December 1 2014: Period of Oscillation

Purpose: To find the period of oscillation for two objects: a semi-circle and a triangle pivoted at different locations.

These two pictures show the apparatus that we used. First is a semicircle pivoted in the middle of the long side, we also did from the top of the curved side. Second is a triangle pivoted from one edge, we also did from the middle of one of the sides.

Explanation: First we are going to find the moment of inertia of each object pivoted at different locations by hand. From that we are going to calculate the period of the system from small oscillation angles. We are going to use two objects: a semi-circle and a triangle, and we are going to have two pivot points for each object.

 This picture shows the calculations to find the center of mass of the semi-circle, we found it to be 4R/3pi.

 This picture shows the calculations that we used to find the moment of inertia of the object pivoted at the two points. We got the long side to be 1/2MR^2, and the curved side to be 1/2MR^2+(1-24/9pi)MR^2.

This picture shows the calculations that we did in order to find the period of oscillation for each pivot point. From the long side we got 0.603s and from the curved side 0.59s

 This picture shows the period of oscillation for the long side by using LoggerPro. We found it to be 0.5994s.

This picture shows the period of oscillation for the curved side by using LoggerPro. We found it to be 0.5991s.

Here, we compared the results that we got by hand and by actually performing the experiment. The % error in both were less than 1%, which was the expected.

 This picture shows the calculations that we did to find the moment of inertia for the triangle at different points. We found it to be 1/24MB^2+1/2MH^2. The bottom part of the picture shows the period that we calculated when the pivot is at one end. We found it to be 0.77s.

This picture shows the moment of inertia when the triangle was pivoted at one of the sides. We found it to be MB^2/24+1/6MH^2. We also calculated the period of oscillation for this case to be 0.74s.

24 November 2014: Mass-Spring Oscillations

Purpose: To determine the relationship between the period of the system with the mass of the object and the constant of the spring.

This is the spring apparatus that we used for this lab experiment.

Explanation: First, each group will find important information such as period of spring with 109g (mass of spring + hanging mass) and constant of the spring by using a motion sensor and LoggerPro, then share the data containing the mass of the spring, mass add to the system, period of spring and spring constant. After that, we are going to calculate the period of our spring with 4 different masses and hopefully find a relationship between mass and period. Then we will use the constants of spring of each group and hopefully find a relationship between the constant of the spring and the period as well.

This picture shows the data collection of each group. The data that we used was from the left side.

 This is the data table that we got from LoggerPro when allowing the system to oscillate.

 This is the graph that we got from the oscillation of the system, we used this to find the period of our spring.

This picture shows the calculations that we did to find the period and the constant of our spring. The calculations are seen at the top of the page.

 This picture shows the data table that we used to find the graph of the constants of each group and the respective period.

 This is what the graph looked like from plugging the values for the constant (K) and period (T) of each group. We used this to compare our calculations made by hand with the real value.

 This is the data table that we used for graphing mass vs. period for our spring. We added different masses and calculated the period in order to create this graph.

This is what the graph mass vs. period looked like. We used this to compare our calculations with the real value.

This picture shows our calculations to find the relationship between the period and mass of hanging object, and relationship between period and constant of spring.

Summary: To conclude this lab, we can say that our results were pretty accurate. From the picture above we can see that for our estimated period with respect to mass was T=1.25m^0.5, and from the experiment we got 1.14m^0.4475, which is close to the calculation that we had. For the period with respect to constant, we calculated to be T=2.07k^-0.5, and from the experiment we got 2.66k^-0.5, which was a little off, but still considered to be close. We can say that our calculations were accurate enough if compared to the results coming from the actual experiment.

19 November 2014: Conservation of Linear and Angular Momentum

Purpose: To determine whether the speed of the system was conserved.

                                                    

The apparatus' for this system are a marble ball, a ramp where the ball is released from, a rotating disk system, a ball catcher and a hanging mass attached to the pulley on top of the rotating disk system.

Explanation: First we are going to release the marble ball from the ramp and let it hit the ground. From that, we are going to calculate the initial speed of the ball. Then we are going to release the ball again from the ramp, but this time the ball catcher in the rotating disk system will catch it. This will make the system accelerate, therefore the hanging mass will start to decrease. The idea is that by having the right information we will be able to determine to moment of inertia of the system. With that we will be able to calculate the final speed of the system.

This is the first calculation, where we released the marble ball and it hit hit ground. We found the initial speed in which the ball exits the ramp.

This calculation shows how we found the moment of inertia of the system. 

 This picture is the angular velocity of the system once the ball hits the ball catcher, we used the average velocity to finish this lab.

This is the calculation that we did to find the new moment of inertia of the system once the marble ball hit the ball catcher, and the final angular velocity of the system.

Summary: We can say that this lab was successful. Because of the fact that we did the lab with the whole class together, we were able to compare all the steps with the other groups. This was really useful, since every calculation we did, we would check to see if the other groups were having similar answers. In the end, when we found the final angular velocity of the system, three other groups had the same answer. This shows that we were in the same pace with them, once we had the same values to conclude this lab experiment. 

November 17 2014: Meter stick with clay

Purpose: To find the height that a meter stick reaches released from a point and hitting a clay in the way. The meter stick is pinned in one of the ends.

This is a picture of our system, as possible to see, the meter stick is pinned by one of its ends, and there is a chunk of clay at the bottom.

Explanation: We are going to use video capture and Logger Pro to compute the information in a computer. After setting up the equipment, we are going to release the meter stick horizontally to its pin and let it hit the clay at the bottom. The idea is to see how high the meter stick will go. We are going to solve the problem by hand because we are able to do that after we measure the necessary information, and then compare to the results that the computer will give us.

 This is the graph that we got from recording the experiment. The y-axis determine the distance that the meter stick raised, but we have to subtract 20 cm from it because the mass of clay was 20 cm above the ground when we did the experiment.

This are the calculations that we made in order to find how much the meter stick would raise after hitting the mass of clay. We approached it by first doing conservation of energy, then conservation of momentum and finally conservation of energy again.

Summary: We can say that the lab was successful because we solved by hand and got 0.238 meters for the height that the meter stick would raise, and when we computed the information, the computer gave us a final height of about 0.280 meters. The error can be given to the friction in the pin and also by air resistance during the experiment. The height that we released can also be a factor that affected the results because we probably didn't release the meter bar exactly in the horizontal position.

November 12 2014: Moment of Inertia of a triangle

Purpose: To use a rotating disk system attached to a triangle and find the moment of inertia of the triangle.

This is a picture of how the rotating apparatus will work. The triangle will be attached to the pulley and the center of the disks. It will also rotate as the hanging mass goes up and down.

Explanation: First we are going to solve it symbolically for the moment of inertia of the triangle. Then, we are going to attach a triangle to the rotating disk system and perform experiments where a hanging mass will be attached to the system. The idea is to find the angular acceleration of the system when the hanging mass going up and down, and from there determine the moment of inertia of the system. We are going to use Logger Pro to compute the data information necessary for us to perform this lab. From that we are going to compare the results done by hand with the ones that the computer give us.

 This is the graph obtained when the triangle was placed horizontally on top of the rotating apparatus.

 This is the graph obtained when the triangle was placed vertically on top of the rotating apparatus.

This calculations show how we found the moment of inertia of the disk. The parallel axis theorem was utilized.

This calculations show the connections between the acceleration of the system, the tension and the torque of the system, therefore allowing us to calculate the moment of Inertia of the triangle.

Summary: This lab can be considered successful because the moment of inertia that we got from the calculations were close to the one obtained from the computer. There was a small difference, which can be given due to to system not being fully without friction, due to air resistance, or even errors that might have been made by us when releasing the hanging mass.

October 29 2014: Angular Acceleration

Purpose: Find the factors that affect the angular acceleration of the system.

This is the rotating apparatus that we used for the lab experiment.

Explanation: After we set up the rotating apparatus, we connected it to LoggerPro for it to read the information. We also configured LoggerPro in the form given in the lab sheet, so we would get accurate data. Then, we attached a string into the rotating apparatus and a mass with certain weight. We ran the experiment a series of times, releasing the mass from a high point, so that the mass would accelerate downwards. We changed the equipment, such as torque pulley, top disk, and weight of hanging mass a couple of times to see how the acceleration would change in each experiment.

This picture shows the data table obtained when we ran the experiment for the first time, our idea was to use the velocity vs. time graph and find its slope, which would give us the acceleration of the system.

 This is the graph velocity vs. time for the first trial.

 This is the graph of velocity vs. time for the double weight of the hanging mass.

 This is the graph velocity vs. time obtained for tripling the mass of the hanging object.

 This is the graph we obtained by changing the size of the torque pulley.

These are the values that we got for each type of experiment. We did six of them, and we got the graph for four of them, the other two we didn't take a picture of the graph because they all look familiar, except for the value of the slope.

Summary: We can say that this lab was successful because we found out that by doubling the mass of the object, the acceleration also doubled, and tripled when we used 3x the hanging mass. When we switched the top steel disk into and aluminum disk, we got an acceleration that was about 5 times faster that with using the steel disk. When using the large torque pulley, the acceleration doubled. By looking at this, we can conclude that there exists a relationship between the torque pulley, the rotating disk and the weight of the object with the acceleration of the system.

10 November 2014: Moments of Inertia

Purpose: Determine the time which a cart slides a ramp attached to a rotating apparatus by using the moment of inertia of the apparatus.

This picture shows the apparatus' that we used for the lab. On top in the rotating disk with cylinders, in the middle is the ramp inclined at certain angle, and on the bottom, the cart.

Explanation: First, we calculated the relevant measurements to find the moment of inertia of the whole rotating apparatus, from that, we used video capture to calculate the angular deceleration of the apparatus. After this, we were able to find the frictional torque of the system. We then, were given a problem where a cart attached by a string to the apparatus was sliding down the ramp by 1 meter at certain angle and we needed to find the time of it. We had to do the calculations before actually setting up the experiment. After the calculations were done and revised by the professor, we set the experiment and ran it a couple times to see if our calculations matched the real life situation.

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This is the data table obtained by using video capture and logger pro.

This graph shows the velocity in the x and y direction as the rotational apparatus spins.

This is the graph of the tangential velocity of the system found by the equation Vt=sqrt(Vx^2+Vy^2)

 These are the calculations that we did to find the moment of inertia of the total system. It came out to be 1.92*10^-2 kg*m^2

This are the calculations that we did to find the frictional torque, the angular acceleration and the time in which the cart would drop 1 meter.

Summary: Unfortunately, our conclusion did not come as expected, because we calculated the time to be at around 9.5 seconds with the ramp angled at 46 degrees, however when we did the experiment, we were getting around 12.5 seconds. This 3 seconds difference is more than the percentage expected, which was around 4%. However, these were the values that we got, and the professor revised the calculations and apparently, they were correct; so the error could be either when plugging the numbers incorrectly in the calculator, which has little chance, given that various members of the group punched the numbers in their own calculator, or some external force changed the system.