# Conservation of energy: Banana leaf marble ramp

Does conservation of energy theory predict the marble speed?

Energy can neither be created nor destroyed. A basic tenet of physics is that energy is conserved. Energy can be changed from one type of energy to another. In this activity the potential energy of a marble in height above the table is converted in linear and rotational kinetic energy.

## Materials

• banana leaf
• marble
• meter sticks
• stopwatch
• ruler

Instructional note: Run the demonstration at this point, generating the data for the table.

h (cm) xv measured (cm/s) y1
0 0
5
10
20
30
40
50

Instructional note: Make a graph of the above data. Note that this is not a time versus space graph. The slope of a time versus distance graph is the velocity. This graph will a square root of height (space) versus velocity (space and time!) graph. The slope is NOT velocity.

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

## Potential energy

Potential energy is energy contained in an object due to its position or composition. Objects at rest a height h above a surface have potential energy. 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 potential energy will be converted to kinetic energy.

## Kinetic energy

Kinetic energy is the energy of motion. In today's activity there are 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. 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. 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 conseration of energy can be expressed mathematically:
PE = KE + RE.

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

After substitution and simplification the relationship between the velocity and the height is given by the following equation.

$\stackrel{⇀}{v}=37.4\sqrt{h}$

## Homework

Using a spreadsheet, take a table of with velocities based on the data gathered in class. Graph the height h against the the measured velocity (speed) y1 and theoretic velocity (speed) y2 on a single graph.

h (cm) x v measured (cm/s) y1 v theoretic (cm/s) y2: $37.4\sqrt{h}$
0 00
5 84
10 118
20 168
30 205
40 237
50 265
• Are the shapes of the curves for the theoretic velocity (speed) and measured velocity (speed) similar?
• Are the predicted values for the velocities close to the measured velocities?
• What factors might be contributing to the differences in predicted and measured velocities?
• Can we disconfirm this theory? Or do we fail to disconfirm the conservation of energy theory based on our data?

## Sample data

[For illustration purposes only]

Data table

h (cm)v measured (cm/s)v theoretic (cm/s)
0 0 0
5 90 83.67
10137118.32
20150167.33
30164204.94
40212236.64
50 264.58