7-June-2017: PHYS 4A--Physical Pendulum Lab

Eric Chong

PHYS 4A--Physical Pendulum Lab

Lab Partners: Ben, Nina Song, and Joel Cook

Purpose: Derive expressions for the period of various physical pendulums. Verify your predicted periods by experiment.

Introduction: Objects with mass have a certain moment of inertia that depends on its radius and the place where the pivot is placed. This is essential to finding out the period for physical pendulums. Physical pendulums are rigid bodies mounted on a horizontal axis, so that it can freely oscillate back and forth. The moment of inertia dictates the tendency for the pendulum to move. If there is a higher moment of inertia, the object will have a larger moment of inertia, then the object will have a larger period. If the object has a smaller moment of inertia, then the object will have a smaller period. This is given through the equation alpha = -(mgr/I)theta. This equation is derived from the torque equation, and it will be explained how through the calculations shown in the pictures later, but we solve for the acceleration or angular acceleration because in simple harmonic motions, the acceleration can be used to solve for the period, since the coefficient next to the theta (or position) is the omega. If we take (mgr/I) as omega^2, then we can solve for just omega, and plug that omega into the equation for period, which is T=2pi/omega. This will help us solve for the period. And as expected, the inertia will affect the numerator of the period equation when we plug in omega, so that means that the higher the inertia the larger the period, or the lower the inertia the lower the period. Through this reasoning, calculation, and analysis, we can solve for the periods of the physical pendulums.

Procedure: First, professor had us do a prelab worksheet with questions about a half circle and a triangle, and the work can be seen here:

Generally, we first found the moment of inertia of an object pivoted at one end. And then we found the position of the center of mass and used the parallel axis theorem to find the moment of inertia of the object at the center of mass. After doing so, we can then use the parallel axis theorem again in order to find the moment of inertia of the object at another pivot. This is what we did for both the triangle and the half circle.

Upon doing the lab, we first measured the radius of the half circle, and the base and height of the triangle. With this, we can use those values to solve for the value of the moment of inertia, and then use that to calculate the value of omega, which can be used to solve for the period by dividing 2pi by omega. Here are our calculations with the measured values:

Half Circle:

(Note: the sin(theta) is approximated to a small angle, so we can treat it as just theta)

Triangle:

Our predicted values are 0.708 seconds for the half circle and 0.674 seconds for the triangle.

With these values, we wanted to test the prediction and oscillated the half circle and triangle. We set up LoggerPro and a motion sensor that can detect the period of the oscillation of objects. Here are the setup, graph, and data:

Half Circle:

Triangle:

As expected, both periods were around 0.708 and 0.674 seconds for the half circle and triangle, respectively. This means that the calculations were correct, and that the small angle approximation is valid.

Conclusion: The experimental results are very close to what we derived. Based on the calculations, the period of the physical pendulum does not depend on the mass of the pendulum, so even if we added the mass of the paper clip it would not affect the period all that much. The masking tape, however, would be expected to have a bigger effect on the pendulum because the radius would be increased, and therefore increasing the moment of inertia. However, we combated this by making the masking tape relatively thin and small.

Overall, the lab is successful in showing how the moment of inertia of an object can be calculated, how the period can be calculated from solving for the acceleration and taking the omega and dividing 2pi by omega. We were able to set up the experiment and obtain data for the periods of both the half circle and triangle to a relatively small percent error. Some sources of uncertainty could come from the radius of the masking tape, the frictional torque from the pivot, and the dimensions of the half circle and the triangle, since they were not completely flat.

31-May-2017: Lab 19: PHYS 4A Lab--Conservation of Energy/Conservation of angular momentum

Eric Chong

Lab 19: PHYS 4A Lab--Conservation of Energy/Conservation of angular momentum

Lab Partners: Nina Song and Joel Cook

Purpose: To predict and measure how high the clay-stick combination should rise after a collision.

Theory/Introduction: The conservation of angular momentum is very similar to the conservation of momentum. The equation is given in the form of Itotal initial * ωinitial = Itotal final * ωfinal, where I is the moment of inertia, and omega is the angular velocity. Using this idea, we can find the total energy of the system and see what form energy it is converted to, such as from gravitational potential energy to rotational kinetic energy, while also combining it with the concept of conservation of angular momentum if a collision occurs. For this experiment, we will be applying the concepts of conservation of angular momentum for the collision of the meter stick and the clay, and the conservation of energy for the transfer of energy between rotational kinetic energy and gravitational potential energy, in order to find how high the clay-stick combination will rise after the collision.

Procedure: To start, we measured the mass of the meter stick and the clay. Both of their masses turned out to be 146.63 g and 34.54 g respectively. Next, we set up an apparatus that looks like this:

(meter stick has nails sticking out in order to help the clay stick)

Before we commence the experiment, we predicted the height at which the clay-stick combination would rise through calculations. We first used conservation of energy of the stick that is angled at 90 degrees horizontally and made it rotate downwards, gaining rotational kinetic energy, until right before it hits the clay. In doing so, we would also have to use the parallel axis theorem to find the moment of inertia, because the pivot has been shifted to the 10 cm mark of the meter stick. We then used the conservation of angular momentum to calculate the final angular velocity. Finally, we used conservation of energy again, but this time with a new moment of inertia, because the clay sticks onto the stick and contributes some moment of inertia, and the new angular velocity, calculated through the conservation of angular momentum. The variable we need to find is theta, the angle. By finding the angle, we can use the value in order to find how high the clay-stick combination rises. Here is what the calculations look like:

The calculations indicate that the clay-stick combination will rise approximately 30 cm.

We next conducted the experiment. We video captured the motion of the stick rotating and colliding with the clay and rising up to a certain height, and plotted the initial position of the clay and the clay's final height. Here is what we found:

The result is strikingly close to what we predicted. The reason that the value is a bit off is most likely that our plotting skills with LoggerPro are not pixel accurate. Since a centimeter is really small, it is possible that even a pixel on LoggerPro could be a centimeter or a half. Nonetheless, we are more concerned with the overall value and how close it is to what we predicted, and looking at this value of 32.45 cm, we are safe to assume that what we predicted was indeed correct.

Conclusion: This experiment showcases the concepts of conservation of energy with rotational kinetic energy and gravitation potential energy, and the conservation of angular momentum. This lab is meant to help us utilize our skills in order to find certain values at certain times while the stick is in motion. And because of this, this lab has high value for our understanding of physics and engineering skills. Some places of uncertainty may lie in the plotting of the graph in LoggerPro, some loss of energy while the meter stick is in motion (most likely due to the fact that there is drag force or friction in between the pivot point and the meter stick), or maybe even the possibility that the meter stick is not exactly 90 degrees when it started (it could have easily been 91 or 89 degrees). Overall, we are safe to say that, due to our predictions being really close to the actual experimental value, the experiment is a success!