The Physical Pendulum

Purpose: The purpose of the lab was to use the moment of inertia of two different shapes in order to try and predict their period.

Procedure: In order to  predict the period we needed to find the torque that the from the pivot point that was acting on the physical pendulums. Using the function Torque = Moment of Inertia * Angular Acceleration we are able to acquire a function for the system taking into account that they are small oscillations. We did this for a semi circle pivoted at the base in the middle and at the tip in the middle. The same thing was done for a triangle that was pinned at one of its sides and in the middle of one of the sides. The triangle was isosceles.

The picture below shows the semi circle we used. The material used to make it was Styrofoam.

The triangle we used is in the picture below.

Data: The data used were the measurements of the circle and triangle.

The graph below is that of the circle when it was at its base. The period we got for this one was 0.5994 s.

The table below shows the period of the top (curved side) of the semi circle. The period of this side was 0.5991 s.

The period of one of the sides of the triangle is shown below.

Calculations: In order to find the period we needed to find the center of mass of both of the semi circle and the triangle and then use the parallel axis theorem and get the moment of inertia at the pivot we choose. Then we could find the torque and get a period of the pendulum after that.

The picture below shows how we got the center of mass of the semi circle.

The picture below shows we got the moment of inertia of the semi circle.

The picture below shows we came to get the period of the whole system.

The picture below the percent error of both the positions of the semi circle.

The picture below shows how we got the moment of inertia of the triangle and the period of it on one side. It came out to be 0.77 s.

The same thing was done to the one in the picture below.

 Summary: Overall the lab was successful because the percent error from the actual value was not off. The percent error for all trials were below 1%. Therefore we can come to the conclusion that the calculations to find the period are very accurate.

Oscillations with a Spring-Mass System

Purpose: The purpose of the lab was to see the relationship between oscillation and a mass-spring system.

Procedure: The whole data gathering was done by the class as a whole. We were divided into two separate sections. The experiment portion was done by groups individually. The equipment was set up as a previous lab with a stand and the spring with a mass at end oscillating above a motion sensor. In order to keep the experiment constant everyone had to have their mass-spring system weigh 109 g. The set up of the experiment is shown in the picture below.

The sensor was used to measure the spring constant like before and the period of the system. This information was shared with the class. Then we measured the period with four different masses. From all this information we could come to a conclusion whether the period and mass have a relationship. We also had the spring constant from the other groups from which we could find if the spring constant and period have a relationship as well.

Data: Most of the data that was taken was by Logger Pro.

The table below is the data that was shared by the entire class. The data that we took for our comparison was on the left side.

The table below was that of when we took the period of the mass-spring system.

The graph below was made from the data in the chart above. The graph will allow us to find the period of the 109 g system.

The table below was of the data we had gotten from the board.

The table below shows the spring constant for each spring and the period respectively. 

The graph below shows the graph from the table above.

The table below is that of the different masses we tried and their periods respectively.

The graph below was that of the mass vs. period.

Calculations: All the calculations done were to find the different relationships between period and mass or the spring constant and to find the period and spring constant.

The calculations below are for calculating the period and spring constant of the spring-mass system. 

Below are the calculations used to find the relationship between period and mass and period and the spring constant.

Summary: Overall the lab was a success because the period and mass as well as the period and spring constant were in direct correlation with each other. After finding the spring constant using the motion detector and the period using the oscillation pattern, we were able to come to a conclusion regarding the movement of the spring. Some errors that could have resulted would have been that the spring changes its moment of Inertia as it is stretched to its mass distribution changes.

Solving for Magnetic Potential Energy

Purpose: The purpose of the lab was to use conservation of energy laws in order to solve for a function of magnetic potential energy for the magnet-cart system.

Procedure: In order to get the magnetic potential energy we first had to set up the air track and the cart on top of it. The air emitted from the air track allowed the system to remain as frictionless as possible. On one end of the air track we had a motion sensor so that we could measure how much the air cart will move. In order to get the actual distance of the magnet from the other magnet on the glider we had to get the track level for it to be in equilibrium. We then let the motion sensor read how far away the magnet on the track was and then we subtracted how far the glider was. This gave us the real distance from the magnet. Then we changed the angle at which the track was placed 8 times so that we could get different readings for the distance the magnets were from each other. This info would then be used to plot a graph of Magnetic Force vs Distance which would give us a value of force. Then we completed the lab by plotting kinetic energy, potential energy and total energy to see whether the system enforces the conservation of energy.

Below is a picture of how the whole set up looked like.

Data: The data we had to record included the angles and distances apart the magnets were.

Below is the data table we had gotten for the different angles and the different distances.

We also plotted the values into Logger Pro in order to get a force and an angle chart. Using the angles and Weight of the glider we were able to come up with the forces.We then used this to make a force and distance graph.

We did decide to get rid of the last point to increase the accuracy of our graph as shown below.

Calculations: The calculations we did were for finding the force of the magnets. In order to do this we needed the mass of the cart and by using its weight and the angle it was elevated with we were able to get the force on the magnets.

The graph below shows what the force vs distance graph looked like after plotting the points.

After plotting the points we took the integral of the curve given by the graph in order to get a function for the magnetic potential energy.

After that we calculated the remaining energies using the given equations.

 Finally, we graphed all the energies.

 Summary: Overall the lab was a success because we managed to find a function for the magnetic potential energy between two magnets and we were able to show that energy was conserved throughout meaning that there is a valid function for the magnetic potential energy. By finding the forces between the magnets and the angles in relation to the weight of the glider we were able to come up with a function for the magnetic potential energy by finding the area under the force vs distance curve. The total energy on the last graph though is not straight and this might have to do with the fact that there was still some friction between the glider and the track. The whole experiment was not also exactly level because of the location of the classroom which was another contributing factor.

Momentum, Impulse and Collisions

Purpose: The purpose of the lab was to see whether the impulse - momentum theorem worked during different types of collisions shown by two carts and a piece of clay.

Procedure: We had three different types of collisions. Two elastic and one inelastic.

 The picture below depicts the elastic collisions. One cart was held in place while the other one was moving. Different weights were added so that we could see whether momentum was conserved even if the mass increased.

The picture below shows the inelastic collision where the moving cart will get stuck to a piece of clay and from there we would use momentum to see what would happen.

We would use the force sensor to get a force vs time graph and take the integral of the area under the curve and as a result that would give us impulse which would then lead us to get the conservation of momentum. We also had some hand calculations to see how far off we were from the actual answer. This is why we set up a motion sensor at the end of the track so we could calculate the velocity of the cart.

Data: The data we had gathered was all input into logger pro. From the motion sensor we acquired the velocity of the car which we used to get our calculations. The force vs time was used to calculate the actual conservation of momentum portion.

Below is the set up the professor wanted to use in order to get data for the lab and set up our graphs.

The graph below shows the result of the second collision. The result of the impulse was -0.2723 Ns when the mass was increased. The red portion represents whatever the motion detector acquired while the blue was the force sensor.

The graph below shows the result of the second collision. The impulse was -0.6706 Ns and this was when the cart was just by itself.

The last graph was that of an inelastic collision. The value of the impulse was -0.2351 Ns.

Calculations: The only calculations we had to come up with were the ones for the actual impulse. All of the values are listed below. We did mv(final) - mv(initial) in order to get the impulse. We then compared that to the impulse we had gotten from the one gotten from the lab. 

Summary: Overall, the lab was successful because the values gotten from the calculations were not far off from the actual values gotten from the lab. Some factors were not taken into consideration tough like friction, air resistance and the strength of the actual push the cart gave. Nevertheless, the lab gave accurate results and proved that the momentum-impulse theorem is accurate.

Work and Spring Relationship

Purpose: The purpose of the lab was to see whether we could prove that the conservation of energy theorem applies to a spring oscillating.

Procedure: In order to see whether energy is conserved we have to find the spring constant of the spring first. In order to do that we have to measure the unstretched position of the spring and then the stretched position of the spring. Then we will let the spring oscillate for about 10 seconds. There is a motion sensor at the bottom of the spring in order to capture the movement of the spring. This will allow us to get the position and velocity which we need for the kinetic and potential energies of the spring and mass. Then we will compare results and come to see whether energy is truly conserved.

Below is what the experiment set up looked like. There was a force sensor attached to the top of the spring in order for the spring to oscillate more efficiently. The notecard was attached to the bottom so that the motion sensor could get a better reading.

Data: The data we needed was captured by Logger Pro which included the position and velocity.

The data table below shows the different types of energies needed to prove that energy is conserved within the spring system. The energy columns are all calculated columns based off of the data gathered.

The graph below is that of the position vs time and velocity vs time. The graphs represent simple harmonic motion because it is a spring oscillating.

Below is the final graph of the energies. The graphs are not completely showing that the energy of the system was conserved. There must have been something wrong with our data because total energy (purple) should have been in a straight line.

Calculations: The calculations were all done in class with the professor. It shows how we came to know what data to acquire and how to solve the functions for each of the energies.

Summary: Overall the lab was not successful because we did not conserve energy within our graphs. This may have been because we did not get the spring constant k correctly. I had a little bit of difficulty knowing how to get it as well as my partner. Otherwise, the rest of the lab was successful because the energies were not exact but they were really close.

Kinetic Energy and Spring Relationships

Purpose: The purpose of the lab was to find the work done by the spring-cart system by using the kinetic energy and force with the position of the cart.

Procedure: The lab consisted of a spring attached to a force sensor and a cart on top of a track. On the other side of the track there is a motion sensor that measure position of the cart on the track. First, we calibrated the force sensor so that we could get accurate readings. Then we moved the cart a certain position back so that when we let go so that we could gather data and plot the data into graphs. Then we would find the area under the graphs in order to get the work.

Below is the set up of how the experiment looked like.

Below is a picture of how the force sensor looked like. 

Below is a picture of how the motion sensor looked like.

Data: All the data was input into Logger Pro in order to get the graphs.

Below is the chart for the different types of data we recorded. It has the force, position, velocity, acceleration an kinetic energy which was a calculated column, meaning that we used different values to come up with the kinetic energy column.

Below are the graphs of position vs time, velocity vs time and force and kinetic energy vs time. We mainly focused on the force and kinetic energy vs time for it was the one that we used to calculate work.

Calculations: In order to find the work done we had to take the integral from a certain point on the graph to the other.

Below is one of the integrals taken of the force and kinetic energy vs position in order to find work. The value was 0.896 Nm.

The next graph is another area of integration which was 0.790 Nm.

The last graph shows the last integral we did for the graph. The value was 0.625 Nm.

Below are just instructions on what the professor expected us to do in the lab.

Summary: Overall the lab was a success because we were able to get a result for the force by integrating the force vs position portion of the graph. Since the work-kinetic energy theorem states that the change in kinetic energy is equal work and the values we got for both were about the same, the lab proved to be successful. The only thing that was weird was the position of the graphs because they were supposed to look like they were increasing but we did not inverse the position and it made the graphs look like the inverse of what they were supposed to be.