# Work, Energy, & Power

Energy comes in many different forms, some of which you might already be familiar with. Just a few of the many types of energy are mechanical energy, chemical energy, electrical energy, thermal energy, and nuclear energy. In this lesson, we will focus on mechanical energy and its conservation, as well as the closely related concepts of work and power. It is important to have a good understanding of kinematics and forces before beginning this lesson, so if you have not already, you should review the notes on Kinematics and Newton's Laws.

Content that appears in a box similar to this is content that applies only to the AP C curriculum. AP B and AICE students can skip any content contained in these boxes.

### Work

- In everyday life, work is generally considered to mean putting effort toward accomplishing a given task. In physics, the definition of work is a bit different; work is done on an object when a force exerted on the object causes the object to move through a displacement.
- More specifically, work is the change in energy of a system resulting from a force that causes a displacement. For example, when you slide a book across a desk, you apply a force to the book, which causes the book to move. In this situation, you have done work on the book, which has caused a change in the amount of energy in the book. It may seem odd to think that a book has energy, but in fact all objects store energy in some form.
- A force only does work on an object if it causes the object to experience a displacement parallel to a component of the force. So, when the book sits on the desk, it experiences a gravitational force and a normal force, but no work is done on the book because it is not displaced. When you push the book across the desk, the force you exert on the book does work because the book is displaced parallel to the force, but the gravitational and normal force still do not do work on the book because they are perpendicular to the displacement, so there is no component of either force parallel to the displacement.
- Work is denoted by the capital letter
*W*, and is found by multiplying the object's displacement by the component of force parallel to the displacement,*W*=**F***d*. The unit for work is the newton-meter (N*m), which is renamed the joule, abbreviated J. - If the force causing the object's displacement is not parallel to the displacement, then you must use trigonometry to find the component of the force parallel to the displacement, which will usually be
**F**cosθ. - Also note that, although work is dependent upon two vectors, force and displacement, work is a scalar quantity; work does not have a direction. However, work can be negative, which occurs when the force is antiparallel (parallel but pointing in the opposite direction) to the displacement. For example, friction does negative work because it acts opposite the direction of motion.
- Work is also negative when the object does work, as opposed to having work done on it. For example, when you push a book across a desk, you do work on the book, so the book experiences positive work, while you exerience negative work.

### Work Done by a Variable Force

- When the force doing work is constant, work is simply the product of force and displacement. However, when the force is not constant, it is not quite as simple.
- The graph above shows the magnitude of a force as it changes over a given displacement. The change in this force is linear, or a straight line. In order to find the work done by this force between positions
*x*_{1}and*x*_{2}, we must find the area of the section between the line and the x-axis. - It is helpful to notice that this area is in the shape of a trapezoid (it may be easier to see this if you turn your head 90
^{o}). So, the area of this section and thus the work done by the force can be found using the area formula for a trapezoid:*A*=^{1}/_{2}(*b*_{1}+*b*_{2})*h*=^{1}/_{2}(**F**_{1}+**F**_{2})(*x*_{2}-*x*_{1}). - You could also accomplish this same calculation by splitting the section into two parts, a rectangle and a triangle, and adding the areas of each.
- Note that area under the graph of a function is the definition of a definite integral. So, work done by a variable force can also be found by taking the integral of force with respect to position, from position
*x*_{1}to position*x*_{2}. This also means that the derivative of work with respect to displacement is force.

### Basics of Energy

- All forms of energy are measured in Joules, J.
- Energy is always conserved. What this means is that energy is never created nor destroyed, the total amount of energy in existence remains the same at all times.
- However, energy can be transferred from one object to another through various methods, and can be transformed; that is, energy can change from one type of energy to another. For example, the chemical energy stored in your food is converted by your body into the mechanical energy that allows you to move.

### Kinetic Energy

- Kinetic energy is, put in simplest terms, the energy of motion. Any object with mass has kinetic energy when it is moving in any direction, and no kinetic energy when it is at rest.
- Kinetic energy is represented with an uppercase letter
*K*or the uppercase letters*KE*. - Kinetic energy can be calculated using the formula
*K*=^{1}/_{2}*mv*^{2}. Therefore, kinetic energy is dependent only upon the mass and velocity of an object, and a change in velocity will affect the object's kinetic energy more than an equal change in the object's mass. - Note that kinetic energy is a scalar quantity, and thus has no direction.
- According to the work-energy theorem, the total amount of work done on a system during a given process is equal to the change in the object's kinetic energy during that process:
*W*_{total}=*ΔK*. - The reason that the work-energy theorem is true is because work is a measure of the energy transferred from one object to another, or transformed from one form to another. In the case of a force causing a change in an object's motion, the energy being transferred to or from the object is in the form of kinetic energy. Since all objects have a constant mass, if an object's kinetic energy increases, then its velocity must also increase, and if the object's kinetic energy decreases, its velocity must decrease.

### Potential Energy

- Where kinetic energy is dependent upon an object's motion, potential energy is dependent upon an object's position, regardless of motion.
- Sometimes, an object that is set in motion gains kinetic energy because the energy was transfered to it from another object, such as when you push a book across a desk: you transfer some of your kinetic energy to the book. But, what about a ball released from rest under only the influence of gravity? There is nothing to transfer kinetic energy to the ball, but yet its kinetic energy increases. Since we know energy is conserved, that kinetic energy has to come from somewhere. In this case, the kinetic energy is transformed from other types of energy the ball already has.
- Though virtually any type of energy can be transformed into virtually any other type of energy through the proper means, in this case it is the ball's potential energy that is transformed into kinetic energy. As the ball accelerates downward, its velocity increases, causing its kinetic energy to rise. At the same time, the ball's potential energy decreases because it is being converted into kinetic energy.
- Potential energy is denoted with a capital letter
*U*or the capital letters*PE*, and may have subscripts to show the type of potential energy. - For now we will focus on a type of potential energy called gravitational potential energy, which, as the name suggests, is caused by an object's position in a gravitational field.
- When you lift an object upward against the force of gravity, you do work on that object, thus transferring an amount of energy to that object. When the object comes to rest at a given height above the ground, that energy is stored as potential energy. Since raising the same object higher in the air causes a greater displacement, the amount of work done on the ball is greater the higher it is lifted, and so we can infer that it also gains more potential energy.
- From this, we learn that the change in an object's gravitational potential energy is equal to the opposite of the amount of work done to lift the ball to that height:
*ΔU*= -_{g}*W*_{by gravity}. Also, since*W*=**F***d*, and**F**=*ma*, and the acceleration due to gravity is -*g*and the displacement is the change in height,*Δh*, of the object, we see that*W*= -*mgh*, so*ΔU*=_{g}*mg**Δh*. - Note that the above equation is true no matter what path the object takes to change its height. So, an object that is lifted straight upward 2 m has the same change in gravitational potential energy as an object that is lifted 1 m, lowered 0.5 m, then raised 1.5 m, or an object that is pushed up a ramp whose height (not length) is 2 m, or an object that travels some curve, as long as the object's final height is 2 m higher than its initial height (assuming all of these objects have the same mass).
- If you define a given height where
*h*= 0 (usually the Earth's surface), then you can find an object's exact potential energy instead of its change in potential energy using the equation*U*=*mgh*.

### Conservation of Mechanical Energy

- Mechanical energy, denoted
*E*, is the sum of an object or system's kinetic and potential energy:*E*=*K*+*U*. - In the absence of friction and other forces that may transfer energy out of a system, mechanical energy is conserved, so the mechanical energy of a system before a given motion or change in motion is the same as the mechanical energy after:
*K*+_{i}*U*=_{i}*K*+_{f}*U*._{f} - When friction is present, mechanical energy is not conserved. Friction causes some mechanical energy to be transferred out of the system and/or transformed into other types of energy. The primary loss of mechanical energy due to friction takes the form of heat and sound.
- When we assume that friction is not present, because of the conservation of mechanical energy we see that any change to an object's kinetic energy will cause an equal but opposite change to the object's potential energy, and vice-versa. So, if an object's kinetic energy increases by 5 J in the absence of friction and other nonconservative forces, then its potential energy must also decrease by 5 J.
- Note that some reference books for the AP C exam cover potential energy curves in relation to Hooke's Law at this point. Because this deals with oscillations, I have decided to save this content for the lesson on oscillations. The important part to know right now is that a curve defining potential energy will appear on a graph using position,
*x*, as the x-axis and mechanical energy,*E*, as the y-axis. For any defined interval between two positions having the same potential energy, the potential energy at any point in this interval is defined by the curve, and kinetic energy is defined by the difference between the highest potential energy in the interval and the potential energy at that point.

### Power

- Power has many uses in everyday language, but in physics, power specifically means the rate at which work is done. So, if two forces each do 10 J of work, but force
**F**_{1}does this work in 1 minute while**F**_{2}does it in two minutes,**F**_{1}has more power because it did the same work in less time. - Power is represented by the capital letter
*P*, and is measured in Watts, W (note that the symbol for work,*W*, is italicized and the symbol for watts, W, is not; Be careful not to confuse the two). Power can be calculated by dividing work by time,*P*=*W*/*t*. - If the force causing work is parallel to the displacement involved, then we can substitute the definition for work into the equation for power,
*P*= (**F***d*)/*t*. Since*d*/*t*is the definition of velocity, we can rewrite this equation to find the alternate definition of power,*P*=**F***v*. - In relation to energy, we can say that work is the amount of energy transferred or transformed, while power is the rate at which this energy is transferred or transformed.
- Note that since power is the rate at which work is done, this defines power as the derivative of work with respect to time. So we now know two ways to differentiate work: differentiating with respect to displacement will yield force, and differentiating with respect to time will yield power. Be careful not to confuse the two.