Rotational Kinematics of a Spinning Beam

 Analysis:

For r = 0.06 m

   

For r = 0.09 m

                        

For r = 0.12 m

For r = 0.15 m 

    

For r = 0.18 m

Using the graphs we were able to determine the velocity by using the formula:

Now, we can compile all of our data into one table:

Graph of Velocity vs Radius  &  Acceleration vs Radius:

1. Use the velocity components to determine the direction of the velocity vector. Is it in the expected direction? 

Yes, the velocity vector is in the expected direction.

2. Analyze enough different points in the same video to make a graph of speed of a point as a function of distance from the axis of rotation. What quantity does the slope of this graph represent?

The slope of this graph represents the angular velocity.

3. Calculate the acceleration of each point and graph the acceleration as a function of the distance from the axis of rotation. What quantity does the slope of this graph represent? 

The slope of this graph represents the angular acceleration.

Conclusion:

How do your results compare to your predictions?

Our group thought that the Velocity vs Radius graph would be positive and linear, and that the Acceleration vs Radius graph would be positive and constant. So, yes our predictions do match the results from the lab. 

Mousetrap Cart Project Slides

 Mousetrap Cart Slides:

https://docs.google.com/presentation/d/1zlmf_Unf6j3Ns-mlvq8xeCbXqNIt-AffCYyyHuuJQHM/edit#slide=id.g3529b6f39cf_2_119

Individual Contributions:

    I started out by helping my group come up with the initial designs. I knew we should have some sort of triangular supports with a rod in the middle to hold the hanging mass. Then, as a group, we came up with a few more ideas like not using the "snap back function" of the mousetrap and also tying each system to a separate axle. After that I made a very simplistic model out of scrap wood, cardboard, and chopsticks to see if it would work, and it did (but not very well). The structure was too weak to support a 500 gram mass, and also the wheels supplied were too flimsy. Then, Enbo designed a full CAD structure that looked great. However, the structure was extremely heavy, to the point that the wheels couldn't even support the design itself, let alone the mass on top. So, I helped dimension the pieces to take most of the unnecessary weight out. After completing the second design, I printed it and then we assembled it. Again there was another issue; the issue being that we needed a heavy mass (500 g) to set the mousetrap, but a light enough mass (200 g), so that the torque produced by the mousetrap was enough to move it back. We came up with two solutions: try and alter the hanging object to find the optimal mass or somehow manage to completely disconnect the hanging mass from the cart. The problem with severing the mass was that our structure was too small to cut a large enough piece out for the mass to fall out horizontally. We needed the 500 g mass to fall horizontally because it was too tall (with the hook connected) to fit under the car, thus stopping its travel. So, I made a totally new CAD design to accommodate the size of the mass. However, we ended up not using the new design (for fear of not being able to print and complete the project in time) and instead drilled a hole out of the base plate using a drill press. 

    After the physical model was complete, I recorded the motion and did video analysis. We had multiple good runs, but I ended up only using the two best ones. Then I developed the work-energy models to incorporate all the different pieces of motion. With that done, we assembled all of the information from previous labs or other sources to find the all of the variables in the equations. Next, I started writing the initial code to initialize the variables, setup some of the design for the cart and background, and also put in the graphs. Enbo took over from there, while I worked on the presentation slides. I looked over the rubric to make sure I had all of the necessary information. I compiled the data from video analysis, labs, equations we used, and anything else pertinent. After most of the code was complete, I tried to help debug some of the issues we were having, like the cart design and loops. Overall, I believe this was a lot of work for two people; however, we both contributed to the project, which made it more manageable. 

Energy Efficiency on Level Track

 Analysis:

Formula to find energy efficiency:

Trials:

The initial velocity of the cart does effect the energy dissipation. This can be seen from the data obtained from the experiment. It is also important to note that the ratio of velocities will never be 1, because the magnetic bumpers are not 100% efficient. We also lose of the energy in the form of friction. Friction does a small amount of negative work on the system. 

Conclusion:

The energy is dependent on the initial velocity of the cart, but we can estimate that the energy efficiency of the magnetic bumpers is roughly 60%. Therefore, we are losing 40% of the initial kinetic energy of the system due to the magnetic bumpers and work done by friction. When compared to our partners group using the inclined track, we can see that their bumpers are even less efficient than ours. They proposed that their work done by friction was greater in comparison; leading to more energy loss. 

Lab: Analysis of the Coefficient of Friction

 Analysis:

Formulas:

1. Using the formulas for frictional force and normal force, we arrived at these results when varying the mass:

2. Graphing frictional force vs. normal force:

3. Using the formulas for frictional force and normal force, we arrived at these results when varying the height (angle):

4. Graph of frictional force vs. normal force:

5. Respective Uncertainties:

The two graphs seem to be quite different. Although both linear, their slopes and y-intercepts are not similar.

Conclusion:

The coefficient of kinetic friction for wood on aluminum, calculated by the slopes of the graphs, was found to be 0.7. This does not match with our prediction (within uncertainty) being 0.2.

Car Velocity Lab

 Conclusions:

How do the predicted velocity and the measured velocity compare in each case?  Did your measurements agree with your initial prediction?  If not, why?  

We used the equation above to determine our predicted initial velocity. The predicted velocity and measured velocity were very similar in comparison. As always, there is uncertainty when using Video Analysis to measure data. The measurements we calculated did agree with our initial predictions within uncertainty. I believe the variation in our data came from the fact that the model we used to make our predictions was "ideal", meaning we did not account for the friction of the system (only the two masses and distance) or the effective mass of the pulley.

Does the launch velocity of the car depend on its mass?  The mass of the block?  The distance the block falls?  Is there a choice of distance and block mass for which the mass of the car does not make much difference to its launch velocity?

The launch velocity depends on: the hanging mass, the mass of the car, and the distance the hanging mass falls before reaching the ground. The mass of the hanging object affects the initial acceleration by the amount of tension it creates. The larger the mass of the hanging object, the more tension in the system, and thus a larger magnitude of initial velocity. The distance the hanging mass falls also affects the initial velocity. If the distance is smaller then the tension being applied only lasts a short amount of time; whereas if the distance (between the ground and the object) is large, then the force applied by the hanging mass will last a longer amount of time. Finally, the mass of the car is also a factor in initial velocity, because if the mass of the car is large, then it will take a considerable amount of force to move the car initially. With a large enough hanging mass at a high enough distance, the mass of the cart would be negligible.

If the same mass block falls through the same distance, but you change the mass of the cart, does the force that the string exerts on the cart change?  In other words, is the force of the string on object A always equal to the weight of object A?  Is it ever equal to the weight of object A?  Explain your reasoning. 

The tension force of the string is not always equal to the weight of the object. In a static system (when the acceleration is 0), the tension force will be equal to the weight. However, in a dynamic system the tension must also account for the objects acceleration and mass.

Was the frictional force the same whether or not the string exerted a force on it?  Does this agree with your initial prediction?  If not, why?

No, the frictional force varies based on the system. The frictional force that the car experiences while the hanging mass is still falling is less than the frictional force the car experiences when the hanging mass reaches the ground. This is because at the point where the hanging mass is still falling, the system has the force of hanging mass (creating the tension that acts on the cart) and kinetic energy of the cart to oppose friction. Once the mass is no longer acting on the system, the friction is only opposing the kinetic energy of the cart, and eventually slows it down. So, we can say that the magnitude of frictional force is changing throughout the duration of motion. This does not agree with our prediction of friction being a constant force.

Hanging Bridge Lab

 

Hanging Bridge Lab

Post Lab Write Up:

1. The graphs should match early on when the weight of the masses is less, but as you increase the masses the graphs start to diverge.
2. The limitations that my group observed were the lack of precise tools. For example taking measurements with a meter stick and "eyeballing" measurements. Our system is also not ideal, meaning that things like the string, the pulley, and how we arrange the masses must be taken into account.
3. The vertical displacement increases as the mass of the center object increases. However, the model created is really only useful for smaller masses (because the curves diverge the more mass you add).
4. My groups predictions somewhat agreed with the results of the lab. As you add more mass to object B the displacement becomes larger. The predicted equation for the model assumes ideal string and pulley, which we know not to be true.
5. No, the pulley doesn't behave in a frictionless way throughout the experiment. The less mass there is the closer the curves match, which means the pulley can be neglected (considered frictionless). However, this does not hold true as you begin to add more mass to object B. Therefore, the pulley can't always be considered frictionless. We also know that string itself is light and shouldn't interfere with experiment. As explained in class, a heavy string is more unpredictable compared to a light string.
6. The information needed would be: length of the walkway, mass of the walkway, mass of object on the walkway, density of rope (cable), effective mass of the pulleys, hanging mass on either side of the walkways.

Project 1 - Projectile Motion Modeling

Individual Contributions:

    For the first part of this project, I modeled projectile motion using Video Analysis and then calculated the uncertainty of the range and initial velocity. Then I found the average spring constant with uncertainty by using multiple masses to stretch the rubber band.  I used the average spring constant to calculate the initial velocity using a Work-Energy equation. Next, I found the max height of the rocket launched at a 90 degree angle by doing 2 trials. I used these measurements to find the initial velocity using a different Work-Energy equation. I combined all of these values into one table that showed each initial velocity using a certain stretch. I made graphs of both of these initial velocities vs. stretch. I then helped to start writing the basic code of the projectile motion. As we progressed, I calculated the value for "C" (using the spring Work-Energy equation) to accurately compute the initial velocity in the code. After the code was finished, I helped to start and finish the project slides. Overall I believe all of my group mates did equal and accurate work to complete our project. 

Project 1 Slides