Lab 15 - Collisions in Two dimensions

 The above picture is the glass table setup that we used to generate collisions. The long metal stand is where we attached our phones to record the collisions in 240frames per second.

 Purpose: The purpose of this lab is to show that in collisions, momentum is conserved even though it can be distributed into two dimensions.

Apparatus/Procedure: We used two glass marbles and a heavier metal ball. One ball was positioned stationary on top of a level glass table, while the other one was thrown at it to create a collision. There was a ring stand positioned directly above the glass table with a clip in order to attach a cell phone. The cell phone was used to record the collision at 240 frames per second. This would allow us to get a clearer picture when we uploaded the collisions to logger pro. We ran one trial with two glass marbles, and another trial with the heavier one colliding into one glass one. We then used logger pro to track the paths of the marbles both before, and after they collided. We set up a coordinate system, and logger pro was able to separate both the paths into x and y components. We then took these values for velocities and checked to see if momentum was conserved. 

Theory: The theory behind this is that momentum is conserved in collisions, regardless of the amounts of dimensions. 

The momentum equations the two collisions: 

m= mass of glass balls : 0.0193kg

3m=mass of metal ball : 0.0667kg

Collision with same mass balls:

Po= pf

m (1.282m/s)+m (0)=m (0.4098m/s)+m (0.6795m/s)

0.0247426kgm/s=0.02102kgm/s

Momentum not conserved along y-axis

Ke before=ke after

Ke before=.5mv^2
Ke=.5*0.0193*1.74^2=0.0292j
Ke after=.5*0.0193*0.42^2 +.5*0.0193*1.05^2
      =0.0123j
0.0292j=/=0.0123j
Ke not conserved
The system lost kinetic energy after the first collision with two balls of equal masses.

m (1.173m/s)+m (0)=m (0.1061m/s)+m (0.7994m/s)

0.0226389kgm/s=0.017476kgm/s

Momentum not conserved along x-axis

Collision with different- mass balls: 
Po=pf

3m (1.332m/s)+m (0)=3m (0.5054m/s)+m (1.490m/s)

0.08884kgm/s=0.062467kgm/s

Momentum not conserved along y-axis

The system lost momentum along the y-axis.

3m (1.871m/s)+m (0)=3m (1.204m/s)+m (0.4905m/s)

0.124796kgm/s=0.08977kgm/s

Momentum not conserved along x-axis

Ke before=0.1764j

Ke after=0.0786

Ke not conserved. 

The system lost about half the ke to heat after the second collision with two ballstat of different mass.

Below are all of the graphs on logger pro with all of the paths of all the balls both before and after collision.

The above screen shots are of the of the  collisions. The top one is the collision with same masses, and the one below is the screen shot of the collision with different masses. 

Conclusion: As it turns out, not even in this lab is momentum perfectly conserved. This is because of the fact that after one ball impacted the other, they would slide, then catch friction and start rolling. As the balls catch friction, they also lose some energy to friction and thus, lose momentum. When the two balls collide, there is also kinetic energy that is converted into thermal energy from the impact. That is another reason why momentum s not conserved.  In reality, there is no collision in which momentum will be perfectly conserved.

My partner for this lab is Elliot Sandoval.

Lab 14 - Ballistic Pendulum

Purpose: The purpose of this lab was to determine the initial velocity of a fired projectile.
Apparatus: Our setup for this lab was a something called a ballistic pendulum. A fixed, stationary gun is aimed at a hanging block of measured mass 0.0805 (+/-)0.001kg. The mass of the projectile is also measured using a scale which results in a mass of 0.0075 (+/-)0.001kg. After the projectile is fired, it becomes lodged in the hanging block and the system swings upwards because of the fact that the block is held up by four strings of negligible mass.
Procedure: The procedure was straight-forward. We placed the projectile in the gun, moved out of the direction of its path, and fired it into the block. The block and ball(projectile) moved together after they stuck and rose to an angle of 16.7 degrees. After we plugged in the values for the masses, and the angle, and the length of the string, which was 0.0206 (+/-) 0.005m, we then were able to calculate the initial firing speed of the gun. We ran multiple trials and figured out that the firing velocity of the ball before impact was 4.73m/s. Afterwards, we cleared a path and moved the hanging block out of the way. We then placed the gun facing the edge of the table. We then placed a sheet of carbon paper on top of a blank sheet of paper on the ground in order to find out where the ball hits the ground after the gun is fired away from the edge of the table. Then, using some simple kinematics and a meter stick, we calculated that the actual firing velocity of the gun was 5.612m/s.
Theory: The theory behind this experiment is that when an object becomes lodged in another, momentum is conserved. So we used the law of conservation to guide us in the right direction. We then set KE=GPE, where KE is before impact and GPE is after impact. However, for momentum to be conserved we had to use the mass of the ball only in the KE formula, and the combined masses of the ball and the block for the GPE formula. In the GPE formula, we multiplied "g" by (L-LcosΘ) in order to get the vertical component of the direction of travel of the block.

Conclusion: What we noticed was that the velocity we calculated from the block moving up after the ball embedded in it was significantly lower than when we just fired it off the edge of the table. This shows that although momentum is conserved in a collision, energy is not. A lot of the energy that was lost was due to heat. The professor showed us a proof on the board by using the formula q=mct. So it is very possible that all of the energy that was lost was converted into heat. This lab was a good representation for how in collisions, momentum is conserved but energy is not. We saw it firsthand, and there is no denying it.
 My partners  for this lab were Elliot Sandoval, Sherri(?) and the person she sits next to. (I forgot to get their names).

Lab 13 - Magnetic Potential Energy

 These are 3 pictures to help visualize what we were working with in the lab. The second picture helps to show how we used varying angles to find the force of gravity, which was also equal to the force exerted by the magnetic field. The 3rd picture helps to show how we recorded position and velocity of the cart by using a motion sensor.  We then used the sensor to take a measurement from the sensor to the reflector, which read 0.340m on logger pro. We then measured with a ruler the actual separation distance between the magnets which was 0.0359m. We then took a difference of the two numbers and used the difference to apply a correction to logger pro when we created a new calculated column. The formula we set was the distance read by the sensor (minus) the difference we calculated. This formula gave us a table with all the actual separation distances that were then used to create the potential energy function.

Purpose: To create a relationship (mathematical model) between distance and potential energy for two magnets of  same polarity that approach each other.
Apparatus: The setup that we used was an air-track with a glider cart that would "float" along. Air that was pushed through holes along the track kept the glider suspended in the air as it moved along the track. There was a magnet on the front of the glider, and another magnet at the edge of the track. They both had the same polarity so that they would repel each other.
Procedure:We turned on the track in order to make the track frictionless. We then lifted one end of the track to a fixed angle, which we wrote down. The angle was measured by a protractor application on my cell phone. We then turned off the track in order to measure the distance between the two magnets. We then increased the angle and repeated the measurement of distance by using digital calipers. As we increased the angle, the distance between the magnets decreased. We then used simple trigonometry, gravity, and the mass of the cart to figure out the force that the magnets applied. We then related each force at each angle with the corresponding separation distance between the magnets by inputting the values into a table, and generating a graph on logger pro. We then used the points on the graph to create a power fit which gave us an equation for the force between the magnets based on distance. We then integrated the equation to give us the equation for the potential energy(U(r)).
U(r)=(0.0001929/1.066)*r^(-1.066)
Mass of the cart was 0.351kg
We then created a calculated column using our new function. We then leveled the track. At the end of the track with the magnet, we placed the motion sensor by way of a ring stand. We then collected data and pushed the glider along the track until it "collided" with the magnet at the end of the track. We then used this newly collected data to generate a new calculated column which was kinetic energy. We then created yet another column called total energy which was a sum of the column with our function and the kinetic energy.
This was a table of our raw data from inclining the air track.

This was our original power-fit function. However, due to to the manner in which we pushed the glider along the track, we needed to remove some points that were too small in distance for the distance between the magnets. Had we collected data with having pushed the glider harder, those points would have been more appropriate.

This is our power-fit after we removed some points in order to better accommodate our collected data.

This is our graph of kinetic energy, magnetic potential energy, and total energy which is the sum of the first two. At first we were having trouble with our graph because our total energy graph was not very flat at the point where the MPE peaks. Then we had to go back and remove some points from our power fit and ended up getting a more appropriate fit. We did not change any data, rather we omitted data that was inappropriate which resulted in a better overall graph.

This is a picture of the above graph, alongside a position vs. time graph for our collected data, and also a velocity vs. time graph.

Conclusion: This lab served as a powerful example that not every knowledge in this world has to be handed down. There exists no laws for magnetic potential energy however, using new technology, we were able to create a model for this specific setup. The proof that our model for magnetic potential energy works is in the graph for total energy. The line for total energy is relatively flat, indicating that the two are equal. It is not perfect however, due to many factors such as air resistance, and the magnetic fields not being perfectly uniform. Although our model is not perfect, it is good enough and shows the importance of ingenuity.

My partner is Elliot Sandoval.

Lab 12 - Conservation of Energy of a Mass-Spring System

 

This is also a sideways view of the setup that we had for this lab. The spring was hung off of the force sensor, with a motion sensor at the bottom.

Purpose: The purpose of this lab is to show how energy is conserved in a mass-spring system.
Theory: The theory behind this is that energy is always conserved within a system. The only time when energy is not conserved is when there is an outside source that absorbs energy, or when it is converted into heat.
Apparatus: The apparatus uses most of the same components from the previous lab involving the cart, and spring. We used the ring stand and used it to extend a rod over the edge of the desk in a horizontal fashion. We then hung the spring off the rod, and hung a mass off the lower end of the spring. On the bottom of the hanging mass we taped an index card so that the position sensor that was directly below it could register its position more easily. We then used a laptop to retrieve all of our data.

Procedure: After we had everything set up, we then pulled on the mass a little bit and let it go. This then caused the spring and mass to oscillate up and down. The motion of the spring and mass was then recorded and used in the laptop. Using the data we collected we then created multiple calculated columns including "elastic potential energy", "kinetic energy", and "gravitational potential energy".

The graph below is KE vs. Position, it shows that as the mass and spring oscillate, the most kinetic energy it has is as it passes through the "natural" resting position of the system. This means that at the point that the mass was, before we pulled on the it, is the point where KE is greatest.

 

 The graph below is KE vs. Velocity. What this graph is showing is that where the velocity was zero, was where the KE was zero. These points occurred at the top and bottom turning points of the mass. This also shows that the KE was greatest at the points where the velocity was the greatest, which was at the "natural" resting position.

The graph below is of Gravitational PE vs. position. What this graph demonstrates is that the point with the greatest GPE is the point furthest away from the motion sensor, which also happens to be the highest point that the mass reaches.

 The graph below is GPE vs. velocity. This graph shows the oscillation of the system. This is shown because although at the bottom there is no more KE because the velocity is zero, and there is zero GPE, the velocity increases again.

 The graph below shows Elastic PE vs. position. What it demonstrates is that the position that is closest to the position sensor is the point with the most  EPE, and that at the highest position of the mass is where the EPE is the smallest.

The graph below shows EPE vs. velocity. What is really being shown here is the oscillation of the mass-spring system.

 These are all of the graphs put together into the y-axis, while the position is on the x-axis. This helps to demonstrate how the spring and mass are able to oscillate. This is because as the EPE loses energy, the GPE gains energy, and vise versa.

The graph above has the same y-axis, while the x-axis has been changed to time. This graph helps to even further illustrate how each energy is transformed into a different one.

Conclusion: It is pretty neat to see real world data provide evidence for things that I have known for years, but only from lecture and from accepting the fact that what I was taught was true. We created our own data and showed the various energies contained within a mass-spring system. We did not position the graphs in a way that made it seem like one energy was converted into another, they just happened. Every position on each graph is placed on the exact second(referring to time) as the others. As you can see from the graph above, the point on the graph where the kinetic energy is at its highest point is where the EPE and GPE are equal, or at their intersection points if the graphs of EPE and GPE were laid on top of each other.

My lab partner is Elliot Sandoval.

Lab 11 - Work-Kinetic Energy Theorem

 This is a sideways side-view of the setup that we had for this lab.

Purpose: The purpose of this lab is to prove that the work done by a stretched spring is equal to the kinetic energy that it generates.

Theory: We are trying to prove that work done by a system on an object is equal to the change in kinetic energy of the object.

Apparatus: Our apparatus for this lab was a metal coil spring approximately a foot long that does work by pulling inwards after it is elongated. We clamped a ring stand to the edge of the table, and attached the spring to the stand. We also placed a cart track in front of the stand-across the table. We then got a cart of mass 0.5439kg and attached a force sensor probe on top of it. The end of the force sensor was then attached to other end of the spring. We then used wooden blocks and track bumpers to get the spring to rest at the same height of the sensor. Using logger-pro, the force sensor, and a position sensor placed on the end farthest away from the ring stand on the track, we were able to calculate the spring constant which turned out to be 18.19N/m as shown on our graph below.

 Procedure: We then generated a new calculated column using the mass of the cart and "velocity" that is inputted into a new column by the position sensor. This new calculated column was the kinetic energy of the cart. We we generated the information for this new calculated column by measuring out how much we stretched the spring with the cart, then letting go of the cart so that the spring can do work on the cart and accelerate it. We then integrated to three different positions and compared the area under the curve (work) to the kinetic energy graph below it.

 Work= -1.029J
Kinetic Energy=  0.929J
~9% loss of energy

 Work = -1.975J
KE= 1.884J
~5% loss of energy

 Work = -2.733J
KE= 2.532J
~7% loss of energy

Conclusion: Based on our graphs and comparisons, it is safe to say that the work done by a spring on an object is equal to its change in kinetic energy. However, the amount of work done by the spring is NOT exactly equal to the change in kinetic energy  of the cart because energy is lost to many factors such as air resistance, and friction between the cart and the track. It was pretty cool to see that the work and the kinetic energy are very close to each other, meaning that our work-KE theorem is correct. The work done by a spring is equal to the change in kinetic energy of the cart, when all other factors are not accounted for.

My lab partner is Elliot Sandoval.

Lab 9 - Centripetal Force with a Motor

Purpose: The purpose of this lab was to create a relationship between angular speed ω and the angle Θ that the string in the apparatus below made with the vertical.
Apparatus: The apparatus that we used was a motor that was attached to a surveying tripod, positioned with the axis of rotation pointing upward and an extension rod. The top of the rod had a meter stick taped to it in a horizontal fashion. The far end of the meter stick had a string tied to the hole of length 187.1cm, with a rubber stopper attached at the loose end. The opposite side of the meter stick had some counterweights taped on.
Theory: The theory behind this lab is intuitive; if you spin something in a lateral circle, the faster you spin it the higher it will move. It is easy to say what will happen, however the challenge that was posed here was to quantify the angular speed based on what angle the string makes with the vertical.
Procedure: When the motor was on and the apparatus was spinning, we had a ring stand on the floor with a piece of paper taped on it. The height of the paper was adjusted until it was at the lowest height required to make contact with the swinging stopper. The height off the ground was measured, and we were able to calculate the angle made by the string by using the dimensions of the apparatus. The relationship we came up with was:
ω=[(g*tanΘ)/(R+L*sinΘ)]^.5
g=gravity
R=length from string to vertical shaft on meter stick
L=length of string

 

This is a table of all the data that we collected and calculated. The column on the left was measured by using a stopwatch. 10 revolutions were timed in order to get an average value, and these average values were compared to our calculated angular speeds (which are shown on the far right column in the table). (Note: for some reason I was having a hard time figuring out how to rotate the table properly).

 

This was a graph with calculated angular speeds, and the measured average values, each on their own axis. Based on the line, there was approximately 15% error in the two different sets of data.
Conclusion: I think that this lab was pretty cool because we were able to calculate the angular speed of the apparatus just by using a free-body diagram and an angle. The amount of error is appropriate for the setup that we had. The angular speed that we measured with a stopwatch was on average 15% lower than what we calculated. This is due to many reasons, such as: air resistance in the string and the rubber stopper as it went around; friction in the apparatus; and the fact that a meter stick was not the best choice for a revolving beam. The air resistance had the most impact on the error because it is very apparent that the differences between the measured and the calculated angular speeds gets larger as the speeds increase.

Lab 8 - Centripetal Force

  This is a top view of the setup that we had.

Purpose: The purpose of this lab was to show how the various independent factors affect centripetal force. These factors include mass, radius, and angular speed.

Theory: We are trying to figure out how angular acceleration and force are affected by various factors.

Apparatus: The apparatus was a large, circular, wooden disk with four wheels under the four outermost points, with a motor at the bottom that drove it in a circle. The force sensor was at the center of the disk, with a string tied to it and a mass tied to the other end of the string. The mass was closer to the edge of the circle so that as the disk spun, it would pull on the force sensor.

Procedure: This was a relatively simple, observational lab. The professor ran various trials with varying angular speeds, varying masses, and varying radius by changing the length of the string. The forces were calculated by the force sensor and computer and written on the board. Everyone had the same data, however everybody generated their own graphs and calculations. This is all of our data, first put into a neat table, and then with varying graphs.

 Force vs. Radius graph

 Force vs. mω^2

 Force vs. rω^2

 Force vs. ω^2

Force vs. mrω^2

 Conclusion: Based on the graphs, the variable that affected the slopes the most is the angular speed. This is due to the fact that in any centripetal force equation, the velocity is squared, be it linear or angular velocity. What this means in real life is that in any rotating object, as the speed is increased, the centripetal force will increase exponentially. So an engine spun at high rpm will generate a lot more stress on rotating components than a similar engine that operates at lower rpms. This also means that a car traveling 2x as fast as another car around a turn will require a lot more grip on its tires to keep it from sliding out of the curve.

My lab partner is Elliot Sandoval for this lab.