Phys4AS17cceron

Tuesday, May 9, 2017

April 26, 2017 Lab: Ballistic Pendulum

Lab: Ballistic Pendulum

Chris Ceron, Amy, John, Ricardo

Goal of Lab

To use conservation of momentum and energy to determine the firing speed of a ball for a spring-loaded gun.

Apparatus used in lab

Theory/Introduction

Within our apparatus there are in theory no external forces acting on our system. This allows for us to calculate the initial velocity using the mass and final velocity of the system. Our equations for momentum are as follows:

Initial Momentum:

$p_{i} = m_{ball}v_{ball_{i}}+m_{block}v_{block_{i}}$

Final Momentum:

$p_{f} = (m_{ball}+m_{block})v_{f}$

Since momentum is conserved, initial and final momentum are equal:

$m_{ball}v_{ball_{i}}+m_{block}v_{block_{i}} = (m_{ball}+m_{block})v_{f}$

Using this equation, we could solve for the final velocity of the ball to later use in our conservation of energy equation:

$v_{f}=\frac{m_{ball}v_{ball_{i}}+m_{block}v_{block_{i}}}{(m_{ball}+m_{block})}$

Since we assume that energy is also conserved, we can derive a formula for the energy in our system:

$KE_{i} + U_{i} = KE_{f}+U_{f}$

where KE is the Kinetic Energy and U is our potential energy.

Before the collision, the energy within our system is all in the form of Kinetic Energy. After the collision, that energy is transferred to potential energy:

$\frac{1}{2}(m_{ball}+m_{block}){v_{f}}^{2} + 0 = 0 + (m_{ball}+m_{block})gh$

The height used in our potential energy can be found by taking the length (L) of the string, and subtracting it by the x component (Lcos $\theta$ ) of the string after the collision.

If we use the final velocity we found through momentum and simply, the result is:

$v_{ball_{i}} = \frac{m_{ball}}{m_{ball}+m_{block}}\sqrt{2g(L-Lcos\theta)}$

To verify our initial velocity, we move the nylon block out of the ball's trajectory and fire the ball at 0 degrees. We can place a piece of carbon paper where we expect the ball to land based on the velocity value found in part one of the lab. This becomes a projectile motion problem. We can use the height at which the ball is fired and the distance the ball travels in the x direction to calculate the initial velocity.

$\Delta y = v_{y_{i}}t+\frac{1}{2}a_{y}t^2 \: \: ,\: \: \Delta x = v_{x_{i}}t+\frac{1}{2}a_{x}t^2$

We use the first equation to solve for t. Since the ball was fired horizontally, the initial velocity in the y-direction is 0. We can substitute t in the second equation and solve for the initial velocity. Since there is no force acting on the ball horizontally, we know that the acceleration in the second equation is equal to 0.

$t=\sqrt{\frac{2\Delta y}{g}} \Rightarrow \Delta x = v_{x_{i}}\sqrt{\frac{2\Delta y}{g}} \Rightarrow v_{x_{i}} = \frac{\Delta x}{\sqrt{\frac{2\Delta y}{g}}}$

To calculate our uncertainty, we look at each calculation separately. For the collision part of our lab, we take the natural log of our equation:

$\ln V_{ball_{i}} = \ln\left(\frac{m_{ball}}{m_{total}}\sqrt{2g(L(1-cos\theta))}\right)$

We can use a property of natural log to expand everything:

$\ln v_{ball_{i}} = \ln m_{ball}- \ln m_{total} + \frac{1}{2}\left(\ln 2 + \ln g + \ln L + \ln (1-cos\theta)\right)$

When we take the derivative, the constants will disappear. Our constants in this case are ln(2) and ln(g)

$\frac{dv_{ball_{i}}}{v_{ball_{i}}} = \frac{dm_{ball}}{m_{ball}}-\frac{dm_{total}}{m_{total}}+\frac{dL}{2L}+\frac{d(1-cos\theta)}{2(1-cos\theta)}$

We can then use the propagated uncertainty equation to find the total uncertainty:

$\frac{dv_{ball_{i}}}{v_{ball_{i}}} = \sqrt{\left(\frac{dm_{ball}}{m_{ball}}\right)^{2}+\left(\frac{dm_{total}}{m_{total}} \right )^{2}+\left (\frac{dL}{2L}\right)^{2} +\left(\frac{d(1-cos\theta)}{2(1-cos\theta)}\right)^{2}}$

To find the propagated uncertainty in our verification portion of the lab, we do the same thing.

$\ln v_{ball_{i}} = \ln \Delta x -\frac{1}{2}\left(\ln 2 + \ln \Delta y - \ln g \right )$

$\frac{dv_{ball_{i}}}{v_{ball_{i}}} = \frac{d \Delta x}{\Delta x} - \frac{d \Delta y}{2\Delta y}$

$\frac{dv_{ball_{i}}}{v_{ball_{i}}} = \sqrt{\left(\frac{d \Delta x}{\Delta x} \right )^{2} + \left(\frac{d \Delta y}{2\Delta y} \right )^{2}}$

Procedure
We begin our lab by recording the mass of the ball we use to fire into the nylon block. We then measure the length of the string that is attached to the nylon block. We ensure that our ballistic pendulum is leveled so that the ball is fired straight into the block. This is important because the ball could lose momentum if the ball hits the sides of the block before being fully embedded. Our apparatus allows for different firing rates, so we chose to use three notches. After the collision, both the ball and block act as one and travel in the same velocity. Due to the collision, the block move in the direction of the ball and goes up a certain height. We can find the height by subtracting the length of the string and the x component of the string as mentioned in the theory/introduction. We repeated this part of the experiment five times and used the average height.

For the verification portion of the lab, we move the nylon block out of the ball's trajectory as mentioned in the theory/introduction. We fired the ball three times using three notches, and used the average distance from the recordings made by the carbon paper.

Data/Calculations

mass of the ball: 0.0076 kg +/- 0.0001 kg
mass of the block: 0.0795 +/- 0.0001 kg
length of the string (L): 0.195 m +/- 0.001 m

Trial #	Angle (degrees)
1	35.5
2	35.5
3	34
4	34.5
5	36.5
	Average: 35.2

Carbon marking from the verification portion of lab

Distance in x direction that ball traveled

Data for the verification portion of lab:

height from ground to the point of fire: 0.945 m +/- 0.001 m

length from ground below point of fire to the center of impact: 3.34 m +/- 0.001 m

Tuesday, May 2, 2017

April 24, 2017 Lab 15: Collisions in Two Dimensions

Lab 15: Collisions in Two Dimensions

Chris Ceron, Amy, John

Goal of Lab

Look at a two-dimensional collision and determine if momentum and energy are conserved.

Theory/Introduction
For our collision, we can split up velocity and momentum into x and y components depending on where our axis is located, and in which direction the objects move before and after a collision:

Before:		After:
	x	y	x	y
m₁	m₁v_1ix	m₁v_1iy	m₁v_1fx	m₁v_1fy
m₂	m₂v_2ix	m₂v_2iy	m₂v_2fx	m₂v_2fy
Total	m₁v_1ix+ m₂v_2ix	m₁v_1iy+ m₂v_2iy	m₁v_1fx+ m₂v_2fx	m₁v_1fy+ m₂v_2fy

Using the table above, we can calculate the total momentum using our x and y components:

$m_{1}v_{1ix}+m_{2}v_{2ix} = m_{1}v_{1fx}+m_{2}v_{2fx}$

$m_{1}v_{1iy}+m_{2}v_{2iy} = m_{1}v_{1fy}+m_{2}v_{2fy}$

Testing the conservation of Kinetic is much simpler. We look at the Kinetic Energy of both objects before and after a collision to ensure they are equal. In our lab, the second ball is initially at rest. The formula for conservation of Kinetic Energy becomes:

$\frac{1}{2}m_{1}{v_{1i}}^{2} = \frac{1}{2}m_{1}{v_{1f}}^{2}+\frac{1}{2}m_{2}{v_{2f}}^{2}$

In our apparatus, we set the y-axis to follow the trajectory of moving ball before the collision. This allows for the center of mass in the x direction to equal 0, and the center of mass in the y direction to increase as the ball continues to move up the y-axis. We can calculate the center of mass using the formulas:

$x_{cm} = \frac{\sum\limits_{i}^{n} m_{i} x_{i}}{m_{i}}$

$y_{cm} = \frac{\sum\limits_{i}^{n} m_{i} y_{i}}{m_{i}}$

Apparatus/Procedure
Our apparatus is composed of a leveled, frictionless glass surface with a stand that allows for a camera to be placed directly on top of the glass. We chose to use Amy's iPhone as it has the option to record and play a video in slow motion.

A high frame rate allows for us to clearly see the balls in slow motion, and plot the ball's position vs. time clearly through LoggerPro. We recorded two different scenarios for this lab: a small marble colliding into a marble of equal size, and a big marble colliding into marble of smaller size.

Once we leveled the glass surface and placed Amy's phone on the stand, we recorded both scenarios and ensures that both marbles had velocities in the x and y direction after the collision. Once the videos were recorded, we imported them into LoggerPro to plot the movement of each ball relative to time. LoggerPro allows for us to rotate our axes, so we moved the y axis to follow the direction of the moving ball before the collision.

To find the initial and final velocities of each marble, we could take the slope of each line on the position graph before and after the collision.

For the center of mass graphs, we used the formulas stated in the Theory/Introduction and created a new calculated column for those values. Using the columns, we could create a graph of center of mass vs. time.

Data and Calculations

Mass of Small Marble = 0.0048 kg

Mass of Big Marble = 0.0193 kg

Graphs for our first scenario (small marble colliding into an equally sized marble)

Velocity of the moving marble (marble 1) before collision

Velocity of marble 1 after collision

Velocity of marble 2 after collision (was at rest before collision)

Velocities of both marbles broken down into x and y components:

	Before:		After:
	x	y	x	y
v₁	0.39170	0	0.1505	0.1505
v₂	0	0	-0.1404	0.1505

Center of mass graphs:

Center of mass for x and y

Center of mass velocities

Momentum broken down into x and y components:

	Before:		After:
	x	y	x	y
m₁	0	0.0019	0.0007	0.0007
m₂	0	0	-0.0007	0.0007
Total	0	0.0019	0	0.0014

Initial Kinetic Energy:
$KE_{i} = \frac{1}{2}(m_{1})(v_{1i})^{2} = \frac{1}{2}(0.0048)(0.3917)^{2} = 0.000368 J$

Final Kinetic Energy:

$KE_{f} = \frac{1}{2}(m_{1})(v_{1f})^{2} + \frac{1}{2}(m_{2})(v_{2f})^{2}= \frac{1}{2}(0.0048)((0.2128)^{2} +(0.2058)^{2}) = 0.00021 J$

Graphs for our second scenario (big marble colliding into a small marble)

Velocity of X2 after collision

Velocity of Y2 after collision

Velocity of big marble after collision

For this portion of the lab, we did not get the initial velocity of the big marble before the collision. We used a similar fit (X2) as the initial velocity.

Velocities of both marbles broken down into x and y components:

Before:		After:
	x	y	x	y
v₁	0.05646	0	0.04083	0
v₂	0	0	0.05646	-0.0106

Center of mass graphs:

Center of mass for x and y

Velocity of big marble after collision

Momentum broken down into x and y components:

	Before:		After:
	x	y	x	y
m₁	0.0011	0	0.0008	0
m₂	0	0	0.0003	-0.0001
Total	0.0011	0	0.0011	-0.0001

Initial Kinetic Energy:
$KE_{i} = \frac{1}{2} (m_{1})(v_{1f})^{2} = \frac{1}{2}(0.0193)(0.05646)^{2} = 0.00003J$

Final Kinetic Energy:
$KE_{f} = \frac{1}{2}(0.0193)(0.04083)^{2} + \frac{1}{2}(0.0048)(0.0574)^{2}= 0.000024J$

Conclusion
The goal of the lab is to use a two-dimensional collision to determine if momentum and energy are conserved. Based on our calculations and data, we see that neither are conserved. This is due to several uncertainties found in our lab. For starters, we assumed that all of the initial energy and momentum is transferred after a collision. We did not take heat and friction during the collision into consideration. Our frictionless glass surface is also not entirely frictionless. The conservation of energy is based on no external forces interaction with the system; however, the friction found on the glass surface and on the collision of the two marbles interact with our apparatus. In our calculations of momentum and energy, we can see that energy is lost because our values before the collision are larger than the values after the collision.

April 19, 2017 Lab 14: Impulse-Momentum Activity

Lab 14: Impulse-Momentum Activity

Chris Ceron, Amy, John

Goal of Lab

Observe and verify the impulse-momentum theorem

Theory/Introduction

The impulse-momentum theorem relates both through Newton's Laws. The definition of momentum is:

$\vec{p} = m\vec{v}$

Impulse is the relation of the sum of forces in respect to time:

$\vec{J} = \Sigma \vec{F} *(t_{2} - t_{1}) = \Sigma \vec{F}*\Delta t$

The theorem states that both are equal. This means that Impulse (J) is equal to the change in momentum:

$\vec{J} = \frac{\vec{p}_{final}-\vec{p}_{initial}}{t_{2}-t_{1}} * (t_{2}-t_{1}) = \vec{p}_{final}-\vec{p}_{initial} = \Delta \vec{p}$

This conservation of momentum also relies and assumes there is no external forces acting on our apparatus. Our apparatus and lab consists of a collision that demonstrates this conservation of energy. In a perfectly elastic collision, we can predict that the speed in which the cart hits the spring is equal to its speed after the collision. However, because we understand that our apparatus and collision is not perfectly elastic, we can also predict that our final speed will be slightly less than the initial.

Apparatus and Procedure

For the first part of our experiment, we have a frictionless cart on a leveled track. We leveled the track by ensuring the cart would stop moving when gently pushed and when simply placed on the track. We placed a force sensor on the cart and a rubber stopper on the force sensor. On the opposite end of the track, we placed another cart with a spring plunger.

The cart on the track collides to the fixed cart by hitting the spring plunger with the rubber stopper on the force sensor. We record the process using the force sensor and a motion sensor to determine the amount of force the cart experiences during the collision and to record the velocity of the cart before and after the collision. We can find the impulse of our apparatus by taking the integral of a force vs time graph during the collision. We can compare this value to our calculated value through the change in momentum of the cart before and after the collision.

For the second part of our experiment (the inelastic collision), the cart collides with clay that is attached to a piece of wood. The rubber stopper on the force sensor is replaced with a screw/nail. The conduct the experiment similarly to the first, except the cart sticks to the clay after the collision, causing the final velocity of the cart to be 0. To compare the values of impulse between the first and second part of the lab, we provided the carts with very similar initial velocities.

Data, Graphs, and Calculations:

Mass of cart with force sensor: 0.677 kg

Graphs for Elastic Collision:

Initial velocity of cart = -0.364 m/s

Final velocity of the cart = 0.246 m/s

Impulse value = 0.3168 N*s

Calculated value:

$J = \Delta p = p_{f} - p_{i} = mv_{f}-mv_{i} = 0.677(0.246 + 0.364) = 0.413 kg*m/s$

Graphs for Inelastic Collision

Initial velocity of cart = -0.324 m/s

Impulse value = 0.1971 N*s

Calculated value:

$J = \Delta p = p_{f} - p_{i} = mv_{f}-mv_{i} = 0.677(0 + 0.324) = 0.219 kg*m/s$

Conclusion

Our initial prediction for the first part of the lab was that energy would be conserved within our apparatus. However, we can look at our initial and final velocity (0.364 m/s and 0.246 m/s) and easily see that there are some uncertainties that caused energy to be lost. One source of error could be the initial point in which we began to record velocity of our cart. We placed the cart close to our motion sensor, which also happens to be the opposite side of the track. Through our velocity graph, we can see a slope in our velocity vs. time graph before the collision occurred. This shows that our frictionless track is not entirely frictionless.

Our values for the second part of the lab were much more similar, 0.1971 and 0.219. This shows the the force sensor recorded more accurately on the second part. This could also show that the rubber stopper could have caused inaccurate readings, likely due to the friction and slippage happening during the collision. We also noticed that the first part of the lab took some trial and error to get the rubber stopper to line up with the spring plunger. It is a possibility that both weren't lined up as needed to have a perfectly elastic collision.

Overall, we were able to produce similar values to verify the impulse-momentum theorem.

Monday, April 24, 2017

April 17, 2017 Lab 13: Magnetic Potential Energy

Lab 13: Magnetic Potential Energy

Chris Ceron, Amy Chung, John Choi

Goal of Lab

Derive an equation of magnetic potential energy using different cases and quantities of kinetic and potential energy.

Theory/Introduction
Unlike Gravitational or Elastic Potential Energy, we do not have an equation for Magnetic Potential Energy. However, we can derive an equation based on what we know about conservation of energies. At the moment that Kinetic Energy is zero, the energy within the system is all transferred to Magnetic Potential Energy to repel our object. Our apparatus allows for this to occur.

Air track and glider

We angled the air track to seven different measurements. As the angle of tilt for the air track gets larger, the glider gets closer to the repelling magnet. With the data acquired we plotted a graph on LoggerPro and applied a power fit to find the equation of Magnetic Potential Energy.

For any system with a non-constant Potential Energy, the Potential Energy "U" is caused by an interaction between force F. We can mathematically establish that the relationship is:

$U(r) =-\int_{\infty }^{r}F(r)dr$

r = separation distance

We can also conclude that the potential energy in our system as the distance between the repelling magnets get infinitely further away is 0.

Using a free body diagram, we found that at equilibrium, the force of the magnet is equal to the x component of gravitational force.

$\Sum F_{x} = F_{magnet} - mgsin\theta = 0 \Rightarrow F_{magnet} = mgsin\theta$

$\Sum F_{y} = N - mgcos\theta = 0 \Rightarrow N = mgcos\theta$

Once we have our equation for Magnetic Potential Energy, we can verify and test the conservation of energy within our system. We placed our glider on the opposite end of the magnetic and provided an initial velocity, which also gives Kinetic Energy. At a certain distance, the Kinetic Energy is all transferred to Magnetic Potential Energy. As the magnets repel and move the glider backwards, the energy is transferred back into Kinetic Energy.

Procedure

We started the procedure by leveling the track. Then we increased the angle and tilt of the track by adding a book on the opposite side of the magnets. For each increment, we gathered the different angles and the separation distance between the two magnets. After seven increments, we plotted Force vs. distance and applied a power fit to derive an equation F(r). As mentioned in the Theory/Introduction, we could integrate the F(r) to find U(r).

$U(r) =-\int_{\infty }^{r}F(r)dr$

After deriving our U(r) function, we can verify the conservation of energy within our system. We first placed our glider closer to the fixed magnet when the air is turned off. To find the separation between the two magnets, we recorded the distance between the motion sensor and the glider.

$separation = position \: reading - (position \: reading - separation_{initial})$

Once we have our separation value, we place the glider on the opposite end of the track. With the air turned on, we gently push the glider and use our motion sensor to measure the distance and velocity. Through LoggerPro, we create a column for Kinetic Energy, Magnetic Potential Energy, and Total Energy of the system. We use a single graph to display each column, which allows for us to verify the conservation of energy within our system.

Data:

Angle (degree)	separation (m)
2.5	0.0239
3.8	0.0232
5.1	0.0212
6.4	0.0199
7.9	0.0183
9.1	0.0176
10.5	0.0163

A phone application was used to measure the angle of tilt of the air track. We used both the long and short side of the phone to measure the angle, and took the average of both.

Uncertainty of angle measurement: +/-.01

Mass of glider: 0.3471 kg

Initial Separation was 0.017 m

As mentioned in the Theory/Introduction, we used our data to plot a graph of Force vs separation (r). We could then apply a power fit and derive a function F(r).

For function F(r) came out to be:

$F(r) = 1.491 * 10^{-5} r^{-2.741}$

Our uncertainty for A was 7.149e-6 (also equals 47.9%)

Integrating our function gave us

$U(r) = -\int_{\infty }^{r}F(r) = -\int_{\infty }^{r}1.491 * 10^{-5} * r^{-2.741} = 8.56 * 10^{-6}r^{-1.741}$

Making a calculated column for Kinetic Energy, Magnetic Potential Energy, and Total Energy and putting them in one graph gave us:

We see that the total energy within the system is not necessarily constant.

Conclusion
Although our results had a large uncertainty, we were able to utilize our apparatus and data to show that energy is conserved within our system. In the graph above, we notice that the total energy of the system before and after the magnets repel each other is in the form of Kinetic Energy. Some sources of error that caused a large uncertainty were the placement of the magnets. For our apparatus, the magnet had to be adjusted and could have been aligned incorrectly. Throughout our lab, we could see the magnets slightly shift.

For the second portion of the lab, the separation value we used could have greatly affected the outcome of our graph. Even a minute change in the separation value would cause a large uncertainty with our Magnetic Potential Energy.