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. 

Conservation of energy and angular momentum

Objective:

• Apply principles of conservation of energy and angular momentum to theoretically calculate the final height of a system (clay+stick) after the stick swings and collide inelastically with the clay.

Physical problem & Approach:

• Basically, we have a meter stick held from a horizontal position and a clay is on the floor. We let go of the stick, it swings and collides inelastically with the clay, then they both continue swings to the final position before swinging back down. The goal of this lab is to find the final height that the system reaches before swinging back. Here is how we approach to solve this problem:
• Step 1: First of all, we apply the conservation of energy before collision. We choose to put gravitational potential energy equal 0 at the center of mass of the stick when the stick is vertical. This means that when the stick is held horizontally, it has a gravitational potential energy. Then when it swings until it is vertical, the gravitational potential energy is transferred into kinetic energy. We apply this concept to figure out the angular velocity of the stick right before it reaches to the bottom of swing.
• Step 2: Then, we apply the principle of conservation of angular momentum during collision to find out the angular velocity of the system which is the combination of the stick and clay. The moment of inertia of the stick itself and that of the whole system are found in order to calculate the angular velocity after collision. 

• Step 3: Lastly, we apply the concept of conservation of energy again to calculate the final position of the system before it swings back down. We choose to put gravitational potential energy equal 0 at the pivot point (near the top of the stick). Given that, we have kinetic energy and gravitational potential energy for both meter stick and clay at start, then kinetic energy is completely transferred into gravitational potential energy when it reaches to the final height. With the value of angular velocity from second step, we can find the angle between its vertical position and its diagonal position. Using the angle to calculate the final height. Here is the diagram.

Calculating theoretical height of the object:

• We first measure dimensions of objects we need for calculation. The meter stick is one meter long, the mass is measured by using a balance. Likewise, the mass of the clay is obtained. The distance from the edge to the pivot point is also measured. Below is the data we obtained.
• Mass of meter stick = 103g
• Mass of clay = 14g
• Distance from the edge to the pivot point = 0.85cm = 0.0085m 
• Then applying the concept of conservation of energy explained in step 1 in our approach to find the angular velocity of the meter stick when it is vertical. Here is the work.

• As we have the angular velocity of the meter stick, we then apply the concept of conservation of angular momentum to find the angular velocity of the system after the collision. Here is the work.

• Lastly, following step 3 in our approach, apply conservation of energy to find the final position of the system. 

• WOW! We have a theoretical value of final position. Now what we need to do is to actually perform an experiment to check how our theoretical value compares to experimental value. The closer these values are, the more accurate we get for both our calculation and experiment.

Set-up:

• Let a metal rod stand on the floor holding by three legs. Secure a horizontal rod on the vertical rod using a clamp. 
• Mount the rotational sensor on the horizontal rod, then nails the meter stick to the rotational sensor to get the pivot point. The other end of the meter stick is wiped with a tape. A clay stands on the floor holding by three paper clips (we use them as legs). A tape is also used to wrap around the clay in order for the inelastic collision to happen between the meter stick and clay.
• Set up a camera far enough to capture the whole experiment. The camera doesn't necessarily see the initial horizontal position of the meter stick; however, it has to capture the final position of the whole system after collision. Connect the camera to Logger Pro.

• Hold the meter stick at its horizontal position, then let it swing down. The meter stick collides with the clay, then both continues to swing to the final position. The video is captured and analyzed in Logger Pro.

Data Collection and Analysis:

• The video is captured, and we start to analyze it by dotting position of the clay after collision until it reaches its final height.

• The origin is placed at the point of collision where the clay stands, the scale is set equal the distance from the the pivot point to the clay. Here is the picture of how we analyze the video.

• The experimental value of final position is equal 0.4842m after analysis.
• Now we calculate the percent error using the following equation:

Discussion:

• From an actual experiment, we get an experimental value of final position is 0.504m, while our theoretical value is 0.554m. The percent error we calculated is 9.03% which is pretty good given so many uncertainties happen in this experiment. Moreover, both values in this experiment are small, thus a small change in both values leads to easy change in percent error. Therefore, it's not surprised to obtain a percent error equal 9.03%.
• There are some sources of uncertainties happening in this experiment:
• Uncertainty in measuring dimensions of the mass of meter stick and clay.
• Uncertainty in dotting points when analyzing the video.
• Friction in rotational sensor.
• A small amount of energy is transferred into sound and heat, not completely transferred from gravitational potential energy to kinetic energy and vice versa.
• Rounding number when calculating theoretical value of final position also affects the percent error.
• The meter stick is assumed perfectly straight, but it is actually not. It bends a little bit at one edge.
• Uncertainty in setting a scale when analyzing the video.

Conclusion:

• In this experiment, we apply the principle of conservation of energy and angular momentum to figure out the final position of the system (stick+clay) reaches before it swings back down. The conservation of energy is applied before and after collision, while the conservation of angular momentum is applied during collision. The concept of moment of inertia and parallel axis theorem are also used during calculation. 
• The theoretical value of final position is found to be 0.554m, while the experimental value is 0.554m. The percent error is calculated to be 9.03%, which is pretty good given errors during experiment. Our good percent error shows that the method we used to calculate the theoretical value is correct. It also refers that our set-up and procedure is "on point" in finding the experimental value.

Finding the moment of inertia of a uniform triangle

Objective:

• Finding the moment of inertia of a uniform right triangle thin plate around its center of mass, for two perpendicular orientation of the triangle.

Physical thinking:

• Approach:
• We need to find the moment of inertia of a right triangle rotates around the center of mass, but we don't approach to solve the problem from the center of mass. Here is why:
• If we rotate around the center of mass, we will miss some piece of the whole right triangle while integrating.
• The limit for integration is not simple. So, we need another approach to solve this problem
• Now, we can approach the problem from the theorem we learn in class which is called parallel axis theorem. Basically, the theorem states that

• Since finding the moment of inertia when rotating around center of mass takes more work and may not work out, we can shift the pivot point to the edge of the triangle. Based on the parallel axis theorem, if we can find the moment of inertia when the pivot point is at the edge and know the distance from the center of mass to the edge, we can figure out the moment of inertia when the pivot point at center of mass.

First, finding the moment of inertia when the triangle rotates around its height (longer leg).

• Choose a small representative disk dm inside the triangle. 
• Write an expression for dm
• Sum all of small disks of the triangle by integrating from 0 to H.
• When we're all done finding the moment of inertia when the pivot point is at the edge, we can apply the parallel axis theorem above to find the moment of inertia when the pivot point is at the center of mass.
• Here is the work.

Figure 1: Deriving an equation for the moment of inertia of the triangle rotating around its height.

Second, finding the moment of inertia when the triangle rotates around its base (shorter leg).

• Doing the same procedure like above, and below is the work.

Figure 2: Deriving an equation for moment of inertia of the triangle rotating around its base.

Now we start an actual experiment to find out the experimental value of moment of inertia. The goal is to compare the value we got from our calculation and from the experiment to see how well we run the experiment.

Set-up:

• The apparatus of the experiment includes the rotation stand, steel and aluminum disk, hanging mass, torque pulley, triangle holder and triangle, and lastly Logger Pro.
• Putting the aluminum disk on top of the steel disk into the rotation stand, make sure that only the top disk rotate this time by letting the air run through the end. 
• The large torque volley is put on top of aluminum disk and connects to the hanging mass. 
• The triangle holder is used instead of a drop pin to hold the triangle.
• Connect the apparatus to the Logger Pro and set up the rotation sensor to read 200 counts per revolution.

Below is the apparatus

Figure 3: Apparatus of the experiment. 

The left one is when the triangle rotates around its long leg (or height)

The right one is when the triangle rotates around its short leg (or base)

• Also using caliper and balance to measure the mass and dimension of triangle, hanging mass, torque pulley and steel and aluminum disk.

Figure 4: Measuring dimensions of components of apparatus.

Procedure:

• First of all, the mass and dimension of components of apparatus are measured. The mass of the triangle is 456g, its height(longer leg) and base(shorter leg) are in turn 14.936cm and 9.844cm. The hanging mass is 25.0g. The diameter of the large pulley is 5.02cm. The mass of aluminum disk is 466g and its diameter is 12.630cm. The mass of the steel disk is 1348g and its diameter is 12.628cm. Lastly, the mass of the triangle holder is 26g.
• We actually do three trials: the first one is without the triangle, the second one is when the triangle rotates around its shorter legs, and the last one is when the triangle rotates around its longer legs. 
• Turn on the air supplier to let the disk rotate, make sure the hose clamp opens so that the bottom and top disk rotate independently. As the disk rotates, the hanging mass starts going down and goes back up again. Logger Pro starts collecting the data since the hanging mass is released. The graph obtained in Logger Pro will be the graph of angular velocity vs. time. By taking the value of the slope, we can find the angular acceleration. However, to minimize the effect of friction torque on the system, we take the average angular acceleration as the mass goes down and back up. Likewise, we in turn find the the average angular acceleration when the triangle is attached. Below is the graph of angular velocity vs. time of three trials.

Figure 5: Graph of angular velocity vs. time when rotating without triangle. 

Figure 6: Graph of angular velocity vs. time when rotating around the shorter leg (base).

Figure 7: Graph of angular velocity vs. time when rotating around the longer leg (height).

Discussion:

• The equation to calculate the experimental moment of inertia is similar to the one we used in angular acceleration lab, so we don't have to derive one more time. Equation (1) is shown below
  

• The average angular acceleration is calculated by summing the angular acceleration when the hanging mass goes down and goes back up, then divide by two. The values are recorded in the table below

Figure 8: Result table of average angular acceleration.

• Using equation 1, we can calculate the experimental moment of inertia of each trial. Below is an example how to calculate it.

• Result table for experimental moment of inertia.

• Then, by subtracting the moment of inertia of the system with the triangle by the system without the triangle, we can find moment of inertia of the triangle for each orientation. Below is an example how to calculate and result table after we're done.

Figure 9: Result table of theoretical moment of inertia of the triangle.

• The experimental value of moment of inertia was found, so we need to calculate the theoretical moment of inertia to compare the result and check how accurate the experiment went. 
• Using the dimensions of the triangle and two equations derived when applying parallel axis theorem above, plugging in the number and find the experimental moment of inertia. Here is the calculation.

Figure 10: Theoretical moment of inertia when triangle rotating around its height.

Figure 11: Theoretical moment of inertia when triangle rotating around its base.

• Last but not least, we calculate the percent error to check whether the result comes out like what we expected (within 10%). If so, we performed our experiment well and two equation we derived are correct. Here is the formula we will use to calculate the percent error (formula (*))

• Apply the formula (*) to calculate the percent error when triangle rotating around its long leg.

• Apply the formula (*) to calculate the percent error when triangle rotating around its short leg.

• As you can see, our percent errors for both orientations are fairly small and within 10%. This means that we run the experimental very well and two equations we derived are correct.

Some source of errors may happen:

1. Uncertainty in measurement of dimensions of components of the apparatus.
2. Uncertainty in measurement of angular displacement due to the rotational sensor.
3. Friction may affect the angular acceleration even though it is already minimized by taking the average angular acceleration.
4. We may let too much air run through the system than we actually need.

Conclusion:

• In this lab, we perform an experiment to find out the moment of inertia of the triangle about its center of mass rotating both orientation: one is around its height, and the other one is around its base. 
• We first successful come up with two equations to calculate the theoretical values of moment of inertia. 
• When the triangle rotates around its base, we find: 

• When the triangle rotates around its height, we find:

• The percent error of this experiment comes out very well. The percent error of the triangle when rotating around its base is 2.32% and that of the triangle when rotating around its height is 0.76%. Both are small and within 10% regardless of errors may happen during the experiment.