# 041 Potential and kinetic energy

## Gravitational Potential energy

Gravitational potential energy is energy contained in an object due to its position or composition. Objects at rest a height h above a surface have gravitational potential energy due to their position. Gravitational potential energy is equal to the mass multiplied by the acceleration of gravity g multiplied by the height.

`PE = mass × gravity × height = mgh`

Note that the acceleration of gravity g is 980 cm/s²

The "potential" part of the phrase refers to the "potential" for the energy to be converted to other forms of energy. In this activity the gravitational potential energy will be converted to kinetic energy. Gravitational potential energy is not the only type of potential energy, there are other forms of potential energy. Springs and rubber bands can store potential energy, as can bonds between atoms.

Letters in physical science can be very confusing. The g above is being used as a variable. In physical science g as a variable means the acceleration of gravity. The letter "g" is also used as unit. The letter g as a unit means "grams." Grams is a measure of the mass m. Letters in physical science are often used with different meanings as variable and units. Only the context can tell you the meaning of a letter.

## Kinetic energy

Linear kinetic energy (KE) is the energy that an object moving across a surface. Kinetic energy is equal to:

$\mathrm{KE}=\frac{1}{2}m{\stackrel{⇀}{v}}^{2}$

where m is mass and v is the velocity.

## Conservation of Energy

The theory that energy cannot be created nor destroyed is called conservation of energy. The word conservation is used in the physical sciences to mean "stays the same" and "no change."

We can trade gravitational potential energy for kinetic energy. RipStik me here, now.

## Conservation of energy: Banana leaf marble ramp

Does conservation of energy theory predict the marble speed?

In this activity an experimental demonstration will be used to generate data in an attempt to confirm or disconfirm an initial guess at how the system might behave. In science, observations and experimental results often occur before a theory is fully developed. The observations and results of experiments often guide the development of the theory. Then the theoretical results predicted by the theory can be compared to the existing experimental data.

Working as a class, we will follow these steps:

1. Make a prediction about the speed of a marble coming off of a banana leaf ramp.
2. Gather data on the actual speed of a marble on a banana leaf ramp.
3. Sketch a graph.
4. Confirm or disconfirm initial theory.
5. Explore alternate possible theory.
6. Use spreadsheet software to generate a graph comparing the actual data to the new theory (homework).

## Materials

• banana leaf
• marble
• meter sticks
• stopwatch
• ruler
• string
• scissors
• tape

Instructional note: Have the students make predictions about the system prior to rolling the marble. "If the marble is released from twice as high, what will happen to the speed?" Run the demonstration at this point, generating the data for the table.

velocity (cm/s) [y]
h (cm) [x]linear predictionactual measured experimental data energy theory prediction $37.4\sqrt{h}$
000

Table one.

Instructional note: Note that this is not a time versus space graph. The slope of a time versus distance graph is the velocity. This graph will be a height (space) versus velocity (space divided by time) graph. The slope is NOT velocity. Time permitting, sketch the linear data. To do this, roll the ball once from a low height and find the velocity. Use that to predict the velocity for twice the height. Check. Continue with predictions and measurements for four times the initial height, and eight times the initial height. Use the actual data from measurements to determine if a linear mathematical model is appropriate.

What follows is the theory. The key point here is not that students will derive these equations, but to show how the theoretical values will be obtained. The students are not trying to construct the theory.

## Energy

Energy can exist in different forms. For the banana leaf marble ramp there are two forms of energy. At the start the marble possesses gravitational potential energy. The amount of gravitational potential energy is directly proportional to the height h of the marble above the table. After the marble is rolling on the flat surface of the table, the marble has only the energy of motion, called kinetic energy.

## Conservation of Energy

The theory that energy cannot be created nor destroyed is called conservation of energy. The word conservation is used in the physical sciences to mean "stays the same" and "no change." All of the gravitational potential energy of height must be present in the energy of the motion of the marble once the marble reaches the table.

## Kinetic energy

As noted above kinetic energy is the energy of motion. In today's activity there are actually two forms of motion, both of which require energy. Newton noted that an object at rest tends to stay at rest. This law applies to both linear motion and rotational motion.

Linear kinetic energy (KE) is the energy that an object moving across a surface. Linear kinetic energy is equal to:

$\mathrm{KE}=\frac{1}{2}m{\stackrel{⇀}{v}}^{2}$

where m is mass and v is the velocity.

Rotational kinetic energy (RE) is the energy of a rotating object. Rotating means spinning. As the marble rolls the marble spins. Rotational kinetic energy is equal to:

$\mathrm{RE}=\frac{1}{2}I{\stackrel{⇀}{\omega }}^{2}$

In this activity the linear velocity (speed) of the marble, v, will be measured. The rotational energy formula is related to mass and velocity by the following relationships for a marble with a mass m and a radius r.

$I=\frac{2}{5}m{r}^{2}\phantom{\rule[-0ex]{2em}{0ex}}\omega =\frac{v}{r}$

The conservation of energy can be expressed mathematically:
PE = KE + RE.

$\begin{array}{}mgh=\frac{1}{2}m{\stackrel{⇀}{v}}^{2}+\frac{1}{2}I{\stackrel{⇀}{\omega }}^{2}\\ mgh=\frac{1}{2}m{\stackrel{⇀}{v}}^{2}+\frac{1}{2}\left(\frac{2}{5}m{r}^{2}\right){\left(\frac{\stackrel{⇀}{v}}{r}\right)}^{2}\\ gh=\frac{1}{2}{\stackrel{⇀}{v}}^{2}+\left(\frac{2}{10}{r}^{2}\right)\left(\frac{{\stackrel{⇀}{v}}^{2}}{{r}^{2}}\right)\\ gh=\frac{{\stackrel{⇀}{v}}^{2}}{2}+\frac{{\stackrel{⇀}{v}}^{2}}{5}\\ gh=\frac{{5\stackrel{⇀}{v}}^{2}}{10}+\frac{{2\stackrel{⇀}{v}}^{2}}{10}\\ gh=\frac{{7\stackrel{⇀}{v}}^{2}}{10}\\ {\stackrel{⇀}{v}}^{2}=\frac{10gh}{7}\\ \stackrel{⇀}{v}=\sqrt{\frac{10gh}{7}}\\ \stackrel{⇀}{v}=\sqrt{1400h}\end{array}$

In the above expression 1400 is in the units of acceleration: cm/s². After substitution and simplification the relationship between the velocity and the height is given by the following equation.

$\stackrel{⇀}{v}=37.4\sqrt{h}$

Note that the mass and radius (size) of the marble is irrelevant to the speed. Note too that the speed is proportional to the square root of the height. This is not a direct relationship, not a linear relationship.

## Homework

Using a spreadsheet, use table one above to make an xy scattergraph using either OpenOffice.org Calc or Microsoft Excel. The first column is the x-axis variable, the other three columns will be on the y-axis. After making the graph, consider which of the two mathematical models best fits the experimental data.

Instructional note: The following chart depicts the measured experimental data and the conservation of energy theory predicted values for fall 2009

In the centimeter-gram-second system energy is in units called ergs. One erg is a very small amount of energy. An erg is an ant push-up.