## Energy

Energy is a term we often hear but may not be familiar with its technical definition. Therefore, before delving into mechanical energy, let us define energy.

**Energy** is a system’s ability to do work.

Now from this definition, we are led straight to "*work", *no pun intended.

**Work** is the amount of energy transferred due to an object moving some distance because of an external force.

Energy and work, both scalar quantities, have the same corresponding SI unit, joules denoted by J.

### Types of Energy

Energy is a broad term that encompasses many different forms of energy. However, within the framework of Newtonian mechanics, energy can be classified as either kinetic or potential.

**Kinetic energy** is the energy associated with motion.

An easy way to remember this definition is to remember that the word *kinetic* means motion. Now the corresponding formula to this definition is

$$K=\frac{1}{2}mv^2,$$

where \( m \) is mass measured in \( \mathrm{kg} \) and \( v \) is velocity measured in \( \mathrm{\frac{m}{s}}. \) However, it is important to understand that this formula corresponds to** *** translational kinetic energy,* energy due to linear motion. Kinetic energy can also be expressed in terms of rotational motion. The corresponding formula for

**rotational kinetic energy**is

$$K_{\text{rot}}=\frac{1}{2}I\omega^2,$$

where \( I \) is the moment of inertia measured in \( \mathrm{kg\,m^2} \) and \( \omega \) is angular velocity measured in \( \mathrm{\frac{rad}{s}}. \)

By contrast, potential energy focuses on position rather than motion.

**Potential Energy** is energy due to an object's position.

The mathematical formula for potential energy varies depending on circumstances within a system. Therefore, let's go through some different forms and discuss their formulas. One of the most common forms is gravitational potential energy.

**Gravitational potential energy** is the energy of an object due to its vertical height.

Gravitational potential energy corresponds to the formula $$U=mgh,$$

where \( m \) is mass measured in \( \mathrm{kg} \), \( g \) is the acceleration due to gravity, and \( h \) is height measured in \( \mathrm{m} \). Note that mass and height are directly related to gravitational potential energy. The larger the mass and height values, the larger the potential energy value will be.

However, gravitational potential energy can also be defined in terms of calculus. The **calculus definition** describes the relationship between conservative forces exerted on a system and gravitational potential energy, \( \Delta U =-\int \vec{F}(x)\cdot \mathrm{d}\vec{x}. \) This integral is equal to the work required to move between two points and describes the change in gravitational potential energy. If we use this in conjunction with our knowledge that gravitational potential energy is equal to \( U=mgh \), we can show how the calculus definition is used to derive the simplest equation for gravitational potential energy:

$$\Delta U =-\int_{h_0}^h (-mg)\mathrm{d}y= (mgh-mgh_0).$$

If \( h_0 \) is set to zero to represent the ground, the equation becomes

$$\Delta U= mgh,$$

the simplest formula for determining gravitational potential energy.

It is important to note that the negative sign of the integral indicates that the force acting on the system is minus the derivative, \( F= -\frac{\mathrm{d}U(x)}{\mathrm{d}x} \), of the gravitational potential energy function, \( \Delta U \). This essentially means that it is minus the slope of a potential energy curve.

Another fairly common form of potential energy is elastic potential energy.

**Elastic potential energy **is the energy stored within an object due to its ability to be stretched or compressed.

Its corresponding mathematical formula is $$U=\frac{1}{2}k\Delta{x}^2,$$

where \( k \) is the spring constant and \( x \) is the compression or elongation of the spring. Elastic potential energy is directly related to the amount of stretch in a spring. The more stretch there is, the greater the elastic potential energy is.

#### Potential Energy and Conservative Forces

As mentioned above, potential energy is associated with conservative forces; thus, we need to discuss them in more detail. A **conservative force,** such as a gravitational or elastic force, is a force in which work only depends on the initial and final configurations of the system. Work does not depend on the path that the object receiving the force takes; it only depends on the initial and final positions of the object. If a conservative force is applied to the system, the work can be expressed in terms of, $$W_\text{conservative}={-\Delta U} = {\Delta K},$$ where\( -\Delta{U} \) is minus the change in potential energy and \( \Delta K \) is the change in kinetic energy.

We also can define conservative forces in terms of calculus as minus the spatial derivative of the potential. Now, this may sound complicated but it essentially means that we can determine what conservative force is acting on the system from the spatial derivative, \( -\frac{\mathrm{d}U}{\mathrm{d}x}= F(x). \) This derivative also can be written in integral form as, \( U(x)=-\int_{a}^{b}F(x)dx. \) which we take to be the definition of potential energy. Let's do a quick example to help our understanding.

If a ball is dropped from a vertical height, we know that it has gravitational potential energy, \( U=mgh. \) Now if asked to determine the conservative force acting on the ball, we can take the spatial derivative.

**Solution**

$$-\frac{\mathrm{d}U}{\mathrm{d}x}= {\frac{\mathrm{d}}{\mathrm{d}h}}(mgh)=-mg=F$$

where \( F=-mg, \) represents a gravitational force that we know to be conservative.

### Conservation of Energy

As we have defined various types of energy, we also must discuss a key concept corresponding to energy. This concept is the **conservation of energy **which states that energy cannot be created nor destroyed.

**Conservation of energy: **The total mechanical energy, which is the sum of all potential and kinetic energy, of a system remains constant when excluding dissipative forces.

Dissipative forces are nonconservative forces, such as friction or drag forces, in which work is dependent on the path an object travels.

When calculating the total mechanical energy of a system, the following formula is used:

$$K_\mathrm{i} + U_\mathrm{i}= K_\mathrm{f} + U_\mathrm{f}$$

where \( K \) is kinetic energy and \( U \) is potential energy. This equation does not apply to a system consisting of a single object because, in that particular type of system, objects only have kinetic energy. This formula is only used for systems in which interactions between objects are caused by *conservative forces*, forces in which work is independent of the path an object travels because the system may then have both kinetic and potential energy.

Now if a system is isolated, the total energy of the system remains constant because nonconservative forces are excluded and the net work done on the system is equal to zero. However, if a system is open, energy is transformed. Although the amount of energy in a system remains constant, energy will be converted into different forms when work is done. Work done on a system causes changes in the total mechanical energy due to internal energy.

**Total internal energy** is the sum of all energies comprising an object.

Total internal energy changes due to dissipative forces. These forces cause the internal energy of a system to increase while causing the total mechanical energy of the system to decrease. For example, a box, undergoing a frictional force, slides along a table but eventually comes to stop because its kinetic energy transforms into internal energy. Therefore, to calculate the total mechanical energy of a system in which work is done, the formula

\( K_\mathrm{i} + U_\mathrm{i}= K_\mathrm{f} + U_\mathrm{f} + {\Delta{E}} \), must be used to account for this transfer of energy. Note that \( {\Delta{E}} \) represents the work done on the system which causes a change in internal energy.

## Total Mechanical Energy Definition

Now that we have thoroughly discussed energy, identified different types of energy, and discussed the conservation of energy, let us dive into the concept of total mechanical energy.

**Total mechanical energy** is the sum of all potential and kinetic energy within a system.

## Total Mechanical Energy Formula

The mathematical formula corresponding to the definition of total mechanical energy is

\begin{align}E_{\text{total}}&= K + U,\\E_{\text{total}}=\text{consatnt}\implies K_{\text{initial}} + U_{\text{initial}} &= K_{\text{final}} + U_{\text{final}},\\\end{align}

where \( K \) represents kinetic energy and \( U \) represents potential energy. Total mechanical energy can be positive or negative. However, note that total mechanical energy can only be negative if the total potential energy is negative, and its magnitude is greater than the total kinetic energy.

## Total Mechanical Energy Units

The SI unit corresponding to total mechanical energy is joules, denoted by \( \mathrm{J}\).

## Total Mechanical Energy Graph

To construct a graph depicting a system's total mechanical energy, let us use an example of a tiny skier trapped inside a snow globe, like the genie in Disney's Aladdin, gliding down an incline where friction is neglected.

At the top of the incline, the skier will have high potential energy because height is at its maximum value. However, as the skier glides down toward the bottom of the incline, their potential energy decreases as height decreases. In comparison, the skier starts with low kinetic energy because they are initially at rest but as they glide down kinetic energy increases. Kinetic energy increases as a result of potential energy decreasing since energy cannot be created or destroyed as stated in the conservation of energy principle. Therefore, the lost potential energy converts to kinetic energy. As a result, the skier's total mechanical energy is constant because kinetic plus potential energy does not change.

## Examples of Total Mechanical Energy Calculations

To solve total mechanical energy problems, the equation for total mechanical energy can be used and applied to different problems. As we have defined total mechanical energy, let us work through some examples to gain a better understanding of total mechanical energy. Note that before solving a problem, we must always remember these simple steps:

- Read the problem and identify all variables given within the problem.
- Determine what the problem is asking and what formulas apply.
- Apply the necessary formulas to solve the problem.
- Draw a picture if necessary to provide a visual aid

### Examples

Let us apply our new knowledge to some examples.

A \( 6.0\,\mathrm{kg} \) ball, initially at rest, slides down a \( 15\,\mathrm{m} \) hill without friction. Calculate the final speed of the ball.

Based on the problem, we are given the following:

- mass,
- height difference.

As a result, we can identify the equation, \( K_{\text{initial}} + U_{\text{initial}} = K_{\text{final}} + U_{\text{final}}, \) and use it to calculate the final velocity of the ball. Note that initial kinetic energy is zero since the ball has an initial velocity of zero and final potential energy is zero because the ball reaches the ground, indicating a height of zero. Thus, we can calculate the following to find the final speed \(v\):

\begin{align}K_{\text{initial}} + U_{\text{initial}} &= K_{\text{final}} + U_{\text{final}},\\ 0\,\mathrm{J} + (6.0\,\mathrm{kg})\left(9.8\,\mathrm{\frac{m}{s^2}}\right)(15\,\mathrm{m})&=\frac{1}{2}(6.0\,\mathrm{kg})v^2 +0\,\mathrm{J},\\ 8.8\times 10^2\,\mathrm{J}&=3.0v^2,\\v^2&=\left(\frac{8.8\times 10^2}{3.0}\right)\,\mathrm{\frac{m^2}{s^2}},\\v&=17\,\mathrm{\frac{m}{s}}.\\\end{align}

Let's try a slightly more complicated example.

A pendulum, shown in Fig. 4, initially at rest, is released from Position 1 and begins to swing back and forth without friction. Using the figure below, calculate the total mechanical energy of the pendulum. The mass of the bob is \(m\), the gravitational acceleration is \(g\), and we can take the potential energy of the pendulum to be \(0\,\mathrm{J}\) at Position 2.

Fig. 4: Calculating the total mechanical energy of a pendulum.

The movement of the pendulum is separated into three positions.

__Position one__

\begin{align}K_1&= 0\,\mathrm{J}, \\ U_1&= mgh=mg(L-L')\\&=mg(L-L \cos \theta)= mgL-mgL \cos\theta\\.\end{align}

The pendulum has zero kinetic energy because it is initially at rest indicating it initial velocity is zero. To calculate potential energy, we must choose the x-axis to be where \( h=0. \) When we do this, we can find the value of \( h \) by using the right triangle seen in the image. The total distance of the pendulum is represented by \( L, \) therefore, we can calculate \( h \) by using the trigonometric cosine function for a right triangle. This function states that the cosine of the angle is equal to \( h \) over \( L,\) allowing us to solve for \( h. \)

\begin{align}\cos\theta &= \frac{h}{L},\\ h&=L \cos\theta\\\end{align}

Therefore, the difference in height between positions one and two,\( L' \) is calculated as follows.

\begin{align}L'&=L-h,\\L'&=L-L \cos\theta,\\\end{align}

which can be inserted into the equation for gravitational potential energy.

__Position Two__

\begin{align}K_2&= mgL-mgL \cos\theta,\\U_2&= 0\,\mathrm{J}\\\end{align}

As the potential energy at this position is zero, the kinetic energy must be equal to the total mechanical energy, which we already calculated in the previous position.

__Position Three__

\begin{align}K_3&= 0\,\mathrm{J}, \\U_3&= mgh= mgL-mgL \cos\theta\\\end{align}

This position is equivalent to position one. The pendulum has zero kinetic energy because it becomes momentarily stationary: its velocity is zero. As a result, the total mechanical energy of the pendulum can be calculated by looking at position 1, \( E_{\text{total}}= K_{1} + U_{1} \), or position 3, \( E_{\text{total}}= K_{3} + U_{3}\).

## Total Mechanical Energy - Key takeaways

- Total mechanical energy is the sum of all potential and kinetic energy within a system.
- The mathematical formula for total mechanical energy is, \( E_{\text{total}}= K + U \).
- Total mechanical energy has SI units of joules, denoted by \( \mathrm{J} \).
- Kinetic energy is the energy associated with motion.
- Potential energy is energy due to an object's position.
- When there are no dissipative forces acting within a system and no external forces acting on the system, total mechanical energy is conserved.
- Graphs for total mechanical energy depict constant total mechanical energy, so wherever kinetic energy increases, potential energy decreases, and vice versa.

## References

- Fig. 1 - Windmill ( https://www.pexels.com/photo/alternative-energy-blade-blue-clouds-414928/) by Pixabay ( https://www.pexels.com/@pixabay/) licensed by Public Domain.
- Fig. 2 - Mechanical energy graph, StudySmarter Originals.
- Fig. 3 - Rolling ball, StudySmarter Originals.
- Fig. 4 - Pendulum, StudySmarter Originals.

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##### Frequently Asked Questions about Total Mechanical Energy

How to find total mechanical energy?

Total mechanic energy can be found by calculating the sum of all potential and kinetic energy within a system.

What is the formula for finding total mechanical energy?

The formula for total mechanical energy is total mechanical energy is equal to all kinetic energy plus potential energy.

How to find total mechanical energy of a pendulum?

The total mechanical energy of a pendulum is found by diving the pendulums path of motion into three positions. Using these three positions, the kinetic and potential energy can be determined for each one. Once this is complete, the total mechanical energy can be determined by added up the kinetic and potential energy of each position.

What is total mechanical energy?

Total mechanical energy is the sum of all potential and kinetic energy.

Can total mechanical energy be negative?

Total mechanical energy can be negative only if the total potential energy is negative, and its magnitude is greater than the total kinetic energy.

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