Energy

## Big Picture

Energy is the measure of an object’s ability to do work (exert a force over a distance) on another system or object. Energy can never be created or destroyed; it can only be transformed into a different type of energy. When somebody says energy has been lost, the person really means that the energy was turned into some form that is unusable. The conservation of energy is one of the fundamental laws of our universe.

## Key Terms

Potential Energy: The potential energy of an object/system is the energy that is stored and ready to be used in the object/ system. SI units: J

System: Any object or group of objects that we are studying. A system where energy is conserved is called a closed system.

Kinetic Energy: A measurement of the energy associated with an object in motion. Kinetic energy can also refer to energy associated with the vibration or motion of atoms or molecules. SI units: J

Work: If a force is exerted over a distance to move an object, the force is said to be doing work. SI unit: J

Conservative Forces: A force in which the work done to move a particle between two points is not affected by the path taken.

Non-Conservative Forces: A force in which the work done to move a particle between two points is affected by the path taken.

Power: The rate at which work is done by a system or object. SI units: J/s or W

## Forms of Energy

There are many different types of potential energy that are differentiated by the way that the energy is stored.

• Gravitational potential energy is based on an object’s position relative to some reference point, and chemical potential energy is stored in the chemical bonds between atoms.
• Gravitational potential energy is the potential for an object/system to move due to the force of gravity.
• Near the surface of the planet, gravitational potential energy $$U_g = mgh$$. For Earth, $$g = 9.8 m/s^2$$.

All moving objects have kinetic energy. $$KE = \frac{1}{2}mv^2$$

• Because the velocity is squared in the formula for kinetic energy, faster objects have much more kinetic energy than slower ones.

## Energy Diagrams

Energy diagrams allow us to visualize how energy is being transferred.

Below is a diagram for a swinging pendulum that shows how energy changes forms from gravitational potential energy to kinetic energy.

## Work

In the work-energy theorem, the work done on an object is always equal to the change in kinetic energy of the object.

• No work is done if there is no motion or if the applied force is perpendicular to the direction of motion.
• If the force is a conservative force, the work done to move an object depends only on the displacement, not the total distance traveled. The work done on an object is equal to the displacement multiplied by the component of the force along the direction of motion.

For example, gravity is a conservative force. When an object’s gravitational potential energy changes, only the object’s displacement determines the work done. The diagram on the right shows how the change in energy is the same whether the red ball takes the green path or the blue path. Gravity does work mgh along either paths.

• Friction is an example of a non-conservative force because the work done actually depends on the total distance the object travels.
• Power is the rate work is done.

Image Credit: CompuChip, Public Domain

## Physics

Energy Problem guide

## Important Equations

$$E_i = E_f$$

$$E_i$$ - initial energy

$$E_f$$ - final energy

$$U_g = mgh$$

$$U_g$$ - gravitational potential energy

m - mass

g - acceleration due to gravity

h - height

$$KE = \frac{1}{2}mv^2$$

KE - kinetic energy

v - velocity

$$W = Fd = \Delta KE$$

W - work

F - force

d - distance

$$p = \frac{\Delta W}{\Delta t}$$

P - power

t - time

## Example

The law of conservation of energy tells us that energy is not created or destroyed. Instead, energy is transformed from one form to another. We can use the conservation of energy to help us solve many types of physics problems.

Tip: When solving problems with gravitational potential energy, pick a reference point that will make calculations easier. The reference point does not have to be the ground.

• In these types of problems, we usually ignore non-conservative forces such as friction and air resistance. If we cannot ignore non-conservative forces, some of the energy in a system will be “lost.” (In fact, the energy is not really lost - it is transferred to another system, usually by heat or sound.)

### Example

An object of mass m begins at rest and starts falling through the air. After falling a distance h, how fast is the object going? Air resistance is negligible.

### Solution

It is possible to solve this using the equations for linear motion, but it is easier to solve by using the conservation of energy. We can set the point where the potential energy PE = 0 at distance h below where the ball started.

$$PE_i + KE_i$$

=

$$PE_f + KE_f$$

begin with an equation showing energy is conserved

$$PE_i + 0$$

=

$$0 + KE_f$$

substitute 0 for the initial kinetic energy and the final potential energy

$$mgh$$

=

$$\frac{1}{2}mv^2$$

substitute in the known values

$$v$$

=

$$\sqrt{2gh}$$

solve for v