To what extent does increasing the height of a ramp affect the final rotational energy of a ball rolling down a ramp?

Physics delves into examining the universe's phenomena such as an object's motion and energy conservation, an observation made based on various physics lessons whereby the concepts of energy and motion are concepts of high significance. I used this observation to satisfy my curiosity about linking these concepts to a sport of personal interest, Motorcross. From a tender age, I always had an interest in motorbikes, which was aroused by my father who also happens to own motorbikes. During a Physics lesson, we had in class on the motion of objects and energy conservation this inquisitive interest in explaining the conservation of energy when transformed from one form to another. I related the topic to the distinctive behavior of motocross motorbikes whose paths involve hitting inclinations and the rider's and bike's mechanical behavior determines both the nature of the ascent or descent. I decided to carry out my investigation on a ball as although motocross and a rolling ball may seem completely different, the underlying concepts of motion and energy conservation are similar.

Realistically following the principle of conservation of energy, if a ball was to be released from a certain height the Gravitational Potential Energy (GPE) should be fully transformed to Kinetic Energy (KE) without any energy loss. However, in this investigation, the concept of rotational energy is introduced but the case should remain the same whereby energy is fully transformed from one form to another without loss when rolling down a ramp. Theoretically, this is practical and is the expected scenario, however, how realistic is it in a real-world setting? This is the intriguing question I am aiming to test and investigate how varying the height of a ramp will influence the rotational energy of a ball. I will further draw my conclusion and link it to motocross and understand the rider's role in controlling the bike's rotation and kinetic energy to facilitate precise and fast landings.

I hypothesized that the height of a ramp is directly proportional to the rotational energy of a ball rolling down the ramp. Realistically, this is because as the height increases, both the GPE and KE of the ball will increase, and by following the principle of energy conservation the rotational energy is also expected to increase.

This is the energy stored by an object due to its vertical position in the gravitational field. (Kinetic Energy and Gravitational Potential Energy, 2020) The GPE of an object is governed by two factors: the mass of an object(m) and the height of the object in the gravitational field (h). In this particular experiment, changes in the height of the ramp govern the GPE of the ball as they are directly proportional.

The formula is as follows:

\[ GPE=mgh \]

Where:
\(m=\) mass of the object
\(g=\) acceleration due to gravity which is \(9.8 m/s^2\)
\(h=\) height of the object

Through the formula, it is derived that as the height of an object is increased its GPE correspondingly increases as well.

Kinetic energy is energy due to motion and is by virtue present in any body in motion. (Kinetic Energy and Gravitational Potential Energy, 2020) It is dependent on two governing factors: the mass of an object(m) and the velocity of an object(v). In this exploration, the motion of the cylinder can be presented in terms of the kinetic energy of the cylinder.

The formula is as follows:

\[ KE=\frac{1}{2}mv^2 \]

Where:
\(m=\) mass of the object
\(v=\) velocity of the object

As the velocity of an object increases, the Kinetic energy also increases which shows the direct proportionality between velocity and KE of an object.

Rotational energy is the kinetic energy of a body moving in a circular/rotational motion. (10.4 Moment of Inertia and Rotational Kinetic Energy - University Physics Volume 1, n.d.)

The principle states that "energy can neither be created nor destroyed but can only be converted from one form to another." (Law of Conservation of Energy - Energy Education, n.d.) It also states that the total mechanical energy(the sum of kinetic energy and potential energy) remains constant so long as there is no external influence such as friction. (Britannica, T. Editors of Encyclopaedia, 2023)

The formula is as follows:

\[ GPE(\text{initial})+KE(\text{initial})=GPE(\text{final})+KE(\text{final}) \]

1. Ramp
2. 2 Wooden stands of different sizes (one small and one big)
3. Meter rule
4. Ball
5. Pencil
6. Thin wooden sticks x 2
7. Tape
8. Carbon and butcher paper
9. Small weight
10. Super glue
11. Thine wire
12. Measuring clamp

1. Glue the two wooden sticks parallel to each other on the ramp to form a path for the ball to roll on in a straight line.
2. Use the two wooden stands of different sizes to prop up the ramp in a position at the edge of the ramp close to the edge of the table.
3. Position the overlay carbon paper on the butcher paper in the expected position where the ball will hit the floor after rolling down the ramp.
4. Tape the thin wire on the edge of the table and tie the small weight to the thin wire and let the weight hang just above the ground.

1. Measure a height of 10 cm from the edge of the table to the corresponding position on the ramp and mark this point with a pencil. Draw a line perpendicular to the tracks formed by the wooden sticks for the point marked.
2. Position the ball on the point marked on the tracks and ensure that the center of mass of the ball is above the line drawn.
3. Release the ball and immediately catch it once it bounces off the carbon paper.
4. Measure the distance(horizontal) between the mark made on the carbon paper from the butcher paper and the suspended weight.
5. Repeat steps 1-4, making increments of 5 cm each time (from 10 cm to 35 cm ).
6. Repeat the steps 1-4 5 times for each increment.

Both the wooden stands and the ramp could contain small nails which could cause major injuries hence, they should be handled carefully and with caution. The super glue could be dangerous if it pours on someone's skin. Gloves should be used when using super glue to avoid any skin damage or injuries. To prevent any hazards caused by the dropping of the rolling ball after dropping, the ball should be caught immediately after it bounces.

Mass of the ball- \(0.098 \pm 0.002\) kg
Radius of the ball- \(0.024 \pm 0.001\) m
Diameter of the ball- \(0.048 \pm 0.002\) m
Dimensions of the wooden sticks- 50 cm × 2 cm × 1 cm
Height of the table from the ground- \(0.97 \pm 0.01\) m

The uncertainty of the values of the horizontal distance covered by the ball was calculated by subtracting the minimum value from the 5 trials from the maximum value obtained and then dividing the result by two. The formula is as follows:

\[ \text{uncertainty}=\frac{\max-\min}{2} \]

The expected velocity was derived from the principle of conservation of energy whereby the potential energy at the top of the ramp is converted to kinetic energy hence the formula:

\[ mgh=\frac{1}{2}mv^2 \]

The formula expected velocity can now be solved from the derived formula below:

\[ v=\sqrt{2gh} \]

Where:
g is the acceleration due to gravity
h is the combined height of the ramp and the radius of the ball

For example, \(v=\sqrt{2 \times 9.8 \times 0.124}=1.56 \pm 0.038\)

The actual velocity was calculated using the formula;

\[ v=\frac{\text{distance}}{\text{time}} \]

The time can be derived from the equation;

\[ d=\frac{1}{2}gt^2 \]

Where:
d is the horizontal distance traveled by the ball which is 9.81 m/s²
g is the acceleration due to gravity

Hence solving for time gives,

\[ t=\sqrt{\frac{2d}{g}} \]

For example,

\[ t=\sqrt{\frac{2 \times 0.51}{9.81}}=0.32 \pm 0.01 \text{ hence } v=\frac{0.51 \pm 0.02}{0.32 \pm 0.01}=1.59 \pm 0.03 \text{ m/s} \]