Conclusion

Now that we have seen the code in action, let's answer the guiding questions from the introduction.

How does the length of the pendulum affect the linear and angular velocity of the cylinder?

The length of the pendulum had a significant effect on the linear and angular velocities of the cylinder. From the equation we derived using impulse in the physics analysis, we can see that as the length of the pendulum increases, the momentum function decreases and the initial height of the bob increases. From our data analysis and code, we could see that as the length of the pendulum increased, the linear and angular velocities of the cylinder also increased. This means that the increase in length increases the value for height at a greater rate than it decreases the value for our momentum function.

How is the energy from the pendulum distributed between the translational and rotational energies of the cylinder?

From the energy equation in the physics equation, we can see that the translational kinetic energy of the cylinder is twice the magnitude of its rotational kinetic energy. This is due to the fact that the rotational inertia affects the value for the rotational kinetic energy. The distribution of energy to the kinetic energies of the object will vary depending on the object's shape. One might think that the radius of the object also affects the rotational kinetic energy since it affects the rotational inertia; however, the radius is always squared in rotational inertias which means that it will always cancel out with the radius squared that is in the angular velocity squared value.

How does the length of the pendulum affect the time of the impulse?

From the equation for time that we derived in the physics analysis, we can see that the length of the pendulum affects the initial height of the pendulum, the momentum function, and the acceleration function. Just like in the answer to the first question, we see height and the momentum function both in the numerator of the equation. Using what we know from the first question, we can say that the numerator increases as length increases. This time however, length also affects the denominator. Looking at our equation for the acceleration function, we can see that as length increases, the denominator also increases. To find out how the effect on the numerator compares with the effect on the denominator, we can add something to our code. Because the program calculates what the duration of the impulse will be (tp), we can add a line of code to print that value after it is calculated. After doing this, we can observe how that value changes depending on the length of the pendulum. If done right, we will be able to see that as the length of the pendulum increases, the duration of the impulse decreases, meaning that the length increases the denominator at a greater rate than it increases the numerator.

How does the length of the pendulum affect the force it exerts onto the cylinder?

With Newton's second law, we know that force is equal to mass multiplied by acceleration. By plugging in the acceleration function we found using a line of best fit in the data analysis, we can see that the force will increase as the length of the pendulum increases. This is because the equation for the line of best fit is a positive exponential function.

Improvement

While this experiment has produced interesting results, it is important to note its flaws. Besides the minor uncertainties produced by things like scaling, parallax, resolution, and frame rate, the important flaw in this experiment is the way the string is taped to the bob. Because the string was taped to the bottom of the bob, the string produced a torque on the bob about its center of mass during the impulse. This caused the bob to rotate or bobble about its center of mass which also affected the position of the string of the pendulum as the hook of the bob would also bobble. Since I wasn't able to analyze the bobbling of the bob in the video analysis, the rotational kinetic energy of the bob was not accounted for. This is the reason for the unusual total energy vs time graphs.

Total energy vs time graphs (for length of 60cm) from code (left) and from trial (right):

A change that could be made in future experiments is to tape the string (the string that attaches the cylinder to the bob) onto the middle of the bob (a point on the bob's greatest cross sectional circumference along a plane parallel to the ground) instead of the bottom of the bob. This would eliminate the bobbling of the bob as the string wouldn't produce a torque about the center of mass of the bob (due to the angle of the force being 180 degrees).