspring = k * x

Such springs are said to follow Hooke's Law. If a spring is not stretched or compressed, then there is no elastic potential energy stored in it. The spring is said to be at its equilibrium position. The equilibrium position is the position that the spring naturally assumes when there is no force applied to it. In terms of potential energy, the equilibrium position could be called the zero-potential energy position. There is a special equation for springs that relates the amount of elastic potential energy to the amount of stretch (or compression) and the spring constant. The equation is

PEspring = 0.5 * k * x²

where k = spring constant

x = amount of compression
(relative to equilibrium position)

To summarize, potential energy is the energy that is stored in an object due to its position relative to some zero position. An object possesses gravitational potential energy if it is positioned at a height above (or below) the zero height. An object possesses elastic potential energy if it is at a position on an elastic medium other than the equilibrium position.

Check Your Understanding

Check your understanding of the concept of potential energy by answering the following questions. When finished, click the button to view the answers.

1. 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?

2. If a force of 14.7 N is used to drag the loaded cart (from previous question) along the incline for a distance of 0.90 meters, then how much work is done on the loaded cart?

Note that the work done to lift the loaded cart up the inclined plane at constant speed is equal to the potential energy change of the cart. This is not coincidental! The reason for the relation between the potential energy change of the cart and the work done upon it is the subject of Lesson 2.

# data[].attributes.attachment_link.description
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.

# data[].attributes.attachment_link.readable_text
previous part of Lesson 1, it was said that work is done upon an object whenever a force acts upon it to cause it to be displaced. Work involves a force acting upon an object to cause a displacement. In all instances in which work is done, there is an object that supplies the force in order to do the work. If a World Civilization book is lifted to the top shelf of a student locker, then the student supplies the force to do the work on the book. If a plow is displaced across a field, then some form of farm equipment (usually a tractor or a horse) supplies the force to do the work on the plow. If a pitcher winds up and accelerates a baseball towards home plate, then the pitcher supplies the force to do the work on the baseball. If a roller coaster car is displaced from ground level to the top of the first drop of a roller coaster ride, then a chain driven by a motor supplies the force to do the work on the car. If a barbell is displaced from ground level to a height above a weightlifter's head, then the weightlifter is supplying a force to do work on the barbell. In all instances, an object that possesses some form of energy supplies the force to do the work. In the instances described here, the objects doing the work (a student, a tractor, a pitcher, a motor/chain) possess chemical potential energy stored in food or fuel that is transformed into work. In the process of doing work, the object that is doing the work exchanges energy with the object upon which the work is done. When the work is done upon the object, that object gains energy. The energy acquired by the objects upon which work is done is known as mechanical energy.
Mechanical energy is the energy that is possessed by an object due to its motion or due to its position. Mechanical energy can be either kinetic energy (energy of motion) or potential energy (stored energy of position). Objects have mechanical energy if they are in motion and/or if they are at some position relative to a zero potential energy position (for example, a brick held at a vertical position above the ground or zero height position). A moving car possesses mechanical energy due to its motion ( kinetic energy). A moving baseball possesses mechanical energy due to both its high speed ( kinetic energy) and its vertical position above the ground (gravitational potential energy). A World Civilization book at rest on the top shelf of a locker possesses mechanical energy due to its vertical position above the ground (gravitational potential energy). A barbell lifted high above a weightlifter's head possesses mechanical energy due to its vertical position above the ground (gravitational potential energy). A drawn bow possesses mechanical energy due to its stretched position (elastic potential energy).

Mechanical Energy as the Ability to Do Work

An object that possesses mechanical energy is able to do work. In fact, mechanical energy is often defined as the ability to do work. Any object that possesses mechanical energy - whether it is in the form of potential energy or kinetic energy - is able to do work. That is, its mechanical energy enables that object to apply a force to another object in order to cause it to be displaced.

Numerous examples can be given of how an object with mechanical energy can harness that energy in order to apply a force to cause another object to be displaced. A classic example involves the massive wrecking ball of a demolition machine. The wrecking ball is a massive object that is swung backwards to a high position and allowed to swing forward into building structure or other object in order to demolish it. Upon hitting the structure, the wrecking ball applies a force to it in order to cause the wall of the structure to be displaced. The diagram below depicts the process by which the mechanical energy of a wrecking ball can be used to do work.

A hammer is a tool that utilizes mechanical energy to do work. The mechanical energy of a hammer gives the hammer its ability to apply a force to a nail in order to cause it to be displaced. Because the hammer has mechanical energy (in the form of kinetic energy), it is able to do work on the nail. Mechanical energy is the ability to do work.

Another example that illustrates how mechanical energy is the ability of an object to do work can be seen any evening at your local bowling alley. The mechanical energy of a bowling ball gives the ball the ability to apply a force to a bowling pin in order to cause it to be displaced. Because the massive ball has mechanical energy (in the form of kinetic energy), it is able to do work on the pin. Mechanical energy is the ability to do work.

A dart gun is still another example of how mechanical energy of an object can do work on another object. When a dart gun is loaded and the springs are compressed, it possesses mechanical energy. The mechanical energy of the compressed springs gives the springs the ability to apply a force to the dart in order to cause it to be displaced. Because of the springs have mechanical energy (in the form of elastic potential energy), it is able to do work on the dart. Mechanical energy is the ability to do work.

A common scene in some parts of the countryside is a "wind farm." High-speed winds are used to do work on the blades of a turbine at the so-called wind farm. The mechanical energy of the moving air gives the air particles the ability to apply a force and cause a displacement of the blades. As the blades spin, their energy is subsequently converted into electrical energy (a non-mechanical form of energy) and supplied to homes and industries in order to run electrical appliances. Because the moving wind has mechanical energy (in the form of kinetic energy), it is able to do work on the blades. Once more, mechanical energy is the ability to do work.

The Total Mechanical Energy

As already mentioned, the mechanical energy of an object can be the result of its motion (i.e., kinetic energy) and/or the result of its stored energy of position (i.e., potential energy). The total amount of mechanical energy is merely the sum of the potential energy and the kinetic energy. This sum is simply referred to as the total mechanical energy (abbreviated TME).

TME = PE + KE

As discussed earlier, there are two forms of potential energy discussed in our course - gravitational potential energy and elastic potential energy. Given this fact, the above equation can be rewritten:

TME = PEgrav + PEspring + KE

The diagram below depicts the motion of Lee Ben Fardest (esteemed American ski jumper) as he glides down the hill and makes one of his record-setting jumps.

The total mechanical energy of Lee Ben Fardest is the sum of the potential and kinetic energies. The two forms of energy sum up to 50 000 Joules. Notice also that the total mechanical energy of Lee Ben Fardest is a constant value throughout his motion. There are conditions under which the total mechanical energy will be a constant value and conditions under which it will be a changing value. This is the subject of Lesson 2 - the work-energy relationship. For now, merely remember that total mechanical energy is the energy possessed by an object due to either its motion or its stored energy of position. The total amount of mechanical energy is merely the sum of these two forms of energy. And finally, an object with mechanical energy is able to do work on another object.

# data[].attributes.attachment_link.description
previous part of Lesson 1, it was said that work is done upon an object whenever a force acts upon it to cause it to be displaced. Work involves a force acting upon an object to cause a displacement. In all instances in which work is done, there is an object that supplies the force in order to do the work.

# data[].attributes.headline
-The highest kinetic energy is at the lowest point of the pendulum's swing

# data[].attributes.subject
-The highest kinetic energy is at the lowest point of the pendulum's swing

# data[].attributes.body
-KE=(1/2m)s
-There is a point of the pendulum swing that potential energy and kinetic energy are equal

# data[].attributes.body
1. It takes work to get the pendulum to move
2. The work is different if you lift the pendulum up fast or slow
3. Work = Fd

# data[].attributes.body
1. There's no power when the pendulum is going down.
2. When the pendulum being pulled  up, there's increasing gravitational potential energy.
3. P=t/w

# data[].attributes.body
The pendulum's mechanical energy is always equal to its kinetic energy.

The pendulum is always using mechanical energy as it is swinging back and forth.

The pendulum's mechanical energy is always equal to its kinetic energy plus it's potential energy.

# data[].attributes.body
the equation is Eg = mgh

# data[].attributes.body
The highest point is where the greatest amount of potential energy is.

# data[].attributes.body
The gravitational potential energy is the greatest when exactly between the highest and lowest points of the pendulum's swing