Potential Energy and Energy Conservation

Principles of Potential Energy and Energy Conservation

As mentioned above, the energy in a system is made up of its kinetic energy, potential energy, and internal energy. In addition to the ability of energy to change forms, the energy in a system can shift from one object to another, like a spring pushing a block. We must consider all objects that make up the system when we find the total energy of a system.

We have defined two types of potential energy: gravitational potential energy and elastic potential energy. In general, for a system to have potential energy, there must be a conservative force acting upon the object. In gravitational potential energy, the conservative force is the force of gravity, and in our spring example for elastic potential energy, it was the spring force. The change in potential energy is proportional to the work done by the corresponding conservative force. Non-conservative forces, such as friction and air resistance, do not contribute to potential energy; they instead contribute to internal energy. We will go into greater detail about conservative and non-conservative forces in a different article.

Relationship between Potential Energy and Conservative Energy

To discuss the relationship between potential energy and energy conservation, let us consider a system in which there are only conservative forces doing work. The total mechanical energy, \(E\), of the system is found by adding the kinetic energy, \(K\), and the potential energy, \(U\), at any moment so that \(E=K+U\). From the law of the conservation of energy, we know that the total mechanical energy in the system is constant. Thus when the potential energy decreases in the system, there is an increase in the kinetic energy.

Consider a ball of mass \(m\) being dropped from a certain height off the ground, \(h\). We will ignore air resistance for this example. We find the gravitational potential energy by using the formula given above: \(U_{grav} = mgh\). The kinetic energy of the ball is given by \(K=\frac{1}{2}mv^2\). We see from these formulas that as the ball approaches the ground, the gravitational potential energy decreases as the height approaches zero and the kinetic energy increases with its velocity.

Total mechanical energy of a ball in freefall, StudySmarter Originals

We can find the total mechanical energy at any point as the ball falls to the ground. Say we find the total mechanical energy at two different points as the ball falls. We can write the total mechanical energy at the two points as:

$$ \begin{aligned} E_1 &= K_1 + U_1 \\ E_2 &= K_2 + U_2 \end{aligned}$$

The total mechanical energy of the system is constant, so \(E_1 = E_2 \). Then we have:

$$ \begin{aligned} K_1 + U_1 &= K_2 + U_2 \\ K_2 - K_1 &= U_1 - U_2 \\ \Delta K &= - \Delta U_{grav} \\ W_{grav} &= - \Delta U_{grav} \end{aligned}$$

The change in kinetic energy of the system, or the work done by gravity, is thus equivalent to the negative change in gravitational potential energy of the system.

In this section, we have only considered energy conservation when there are conservative forces. When there are non-conservative forces acting on objects in the system, we must consider the work done by these forces. The work done by these forces is equal to the negative change in internal energy, \(W_{nc} = -\Delta E_{int}\). If we do not ignore air resistance in our example above, we must also include the change in internal energy. The total work done in the system is the sum of the work done by gravity and the work done by the non-conservative force, which is air resistance in this case. We can write it as:

$$ \begin{aligned} W_{net} &= W_{nc} + W_{grav} \\ - \Delta E_{int} - \Delta U_{grav} &= \Delta K \\ 0 &= \Delta K + \Delta E_{int} + \Delta U_{grav} \end{aligned}$$

From this equation, we see the law of the conservation of energy. The total change in energy of the system is zero.

Applications of Conservative and Potential Energy

We see many applications of energy conservation and potential energy every day in our lives! Some applications include the following:

• Jumping on a trampoline is a good example of turning potential energy into kinetic energy and vice versa. The trampoline gets stretched when you land on the trampoline, and the elastic potential energy is transformed into kinetic energy as you fly into the air. Gravitational potential energy increases as you go higher while kinetic energy decreases.
• When a car on a roller coaster goes down a hill, its gravitational potential energy transforms into kinetic energy. If the car goes up another hill, the kinetic energy becomes potential energy as it goes up the hill.
• When you shoot an arrow, the elastic potential energy from the bowstring converts to kinetic energy as the arrow flies through the air.

Shooting an arrow, Pixabay

Examples of Potential Energy and Energy Conservation