Jumping on a trampoline is an activity that kids love to do in the summer! Along with being a fun activity, a child jumping on a trampoline can be a great example of potential energy and the conservation of energy. In this article, we will define potential energy and energy conservation. We will discuss the principles that govern potential energy and energy conservation and how they relate to each other. We will then go over the applications of each as well as do some examples.
Description of Potential Energy and Energy Conservation
In the article, "Kinetic Energy", we discuss how the kinetic energy of a system is related to the motion of an object and is independent of position. Now, we will discuss a form of energy that does depend on position. Potential energy is energy that is related to the position and internal configuration of two or more objects in a system. Examples of potential energy that we will focus on are gravitational potential energy and elastic potential energy.
Potential energy: the energy that is related to the position and internal configuration of two or more objects in a system.
The gravitational potential energy of a system is related to how high an object is off the ground and the weight of the object. When finding the gravitational potential energy of a system, you only need to consider the vertical change in position of the object since the force of gravity is a vertical force. If the object changes position horizontally, it will not affect the gravitational potential energy. We find the gravitational potential energy of a system using this formula:
$$ U_{grav}=mgy $$
In this equation, \(U_{grav}\) represents the gravitational potential energy, \(m\) is the mass of the object, \(g\) is the acceleration from gravity, and \(y\) is the height of the object off the ground.
It is important to note how our coordinate system is defined when finding the gravitational potential energy of a system. The change in vertical position might not always just be the height of the object off the ground depending on where we choose to define y = 0.
The elastic potential energyof a system is energy that can be stored in a stretchable object, like a rubber band or a spring, and used later. For the example of a spring, we find the elastic potential energy using this formula:
$$U_{el} = \frac{1}{2} k x^2$$
In this equation, \(U_{el}\) represents the elastic potential energy, \(k\) is the spring constant, and \(x\) is the distance the spring is either compressed or stretched.
Kinetic energy and potential energy are two forms of energy that contribute to the total energy of a system. Internal energy also contributes to the total energy. The internal energy of a system refers to the microscopic changes in energy of an object, like how friction causes an increase in temperature on an object being pushed. Energy conservation tells us that if we add together all of the changes in energy in a system, the sum is always zero. Thus the total energy in a system cannot be increased or decreased because energy cannot be created or destroyed. Although it cannot be created or destroyed, a form of energy can transform into a different form, like the gravitational potential energy of a falling object turning into kinetic energy.
Energy conservation: the total energy of a system is constant; energy can change forms, but it cannot be created or destroyed.
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.
Total mechanical energy: the sum of the kinetic energy and the potential 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
Find the change in gravitational potential energy of a \(0.2\,\mathrm{kg}\) rock that is picked up from the ground to a height of \(1.5\,\mathrm{m}\).
The change in gravitational potential energy is given by:
$$ \begin{aligned} \Delta U_{grav} &= U_2 - U_1 \\ &= mgh_2 -mgh_1 \\ &= mg(h_2 - h_1) \end{aligned}$$
If we define the ground to be \(y=0\,\mathrm{m}\), our initial height is zero, \(h_1=0\,\mathrm{m}\). The final height is \(h_2=1.5\,\mathrm{m}\). So, the change in gravitational potential energy is:
$$ \begin{aligned} \Delta U_{grav} &= \left(0.2\,\mathrm{kg}\right)\left(9.8\,\frac{\mathrm{m}}{\mathrm{s}^2}\right)\left(1.5\,\mathrm{m} - 0\,\mathrm{m} \right) \\ &= 3\,\mathrm{J} \end{aligned}$$
A block attached to a spring is stretched a distance, \(x\), from its equilibrium position on a frictionless, horizontal surface. It is released from rest and travels to the left towards the equilibrium position. Use energy conservation to find the relationship between \(x\) and the velocity of the block, \(v\), when it returns to the equilibrium position. Ignore the mass of the spring.
A block on a stretched spring returning to equilibrium position, StudySmarter Originals
Let us find the total energy of the block when the spring is stretched and when the block reaches equilibrium position. The system is made up of elastic potential energy from the spring and kinetic energy when the block is moving. When the spring is stretched, there is only elastic potential energy since the block is not yet moving. Thus the energy when the block is stretched is \(E_1=\frac{1}{2}kx^2\). When the block reaches the equilibrium position, there is only kinetic energy because \(x=0\). So, we have \(E_2=\frac{1}{2}mv^2\). Since we only have conservative forces, the energy will be constant at all positions so that \(E_1=E_2\). This allows us to find the relationship between \(x\) and \(v\):
$$ \begin{aligned} \frac{1}{2}kx^2 &= \frac{1}{2}mv^2 \\ v &= \pm \sqrt{\frac{kx^2}{m}} \end{aligned}$$
Potential Energy and Energy Conservation - Key takeaways
- Potential energy is the energy that is related to the position and internal configuration of two or more objects in a system.
- There must be a conservative force doing work for there to be potential energy in the system. The work done by the conservative force is equal to the negative change in potential energy.
- The gravitational potential energy of a system is related to how high the object is off the ground and the weight of the object. The formula we use to find it is \(U_{grav} = mgy\).
- The elastic potential energy of a system is energy that can be stored in a stretchable object, like a rubber band or a spring, and used later. The formula we use to find it is \(U_{el} = \frac{1}{2}kx^2\).
- The internal energy of a system refers to the microscopic changes in the energy of an object.
- A system is made up of kinetic energy, potential energy, and internal energy.
- Energy conservation tells us that the total change of energy in a system is zero. While energy cannot be created or destroyed, it can change form and be transferred from one object to another.
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