Investigation of the moment of inertia of hollow cylinders

By observation, the final velocity at which the cylinders roll down the slope decreases as the diameter of the hole increases in the hollow cylinder.

1. Set up the apparatus as shown in Figure 3.
2. Turn on the velocity sensor and the data logger.
3. Let go of the cylinder at the top of the slope from the rest.3
4. Turn off the velocity sensor and the data logger after the cylinder rolls off the slope.
5. Repeat steps 1 to 4 for 3 times to obtain 3 trials for the experiment.
6. Repeat steps 1 to 5 for the remaining 9 cylinders.
7. After the experiment, collect and process the data for further analysis.

From Figure 4, we can see that as the diameter of the hole in the hollow cylinder increases, its average final velocity in rolling down the slope decreases at an increasing rate. A parabolic curve is used to fit the data as a parabolic curve resembles the trend suggested by the data, and a curve with an equation of y = – 0.0026x2 – 0.011x + 1.4699 is yielded.

From Equation 7, the theoretical final velocity at which a hollow cylinder rolls down a slope can be calculated. The hollow cylinder with a hole of a diameter of 9 cm is used as an example.

\(v=\sqrt{\frac{4gh}{3+(\frac{R'}{R})^2}}=\sqrt{\frac{4\times9.81\times1.50sin(10')}{3+(\frac{9/2}{10/2})^2}}=1.64ms^{-1}\ (3sf) \)

(∵ The slope has a length of 150 cm and is put at 10∘ to the table, ∴ ℎ = 1.50 sin(10° ))

Similarly, the theoretical final velocities of all the other cylinders can be calculated. The results are shown in Table 1 in Appendix 3.

To compare the calculated theoretical values with the experimental data, the experimental and theoretical relationships between the final velocities, the diameters of the hole in the hollow cylinders, and the Ratio Factors will be graphed. These are shown in Figures 6 and 7.

The final velocities which the cylinders reach at the end of the slope is calculated by averaging the final 10 velocity data points recorded by the velocity sensor (Note that the velocity sensor records data at a frequency of 20 Hz). The data collected when the diameter of the hole of the hollow cylinder is 0 cm is used as an example.

\(Final \ velocity(v)=\frac{1}{10}\times(1.42+1.55+...+1.29+1.57)=1.47ms^{-1}(2dp)\\∆v=\frac{1}2\times(1.57-1.35)=0.11ms^{-1} (2dp)\)​​​​​​​

Similarly, the final velocities of Trial 2 and Trial 3 can be calculated. The three final velocities calculated is then averaged to one value.

In Trial 1,\(v=(1.47\ ±0.11) \ ms^{-1}\); ​​​​​​​In Trial 2,\(v=(1.48\ ±0.10)\ ms^{-1};\) In Trial 3,\(v=(1.45\ ±0.12)\ ms^{-1}\)

\(Averaged \ final\ velocity\ (v_{avg})=\frac{1}3\times(1.47+1.48+1.45)=1.47ms^{-1}(2dp)\)​​​​​​​

\(∆v_{avg}=\frac{1}2\times(1.58-1.33)=0.13 ms^{-1} (2dp)\)​​​​​​​

Similarly, using the above method, the final velocities and the averaged final velocities of all the cylinders can be calculated. The results are shown in Table 3.

From Equation 4, the expression of the final velocity of a cylinder rolling down a slope is derived. The expression showed that given that the acceleration due to gravity and the vertical height of the slope remains constant, the final velocity is proportional to the inverse of the root of the square of the ratio between the inner radius and the outer radius of the hollow cylinder added by 3 (This will be named as the Ratio Factor in this report). The proportionality is shown in

Equation 8. The proportionality of the final velocity to the Ratio Factor.

\(v=\sqrt{\frac{4gh}{3+\left(\frac{R'}{R}\right)^2}}\Rightarrow v\propto\frac{1}{\sqrt{3+\left(\frac{R'}{R}\right)^2}}\)

To see if the data obtained from the experiment fits with the theoretical model of this experiment, the data obtained will be plot against the Ratio Factor of the cylinders. If the data produces a linear relationship with the Ratio Factor, it means that the data obtained fits the model.

The Ratio Factor of the hollow cylinders has to be calculated. The hollow cylinder with a hole with a diameter of 9 cm is used as an example.

​​​​​​​\(Ratio\ Factor (P)=\frac{1}{\sqrt3+(\frac{9/2}{10/2})^2}=0.512(3sf)\)

\(\frac{∆P}{P}=\frac{1}2\times[2\times(\frac{ ∆R}{R}+\frac{ ∆R'}{R'})]\\ \)

\(∆P=0.005\)

Similarly, the Ratio Factors of all the other cylinders can be calculated. The results are shown in Table 1 in Appendix 2.

The average final velocity is then graphed against the Ratio Factor of each of the cylinders. The relationship is shown in Figure 5.

When I was 8 years old, my primary school science teacher performed an experiment in class. She showed us that a cylinder with a hole in it took longer to roll down a slope than a cylinder without a hole made from the same material. This intrigued me as I thought that the speed at which an object falls is independent of its mass, and thus the experiment lingers in my mind to this day. Therefore, for this IA, I will be investigating and researching the reasons behind this phenomenon and replicating the experiment with a much higher level of detail.

From Figure 5, we can see that as the Ratio Factor increases, the final velocity increases linearly. A very high R² value of 0.9875 is produced. Furthermore, we can see that the slope of the best-fit line falls between the maximum slope line and the minimum slope line (0.5682 < 4.6193 < 10.428). This means that the data obtained from the experiment fit the theoretical model of the experiment.

How does the diameter of a hole in a hollow cylinder affect the cylinder’s final velocity when rolling down a slope?

The moment of inertia of the cylinder increases as R’ increases, which would increase rotational kinetic energy and decrease the translational kinetic energy. Therefore it is hypothesized that as the R’ of a hollow cylinder increases, its final velocity in rolling down a slope will decrease.

Safety precautions such as wearing gloves and protective goggles must be taken during the creation of the cylinders. Furthermore, leftover wood from the construction of other products is used for the creation of the cylinders to prevent further cut-offs of trees upon taking ethical and environmental factors into consideration.