Lab #14 2D Collisions

Purpose: To observe and collect data on two dimensional collisions to explore the conservation of momentum and energy. For this experiment we will conduct two separate runs. One with two metal balls of equal mass and the second with one metal ball and one glass marble.

Procedure: We set up a camera directly above a glass table(so that there would be very little friction) to capture video of the collisions. These videos can then be analyzed in Logger Pro frame by frame.



For our first run we used a metal ball and a marble. We had one ball stationary and rolled the other one at an angle so that they would collide and bounce away in different directions each with its own new velocity and direction.

Using logger pro we went through the video frame by frame plotting points for each ball's position. We then created calculated columns for the x and y components of each ball and plotted each component on a distance vs time graph. We also set up a linear fit for each component before and after the collision.



Using each linear fit to find each components velocity before and after the collision we now calculated the momentum each ball and set up conservation of momentum equations for both the x and y components to see if momentum really was conserved.



For our second run we used two metal balls and conducted the same steps; capturing video, plotting points, analyzing linear fits to get velocities, and finally setting up and calculating conservation of momentum equations.





Our data safely confirmed the conservation of momentum during these two collisions, but what about energy? was it also conserved? To verify this we had to use our plotted points to set up new calculated columns for the kinetic energy(1/2*m*v^2) for the components of each ball and plotted each one on a graph of energy vs time.





We observed that for the energy to be conserved before and after the collision the potential energies for the x and y components acted similarly while the kinetic energy acted in an inversely proportional way.

Though our experiment had some small degrees of error this is most likely due to fundamental errors with the experiment. Such as the glass table not being completely frictionless and the curved camera lens distorting our video slightly.

Overall our experiment was a success in showing how both momentum and energy are conserved in our 2D collisions.

Lab #13 Impulse - Momentum

Purpose: To explore both impulse and momentum by using a system in which can examine both elastic and inelastic collisions.

Definitions/Equations:
Momentum = Mass * Velocity ( p=mv )

Impulse = Force applied * Time (J=F*t)

Impulse without any external forces conserves momentum thus allowing us to use
Impulse = the change in momentum = intergal of (F*dt)

Procedure:
We set up a metal cart on a track that when pushed alongst the track it would collide into a plastic cart positioned so that the spring and bumper in it bounces the metal cart back. We also set up a motion sensor on the other side of the track to record position and time information for us and also set up calculated columns for both velocity and Force on the cart. We then ran this expirement twice, once with only the cart and the second time with added mass





Once we took data we created both Velocity vs Time and Force vs Time graphs, focusing on the velocity before and after the collision. We then integrated the area within the part of the force graph where the collosion occurs.

Inelastic collision: For the second part of the experiment we had a very similar set up except instead of colliding with the plastic cart and bumber it will have a nail that will srick into a block of clay mounted to an upright wood with a base that wil be underneath the track itself.

 We finally calculated the theoretical values each expirement had, by multiplying each mass by the change in velocity( Vafter - Vinitial ) it experienced due to the collision.



Conclusions: Comparing our Theoretical and Actual values for the impulse in each experiment we can safely agree with the theorem that impulse = the change in momentum an object experiences. Our data might be slightly off due to experimental flaws such as the cart experiencing a frictional force and slowing down on its own, but overall we now better understand how impulse, momentum, velocity and mass are connected in collisions.

Lab #12: Magnetic Potential Energy

Purpose: To verify that conservarion of energy still applies for a system involving magnetic potential energy.

Set Up: Place an Air track on table and make sure to level it using something like notecards. Make sure there are magnets on the end supports of the air track and also on the glider. The vacuum will provide a force through the holes in the track and keep the glider afloat so we can oppose any drag due to the glider touching the track.



Procedure:
1) Because we have no set equation for magnetic potential energy we must build our own for it. We will use the relationship of potential energy being equal to the negative integral of force over distance.

2) If we raise one end of the air track the glider will slide and find an equilibrium point at which the magnetic repulsion is equal to the parallel gravitational component.



3) We stacked different heights of books underneath one end of the track and measured the distance between the glider and the opposing magnet and also recorded each new angle due to the raising of one end.



4) To plot this data in logger pro we first had to convert the angle measures into radians so that we could calculate each force using trig functions in logger pro.



5) Once we graphed each force vs the separation distance we used a power fit(Ar^B) to calculate a coefficient and exponent to be able to later integrate.



6) Now that logger pro has given us our A and B we can find the negative inegral of the force over distance to calculate an equation for magnetic potential energy.

Lab #11: Conservation of Energy - Mass-Spring System

Purpose: To explore the conservation and transfer of energy of a vertically-oscillating mass-spring system.

Procedure:

1) We need to identify all the different forms energy will take in this experiment. We know that th hanging mass will have both a kinetic energy and a potential energy. However, the mass itself is not the only thing interacting in this eperiment. The spring itself will also have its own KE and GPE, but because it is a spring it has a third energy form of elastic potential energy. Below are the equations we used for each:



2) We set up a column style apparatus with a force sensor attached to the top of our clamped bars, a spring with a hanging mass attached, and at the bottom we have a motion sensor reading upward. We then calibrated and oriented all the equipment as needed.



3) The only variable we will have to do extra work to find is the spring constant. To do this we slowly stretch the spring downward and plot the force vs distance stretched in logger pro. Once this is complete we simple have logger pro calculate a linear fit to the graph whereas our slope is equal to our spring constant.

4) Now we used our equations to set up calculated columns within logger pro and simply release the system from a sthertched position and have logger pro collect data for each seperate energy.



5) Observing each different energy oscilate we can see how the energy is trasfered from one type to another as they fluxuate in opposite paterns.

6) Our column and graph for the total energy stored in the system was reassuring that the energy is conserved.

Lab #10: Work-Kinetic Energy Theorem

Experiment 1: Work Done by a Nonconstant Spring Force

Purpose: To calculate the spring constant by using Hooke's Law



Procedure:
1) We first calibrated a force sensor by zeroing it with no mass hanging and then added 500grams setting the sensor to read 4.9N.

2) We set up a track on our table, a cart attached by a spring to the force sensor, which is held horizontally on one end of the track by a clamp and bar. On the other end we have a motion sensor to read the distance the cart will be pulled.

3) Using logger pro we collect data from both the force and motion sensors and plot them to create a graph of force vs distance.



4) Using a Linear fit we find the slope(spring constant) to be 5.724 N/m.

5) By measuring the area under the curve(integrating) we have now calculated the work done by the spring, .05 N*m

Experiment 2: Kinetic Energy and the Work-Kinetic Energy Principle

Procedure:
1) First we must create a new calculated column for kinetic energy of the cart, .5mv^2

2)Using the same set up as before we will now Release the cart from a stretched position and allow logger pro to graph both KE and Force against the distance.

Lab #9: Centripetal Force with a Motor

Purpose: To come up with a relationship between the angle that the hanging mass makes with the vertical and the angular speed of the apparatus.

The apparatus:
The motor mounted to the top of a tripod spins a pole with a two-meter stick with a hanging mass at one end. This apparatus is similar to a carnival "swing-merry-go-round".







Procedure:
1. Before turning the motor on we measure everything that will remain constant troughout the experiment: The height of the apparatus(H=2m), the radius where the hanging mass is attatched(R=.98m) and the length of the string for the hanging mass(L=1.645m).

2. We then drew force diagrams and symbolically solved for both the angle and time. (note* the height we will measure is from the ground to the rotating height of the mass denoted by the lowercase "h").

3. After deriving the formulas we used we came together as a class to observe the apparatus spin at different speeds. For each speed we timed ten rotations and divided that by ten to calculate the time for ohe rotaion. Each speed also provided us with a new height "h" that we measured by slowly raising a piece of paper on a ring-stand until the mass barely nipped it, pulled it aside and measured the height. We collected data for each of the six runs we had at increasingly greater speeds.

4. Once the experimental data was collected we went back to our equations and plugged in "h" and calculated our theoretical values of both the angle and time.

5. To compare our actual to our theoretical results we plotted them agains one another and had logger pro insert a line of best fit to show us how accurate our process was.

Lab #8: Centripetal Acceleration vs. Angular Frequency

Objective: To determine the relationship between centripetal acceleration and angular speed.

Apparatus: For this demonstration there was a wheel and an electric motor set up to spin a rotating platform. To measure the centripetal acceleration there was a accelerometer attatched to the top ot the platform facing inward toward the center and a photogate to measure the period of each rotation to find angular velocity.

Procedure:
1. We set the power supply to 4.8, 6.2, 7.8, 8.8, 10.9, and 12.6 volts and collected data and plotted acceleration vs. time for each setting.

2. Our data unfortunately shows acceleration to not be constant but instead bounce up and down, to remedy this we simply told logger pro to calculate the average acceleration for each setting.

3. Because our goal is to compare centripetal acceleration and angular frequency we used our equations of v=w/r and a=rw^2 to calculate two new colums in logger pro.

4. Finally we plotted Centripetal Acceleration vs W^2

5. To complete or comparison we asked logger pro to give us a linear fit, which had a correlation of .9999! The most important part was to notice that when we compare the acceleration to W^2, our slope is .1371 which is very close to the actual radius of the wheel (.138m).