Physical Pendulum

Objective:

• Derive expressions for the period of various physical pendulums. Verify the predicted periods with experiment.

Physical problems:
This experiment can be divided into five small parts:

• Part 1: Find period of a solid ring, mass M, with outer radius R and inner radius r
• Part 2: Find period of isosceles triangle, base B, height H, oscillating about its apex.
• Part 3: Find period of isosceles triangle, base B, height H, oscillating about the midpoint of its base.
• Part 4: Find period of a semicircular plate of radius R, oscillating about the midpoint of its base.
• Part 5: Find period of semicircular plate of radius R, oscillating about a point on its edge, directly above the midpoint of the base.
• Compare all of theoretical values of periods with experiment.

Physical principles:

• In order to find the period in oscillation problem, we can find omega, then find period by getting 2pi divided by omega. In oscillation, omega could be found by figuring out the relationship between a (linear acceleration) and x (linear distance) or between alpha (angular acceleration) and theta (angular distance). These relationship can be found by applying the second Newton Law or torque equal inertia multiplied by alpha.
• Therefore, the first step to approach oscillation problem is to figure out how many forces applied to the system. When we know forces, we can apply either second Newton Law or torque to find relationship between a and x or alpha and theta.
• By then, find omega and period.

Finding theoretical values of period for five parts:
Part 1: A solid ring

• Analyze how many forces applied to a solid ring that makes it oscillate. Gravity is the only force applied to the system.
• Apply torque equal inertia multiplied by alpha to find the relationship between alpha and theta. In order to do so, we consider problem where theta is "small", which makes sin(theta) also equal theta. Moreover, we need to find moment of inertia of a solid ring. Don't forget to find the center of mass, either.
• With the relationship between alpha and theta, we manage to find omega, then use it to find period.

Figure 1: Finding the period of a solid ring oscillating.

Part 2: Isosceles triangle oscillating about is apex.

• Gravity is also the only force makes the triangle oscillate. Gravity applied to triangle right at the center of mass, therefore we need to find center of mass of triangle first.
• Apply definition of torque again to find relationship between alpha and theta. Moment of inertia needs to be found to use equation above. Then, using the relationship between both of them to find omega, then find period.

Figure 2: Finding center of mass of triangle 

Figure 3: Finding inertia of triangle pivoted at its apex

Figure 4: Finding period of an isosceles triangle oscillating around its apex.

Part 3: Isosceles triangle oscillating about the midpoint of its base.

• Same idea! First, find center of mass and inertia when the pivot point is at the midpoint of is base. Then apply definition of torque to find relationship between alpha and theta. From that, find omega and period.

Figure 5: Finding inertia of triangle pivoted at the midpoint of its base

Figure 6: Finding period of an isosceles triangle oscillating around the midpoint of its base.

Part 4: A semicircular plate oscillating about the midpoint of its base.

• Same idea! First, find center of mass of semicircular plate. Then find the moment of inertia when the pivot point is at the midpoint of its base. Then apply definition of torque to find relationship between alpha and theta. From that, find omega and period.

Figure 7: Finding center of mass of semicircle

Figure 8: Finding inertia of semicircle pivoted at the midpoint of its base.

Figure 9: Finding period of a semicircular plate oscillating around the midpoint of its base.

Part 5: A semicircular plate oscillating about a point on its edge, directly above the midpoint of the base.

• Same idea! First, find center of mass of semicircular plate. Then find the moment of inertia when the pivot point is at the a point on its edge, directly above the midpoint of the base. Then apply definition of torque to find relationship between alpha and theta. From that, find omega and period.

Figure 10: Finding inertia of semicircle pivoted at the top of its rim.

Figure 11: Finding period of an isosceles triangle oscillating around the top of its rim.

Here is the result table for theoretical values of period of various physical pendulum.

Now, we need to set up the experiment to check our theoretical values of period.
Procedure:

• For part 1 only:
• Setting up two physical stands on the table. Secure a photo-gate at one end of the clamp and connect the other end to the physical stand.
• Use the other physical stands to hold the solid ring. Attach a tape to one end of the solid ring such that it oscillate around the photo-gate on the other physical stand.
• Connect the photo-gate to the Logger Pro, set Logger Pro into pendulum timing to measure the period. Start collecting appropriate data.

Figure 12: Apparatus of part 1

• For part 2,3,4, and 5.
• Cut out isosceles triangle and semicircle. Measure the appropriate dimensions of each one.
• Let a big vertical rod sit on the floor held by three legs. Use two C-clamp to secure two horizontal rods into vertical rod.
• Attach the photo-gate into the bottom horizontal rod by using another C-clamp. Attaching a pivot to the appropriate location in each object. Then use a tape to attach the shape with the top horizontal rod.
• Pull the shape slightly to one side, and then let go. Record the period in Logger Pro.

These pictures are apparatus of finding the period of semicircle pivoted at different point. Same apparatus used for finding the period of triangle when the semicircle is replaced by semicircle.

Data Collection and Analysis:

• The period is part 1 is recored separately from these last four parts. Here is the period of a solid ring.

Figure 13: Experimental period of a solid ring

• Here is one example of recording period for the rest four parts including 2,3,4 and 5.

Figure 14: Finding period of the semicircle pivoted at the top of its rim

• Here is the result table of five parts:

• We compare the actual and theoretical values by finding the percent error. Below is one example.

• Here is the result table of percent error:

Discussion:

• Looking at the result table, all percent errors are within 1% which is pretty good. Within 1% error is reasonable for physical pendulum. The error occurs in this experiment is due to friction at the pivot point and sometimes the object moves back and forth, not side to side. Moreover, the uncertainty in measurement also contributes to the percent error of this lab. Last but not least, rounding number when calculating theoretical period also affects the result of percent error.

Conclusion:

• In this lab, we manage to find period of various objects oscillate at small angle. These objects include a solid ring, an isosceles triangle, and a semicircular plate. The theoretical values of period is similar to the experimental values which is proved thanks to percent error. All percent errors are within 1%, like what we expected. Given some small errors and uncertainties in this experiment, we still performed a wonderful lab and obtain the result we wanted. WOW! Here we go. The last lab and the end of semester. THANK YOU PROFESSOR WOLF FOR A GREAT SEMESTER.