08. April. 2015: Conservation of Energy

Purpose: The purpose of this lab is to show that conservation of energy applies even to a mass-spring system.

Materials: spring, clamps, mass, motion sensor, force sensor, ruler, logger pro.

Apparatus: You will have a spring hanging down from a force sensor. The spring will hang at some height (H) and the end of the spring will have some height (y) to the motion sensor below.

Procedure:
1. Calibrate the force sensor. Take the height (y) from the motion sensor to the end of your spring. After that take the height (y) of the spring with some 200 gram mass. You don't have to use a ruler because you have your motion detector set up so it will take the height for you. However, you might want to take a ruler to just make sure.
Ours was off by +/- 1 cm.
2. Record the position and velocity graph as you let the spring jump up and down with a 250 gram mass. We did not use 250 grams instead we used 100 to avoid the masses falling off. From the position and velocity you have recorded you can use the equations that we already know..

KE = 1/2 (mhang + 1/3 mspring) v^2
GPE = mass/2 * g * y
Eleastic PE = 1/2 * k * y^2

This was our graph of KE, GPE, and EPE. It is wrong as you can see because this experiment is about conserving energy. That should mean that if GPE,KE, and EPE were equal to each other our total should be a straight line but it wasn't. In our experiment, we did so many things wrong. We moved the motion sensor and we used different masses for our position. We pulled it down odd too so our masses would always change the boyancy of the spring since the masses would be bouncing around as well.

Conclusion: 

     To start off, we calibrated the force sensor wrong because of the fact that we set 499 to be the force with 500g. This is wrong because weight = mg so we did not take into account gravity and that it was the wrong units. Onto the actual data collection, we didn't get a graph that came out to look nice because the masses were always bouncing around so there were hiccups in our curves. Another thing we did wrong was we moved the motion sensor a couple times because we used the one that has a nail on the other side so it was unsteady on the floor. We ended up putting it on a block to keep it steady. However, this was in the middle of our experiment so we did not remeasure the height like we were supposed too. 

The point of this lab was to show that the spring moves with some velocity at some height with some spring potential energy. The tricky part was to treat the mass of the spring and the mass that you let hang differently and not as a whole. We also treat the position very funny. Lets say you have a 6 meter spring compressed, you can say the spring compresses 3 meters if you put a weight double it. But let's say you stretch it with the heaviest mass, it would be 12 meters uncompressed which means that the center of the string is 6 meters. So when we do the elastic potential energy it is the change in y which is what we should have calculated subtracted to the position. It was to show that the energy the spring exerts was supposed to be conserved which means equal. However, our graphs came out very different and wrong because of the things that I mentioned above. Somehow, our energy was not conserved probably due to wrong measurements.

01.Apr.15: Centripetal force with a motor

Purpose: We will come up with a relationship between angular speed and the theta.

We already had the apparatus set up for us, it looked like this:

Materials needed: machine that turns, a ruler to hang the string with a mass at the end

Procedure:

We predicted equations as well as things that were constant. Some things that we knew would not change: length of the string, radius 1, and height of the machine. Some things did change such as radius 2, the angle of the string, the height of the object to the ground. All of these depend on how fast these objects are moving. We also set up the sum of the force equations on the bottom picture.

This is the equation that we came up with to solve for theta. Notice that little h will be measured when we do the experiment and everything else stays constant.

If you were a little confused by my variables above you can refer to this picture for how we labeled the apparatus. We did the sum of forces in the y direction as well as the x direction. In the x direction we set it equal to mrw^2 because the acceleration is in the x direction.

1. Measure the height (of the whole apparatus), radius(from the center to the string), and the length of the string. These measurements stayed constant throughout the whole experiment. Keep in mind that when the thing is spinning (the faster or slower it goes) theta changes.

2. We recorded six trials altogether each time spinning the whole thing a little faster. Every time we moved it faster we had a new time per revolution and a new height at which the object hit our "piece of paper) each time.

Professor Wolf controlled the apparatus as well as measured the height. The object would hit the paper slightly and every time we made it faster the height would increase. We slowly moved the paper up and up each time and when the mass hit the paper, we measured the height. We timed 10 rotations and just divided it in order to rule out some hiccups in the rotations.

3. Using logger pro, we want to find the relationship between angular speed and T

We combined the two forces by dividing it (sum of the forces in the x direction // sum of the forces in the y direction) so we get our new equation tan theta = r*w^2 / g. We want to solve for w (in the orange) and we plugged our own calculations of (theta) and (r) so that leads to the bigger equation you see on the right of the equation. We also did another equation with T and we found that by substituting w=2pi/T so that's where we got the Tequation in blue.

4. To find the relationship we put in all these equations into Logger Pro and then we graphed the w v angle and we got 

We fit a Ax line to it and A should have been about .99. The correlation between the two is linear and almost perfect. Which means they are proportional.

Conclusion: 

The purpose of this lab was to find the relationship between angular speed and theta. What we found was that when theta is bigger it goes faster. The angle is always changing with respect to speed or how fast this apparatus was moving. There was uncertainty in the lab because one, sometimes when the object was spinning it wasn't the same height on each side so on one side it was slanting. So the height there was definitely some uncertainty. Second, the last trial that we did the height measuring device that we had set up did not reach the actual height of the object so we added another ruler to the measuring device and the only way we did that was to use a clamp and add another ruler. This added some uncertainty because we weren't sure if the ruler started out at zero to the height. Another uncertainty in measurements was that it hit the paper at different distances. We tried to move it so that once it hits the paper we took that height but sometimes it would skim the paper sometimes it would hit the side of the paper and sometimes it hit in the middle of the paper which left us with some uncertainty. 

15. April. 2015: Impulse Momentum activity

Purpose: We want to prove that impulse is equal to the change in momentum.

Equations to know:

Impulse(J) = Force*Time
Momentum (p) = mass*velocity
Impulse(J) = Change in Momentum (p) = m(v(f)-v(o))
Also know that since Impulse is force * time it can also be represented by the area under force vs. time graph.

Set up:

track
clamps
cart
motion sensor
force sensor
cart with spring
balance
logger pro


Set it up so that when the moving cart runs into the clamped cart the force sensor is the one pushing into the springy part.


Procedure:
First I calibrated the force sensor and then took the weight of the entire cart ( with force sensor ). The mass is something we are going to need to plug into our equations. We are going to be looking at velocity v time and the force v time graph. I used the Impulse and Momentum file in logger pro which has the force and motion data a 50 data points per second and it also already set the positive direction on the force probe and motion detector as well. After that give the cart a little push and make sure you record little before and after collision.

In this picture we took the integral of the collision from the Force v time and as we recall that gives us the impulse. Now we want to look at the minimum and maximum of the velocity v time graph which is data to find the change in momentum. SO let's look at the equation:

Impulse = change in momentum = m ( vf-vo )  = .76 (.5125 - (- .5545) =  .818 and our impulse was .814

I think the difference was the from what integral to what integral. I could not align the times to be just right so it was kind of off from each other. But overall, if I were to find the exact time of both graphs it would match.

We ran this experiment again but we replaced the car(with spring) with a block with clay. So that when the cart runs into the clay it will just stay there.

Predictions I made before experiment: I predicted that the impulse was going to be larger than the experiment we did before because in this one velocity final is equal to zero. I predict that even under this circumstance that the impulse and  change in momentum will be equal to each other. 

So we used the same logger pro file and we ran the cart into the clay. It just stayed after that and this was our graph: 

Lets test out to see if impulse is still equal to change in momentum: 

Impulse = change in momentum = m ( vf-vo )  = .76 (.01087-(- .4058) =  .316 and our impulse was .309

So this confirms that the impulse momentum theorem is true. I was wrong the impulse is actually less than if there were to have a v(final) not equal to 0 at the end. BECAUSE when v is zero it means that the cart has fully compressed into the "spring" and the force would be at its greatest.

Conclusion: 

In this lab, there was errors probably in mass. The directions said to add weight to the cart (I'm guessing to have a better reading on the force sensors) however we didn't add any extra weight and it would have messed up our consistency if we did the clay one with weights and the other one without. Questions to think about was what was the net force exerted on the cart just before it starts to collide? I think the net force was around zero because if you look at the graph before the collision it went up down up down but all near zero so adding up the areas of those would have been around zero. The magnitude of force exerted on the cart was when it was fully compressed and velocity was zero. I think the net force after the equation was also zero. And that the time that the collision takes is almost half a second from our graphs. 

15. April. 2015: Magnetic Potential Energy Lab

Purpose: Show that conservation of energy applies even with our own made up equation (from our experiment of course no, you can't just choose any random equation)

Materials
Air track (one that if you turn it on the glider is barely moving), glider (with a magnet that repels to the end of the air track), angle measurement, logger pro, balance, ruler, books

Procedure:
BACKGROUND INFO: So first of all we do not have an equation for the potential energy of a magnet. So that's what we have to find. BECAUSE when our cart is moving towards the magnet we agree that there is some kinetic energy and when it reaches the magnet that repels each other it heads back in the same direction which is from the potential energy of the magnet. To do this we will look at the angle the cart is at and the distance of how close it gets to the magnet.

1. setup:

Okay, so first we started with a lower angle then measured the distance between the two magnets (one on the end of the air glider and the other on the end of the track). Each time we added around 4-5 books in order to get a higher angle which we wanted to be around 5 degrees more than the last data point. We put this data straight into our logger pro graphing page. Typically we want more than 5 data points. In our lab the book stacking got really high and one person was holding the air track so it might not fall off, one person was measuring, and one person was holding the books. 

What we got from our data was the angle and x (distance). We know that these equations like radians more than degrees so we should have calculated radians first but in the picture we did it after. From that we know how to find the force of the magnet by the equation m*g*sin(radians). And now you are ready to graph.

From our graph, we took the power fit and got ourselves an equation of the magnet. As you can see as the distance gets bigger the force between the magnet got smaller.Our power fit through the line gave us F= .000508x^(-1.721). Now that we had our force equation does not necessarily mean that we have our potential energy function. Keep in mind this is what we are trying to find.

So from common knowledge we know that potential energy is negative the force of the integral with respect to "R" we changed our x value to r. It's just a variable. So we calculated for our potential energy by taking the integral. This is the potential energy of the magnet. We have taken a random magnetic force that we did not know the equation of and made an equation.

BACKGROUND INFO: so now that we have the potential energy we should have some way of verifying that this is correct. So we will now take out a motion sensor and have it at the end of the magnetic side. This has to be kind of behind the motion detector by some distance. Which means we do have to find the "new" distance.

First: take all of the books under the air-track and have it leveled so that the glider is barely moving (in a perfect world it wouldn't move unless you give it a push). The motion detector should be set up like mentioned above. We will give the cart a little push and record on logger pro. The cart should edge up to the magnet but not touch and then it should start going back the way it came from. 

The graph should have a position v time and also a velocity v time. The position vs time should go down then slowly curve back up like a parabola. From this we get data points yay. 

Okay so the graph originally gives us time, position, acceleration, and velocity. We can calculate Kinetic Energy by the equation 1/2 * m * v^2. The new x from what I mentioned above. The Potential energy is the equation that we have calculated before. The total is the kinetic energy + potential energy. We want a graph of the kinetic, potential, and also total. 

the total is not as "constant" as we would like it to be. Our ideal graph is if the total energy was at least kind of constant. However, we realized that our potential energy spiked up a bunch so we messed with the power so if we kind of changed our power the potential energy spike would be lowered.

If I changed the power by .05 our potential energy lowered significantly. This is due to the error in the lab which is a lot. A little bit off can mean a significant change in our graph.

Conclusion: 

There was a huuuuuge room for error in our lab. We would change data by .05 and it would make a significant change in our graph. I assume this is from 1. the friction on the air track. (most of the gliders slid on the air track when it was ideally supposed to stay still) 2. the magnets were different which meant that the distances could have been different each time they were next to each other. 3. sometimes the glider would go past the magnet so the magnet would push the cart off of the track (we always tried to redo it if this happened. Although our lab did not come out as ideally as we wanted it too. It was close enough for us to see that even with a magnet conservation of energy still holds true and that the potential energy of the magnet was the (same-ish) as the kinetic energy. The kinetic energy went down as potential energy went up and as potential energy went down kinetic energy went up.

06. Apr. 2015: Work-Kinetic Energy Theorem

 What is the Work- Kinetic Energy Theorem?

The work done by the net force on a particle equals the change in the particle's kinetic energy.

W(total) = K(2) - K(1) = change in K

Purpose: We will calculate the work done in a stretched spring through the force vs distance graph. We want to prove that the area under the force v position graph is equal to the work done in the spring graph.

Materials: ramp, cart, motion detector, force probe, spring, balance 

SET UP: 

The force sensor is attached to the spring which is attached to the cart. The calculator was there so that we can make the spring straight and unstretched. We added masses on our cart, however we ended up taking it off for our experiment. It doesn't matter whether you have masses or not. If you do, you just have to take in account that the mass of the cart is different.

On the other side of the track is a motion detector to monitor the position of the cart. 

Procedure: 

1.  set up :) of course.

2. Start recording on logger pro and give your cart a little push. This will generate a force vs distance graph. The slope of that graph will give you the spring constant. 

3. Now we will pull the cart close to the motion detector but not too close or else the motion detector won't read it. Then record and let it go. This will give you another force vs. distance graph.

3. Drink a glass of water because it gets a little confusing right here. 

4. Data should include time/position (from your graph). 

Velocity is from the derivative of "force", "time." 

Acceleration is the second derivative of the force v time. 

KE (kinetic energy) is following the formula 1/2 * m * v^2

Force should have been on there already. 

Our Fadj (Force adjusted) is because of the fact that our graph did not start at 0 so we subtracted it by the difference so that the beginning of our graph would be zero.

It would look like this: 

DATA

5.  After you have this, you should graph the position vs force graph and the position vs. Kinetic energy graph on the same screen.

6. If you take the integral of your position vs force graph the integral under the curve should equal to your KE or close to it. Ours was a little off but we took more and more area it started getting even closer and closer. 

This is the first integral

Second integral

Third Integral

Conclusion

Through this experiment we saw that the work done at any position is the same as the kinetic energy at any position. There were ways to mess up this experiment. One, there could have been friction on the track. Second was the force position graph did not start at zero (which was what we had). Make sure the position is zero and that your force sensor is reading it right. After we got rid of that data the values were a bit closer. So the work and kinetic energy theorem holds true that it is equal. 

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There is another part to this experiment where we watch a video of how they would have done the same experiment but in the olden days. 

The professor has a machine pull back on a rubber band that is attached to an unknown mass. The cart passes through two photogates a 15 cm apart. Since we know the distance and the time of the cart passing through both we can calculate the final speed and kinetic energy of the cart. 

The woman also pulls on the board when the data is printed on a piece of paper. She supposedly pulls on it with a constant rate. She repeats the experiment about 4 times.

It had a bunch of squiggles but the squiggles made shapes that we did know how to find the area of. 

So we broke it down into shapes and found areas and added it up. 

Conclusion: 

This was just an addition to the previous lab, however the thing is this one had much more uncertainty than our labs. Which I thought was the point of this lab. The woman pulled the board at what she wanted to be constant speed but it wasn't. There was also friction in the board when she was pulling the lines were always off. And she took the best shape that the squiggly lines made and took the area of that like we did. The squiggly lines were in about three different places so we assumed the middle of the three was the most accurate. 

18.Mar.15: Centripetal Acceleration vs. Angular Frequency Lab

Purpose: The purpose of this lab is to figure out what is the relationship between centripetal acceleration and angular speed by looking at the acceleration vs angular speed graph that was represented by a spinning rotating disk with an accelerometer. 

To set up this lab, Professor Wolf had it set up for us but it was a rotating disk attached to a spinning wheel that we could change how fast it went. There was also a photogate on the side. (It is a sensor where if there is something that comes in between it it takes the data. 

BACKGROUND INFORMATION 

An object that moves in a circle that is rotating, it is constantly changing velocity. 

WHAT?! How does the velocity change? 

It changes because the direction of the object is constantly changing which means that it is accelerating. This is centripetal acceleration and there is a centripetal force that pushes an object inward to the center. 

The angular speed of an object is measured by taking radians divided by time (sec). 

SIDE NOTE: linear velocity = radius * angular speed

The angular acceleration is measured by v^2/ r or (angular speed)^2 * r. 

The lab itself was really fast so PROCEDURE

1. Since we wanted to find how much time it takes to make one rotation and the acceleration. We 

started off with 4.4 volts and moved on to however many data points you want. We took data and it would give us time and how many rotations. We took a chunk of time vs a few rotations and just divided it in order to find the time for one rotation just to make sure if one rotation was off then we'd have an average of many. 

The mass of the object was .1389 and you'd need this later to find the Force which is mv^2/r or in other words m * (angular velocity)^2 * r  so I wrote down the data.

Professor Wolf read this out for us to write down so the whole class had the same data. We would subtract the time(final) - time(start) and divide it by however many rotations we decided to get the data points from. That would give us the time per rotation. We already have the acceleration due to logger pro for each data point.

Here we graphed our acceleration vs angular speed which gave us a value of close to 1 (.997) which means that there is a linear correlation between the two. 

Conclusion:
Errors in this lab could have occurred with friction of the rotations. There was definitely friction sometimes which meant that there could have been little errors in the reading. We reduced this error by taking many rotations and taking the average rotation for those. The point of this lab was to find the relationship between acceleration and linear speed and we did that by calculating the angular speed and graphed it with respect to time. We found out that the correlation between the two is proportional.

23.Mar.15: Trajectories

Purpose: We will set up a scenario that resembles a ball being thrown off a cliff and predict where it lands.

Materials:

• Carbon paper
• Ring stand
• Clamp
• paper
• Steel ball
• meter stick
• Angle Degree measurement device
• Wooden board (Part 2)

You will set it up like this however, in the first part you will not have the wooden board there. 

Procedure: 

1. Make sure you mark where you let go of the ball because you are going to keep using that distance. Watch approximately where it lands on the floor and put the carbon paper over a white sheet of paper there. After 3-4 trials measure where it lands in the x direction and in the y direction. 

OPTIONAL: use a degree measurement to make sure your meter stick is straight in x and y direction. 

2. From this calculate the speed of the ball using the trajectory equations. Make sure to separate the x and y component because they have different velocities, accelerations, and height. Only their times are dependent on each other. (This is JUST like homework problems except we actually came up with the numbers from our experiment) :) 

our speed turned out to be 1.41 m/s

First we used the y motion to find time using x=V(initial)t + 1/2(a)(t)^2

where v(initial) is zero. After we found time we used the same equation but with the x side now to find the v(initial) which equals v(final)

3. Now add the wood cardboard like the picture shown above. Drop it a couple times to approximate where it will land and tape carbon paper to the board. You will then have to find the angle and using calculations determine your value of d (distance). This is where the most trouble came out of our lab because different angle measurement tools gave us different angles. We decided to just use the one that made most sense to our data; however, there is a big uncertainty because the measurements were about 10 degrees apart. 

our calculation for distance

4. Then calculate the change in distance. We started off with the 1/2*gravity*time^2 because Vinitial is zero. and set that equal to distance * sin theta which is the length of the board. We also found that x = vo * t which equals to the distance * cos theta which is the length of the floor. SO we solved for time in each equation and set those equal to each other because time is dependent of each other throughout. We wanted to solve for distance. After solving for distance we calculated the uncertainty.

5. We measured our actual value of x, y, and theta from our experiment. The dx, dy, and dtheta came from uncertainty that we could have been off by by our measurements. In other words it is the +/- of our measurements. You'll see that all of this was listed on the left hand side of our board.So we began with the distance formula that we used to solve. We took partial derivative to each variable (x, y, and theta) which gave us our dd equation in the orange. After coming up with our change in distance formula we plugged in the numbers which is in the green at the bottom of the board. Our distance has an uncertainty of +/- .0159.

Conclusion/Errors:

The biggest error that our lab made was the angle. We measured it with three difference devices and got three (almost totally) different angles. We decided to go with the angle that gave us the best result which was 49.2 but that could have been the wrong angle. Another error was the measurement of the ball. The ball landed some-what in the same spot but it was scattered. So we took the middle one and +/- those few meters for uncertainty.