6th April 2015: Conservation of Energy Lab

Lab 11: Conservation of energy with a Spring

Purpose: The purpose of this lab is to essentially prove the theory of conservation of energy using a spring and a mass to show the oscillations of motion as a weight is applied to one end of the spring and a force sensor is calculating the energies as the move back and forth.

Spring constant = K

Procedure:  We first began by getting a spring, clamps, a motion sensor, force sensor, and a small mass. We use the clamps to attach a rod to our desk which went up some height and then an additional bar was used to clamp a pole above the spring so that a force sensor could be placed above the object and measure the forces. On the floor was motion sensor that was picking up the displacement of the object and spring as they oscillated. By knowing the total change in distance from the motions sensor and the force that the mass exerted on the spring we came up with a value of 14.86 for the spring constant of our mechanism.

This is a picture of what the mechanism looks like in class with the spring
hanging from the force sensor. 

After we found the spring constant we now need to find the forces of potential, kinetic, and total energies. By using the motion sensor for calculating the displacement and the force sensor to calculate the force, we were able to solve for our energies by inputting calculated columns into logger pro. 

Kinetic Energy (KE)= 1/2mv^2

Potential Energy mass (PE)= mgh

Spring PE= 1/2kx^2

KE spring= 1/2mv^2

PE gravity spring= mgh

Total Energy= Sum of (initial) KE+PE=PE+KE (final)

By inputing the formulas we came out with several data points that made up several energy curves as follows: 

These are all of the energies that we solved for
by using the different formulas we were given.

By solving for the calculated equations above we got the results from the energy graph in the picture above. With the groups knowledge of conservation of energy, we know that the potential energy of both the mass and the spring should be equivalent. We know that the have a relationship which they mirror each other meaning that if PE of the mass is increasing the PE of the spring should due decreasing and vise versa. In our data we notice that the graphs in fact do correspond with each other and total energy remains constant throughout our graph for the entire experiment. Since we found out the energies and matched those results to our predictions and knowledge of what we know we find that conservation of energy does support the spring and the mass. All in all our experiment was successful in proving conservation of energy within our system.

13th April 2015: Magnetic Potential Energy

Lab 12: Magnetic Potential  Energy with a Cart

Purpose: To prove that there is conservation of energy within the system of an unknown magnet and a glider that is on an air track. By proving an equation for magnetic potential energy then we can prove that the system does have conservation of energy.

Procedure: The materials consisted of a large frictionless air glider and an air track. Attached to the air track was an inverted vacuum which allowed the gilder to be frictionless(almost). We also used books in order to give an angle to the the air glider. On top of those materials we also used magnets which we put at the end of the air glider and the opposite end of the track in order to have a repulsion as the glider glided down the track. We also used logger pro in order to set up a motion sensor that we used to measure the displacement of the glider as it moved closer to the magnet on the opposite side.

These are the materials we used in the experiment the motion sensor is not visible in the picture but is off to the right. 

After we had our track set up and running. We ran several trials which consisted of different angles and each time we moved the angle up we measured the force. Newtons second law of F=(Mass)(acceleration), was used to solve the force of the glider moving on the track, we derived an equation of f=mgsin(ø). We then used logger pro and used the five different forces from the five new angles to create a calculated column, then we compared it to the distance the magnets repulsed. This separation distance vs. force graph was carefully examined, by finding the curve fit of the graph using a power equation, we were able to establish an equation for the magnetic potential force of our contraption. 

Our trial with several books
so that we can increase the angle of our mechanism,

This is the power curve fit to our force vs. separation
equation. Power-fit= y=Ar^B

After we had found the magnetic potential energy equation we made more calculated problems so we could try to prove the conservation of energy. First we calculated a column for the the total energy, which consisted of the two new calculated columns we made for both potential and kinetic energy. We know that the initial potential energy and the final kinetic energy is equal to the total energy of the system. And in our case we are trying to solve for the magnetic potential which we did and to prove it has conservation of energy we had to set our magnetic potential energy equal to the total energy which is potential initial energy plus final kinetic energy. 

This is a graph of all of the calculated columns.
The top is total energy, red is potential, and orange is kinetic.

When examining and analyzing our results we find that there is an error in our data, with the total energy being equivalent to .095 jules approximately, we find that our data is off. For one, it should have the same position as the kinetic energy graph and mirror its every curve. There was most likely an error when we first ran the trial and did the power fit. The most reasonable answer to the error in our graph was the power fit curve which was not perfectly fit to our data because the surface wasn't perfectly frictionless. And the force which we applied to the glider in order for it to come into repulsion contact with the magnet may have been a different applied force every time causing our data to be inaccurate. All in all mathematically we found that energy was in fact conserved, but we could have ran better trials to assure that our data came out to be more sufficient. All in all we had a successful experiment in proving magnetic potential energy does in fact have conservation of energy.

8 April 2015- Work Energy-Theorem Lab

Lab 9: Work Energy-Theorem

Purpose: We examined the proven conjecture of the work-energy theorem that is equal to the kinetic energy graphs area under the curve which is the total work being done.


Procedure/Materials: In order to begin the experiment, we first began by using a metal track that lay across our desk which we placed a cart on and attached a spring to. At one end was the motion sensor which was calibrated to zero, on this same end the cart has a spring attached to the sensor that is unstretched but when we do stretch it the motion sensor will record the data. As soon as we calibrated our motion sensor we began the experiment which involved us pulling the cart back a certain distance. After doing so this distance was measured and then we began recording our data using logger pro. As we released the cart logger pro began measuring data on the force vs. time graph.

This is the force vs. time graph. 

This is a picture of the experiment we did in class to do the force vs. position graph. 

After recording the data we added another column in order to incorporate the calculated kinetic energy of the system deltaKE= .5mv^2. We then used this formula to compare and contrast the value of kinetic force vs time and the total kinetic energy vs time over that same exact time interval, there was some small error. We had to use three separate graphs to show the three different points we tested for each kinetic energy. 

Kinetic energy is .165 and the integral under the force vs time diagram is .152

Area under the curve .344 vs. kinetic energy of .365

area under the curve .453 vs. Kinetic energy .432

These three different intervals are the work energy theorem which is represented by force x time. The area of this integrated curve should be equal tot he kinetic energy with respect to time. As logger pro compared the two graphs we found that this relationship is hared between work energy theorem and kinetic energy, although we had an error. Our data was only of by 1 or 2 percent which could have easily occurred when the motion sensor was measuring data or not zeroing our equipment correctly, leading us to the conclusion that we had a successful lab in learning about the work energy theorem and its relationship with the area under the curve being equal to kinetic energy.

01 April 2015- Centripetal force with a motor

Lab 7: Establishing a Relationship between Angular Speed and Angle of Rotation


Purpose: The purpose of the lab is to find a relationship between angular velocity(w=mv^2) and the angle that the object is rotating at represented by theta (ø).

Procedure/Material: In this experiment, a mechanism was created by attaching a long metal pole to a large tripod which controlled a wooden stick that spun in a circular motion at the very top. This wooden stick ran parallel to the ground. A string was attached to one end of the stick which was sent into a circular motion around and around as it was powered by an electric generator which allowed a constant speed for the conical pendulum to rotate. We did this over again with different wattage to give a large speed which we would calculate theoretically and experimentally to prove our relationship.

Before we start with the experiment and run several trials changing the angular speed, we must first measure the total heigh of the mechanism which is taken to be 2 meters tall. We also needed to know the length of the string because it stays constant at 1.654 meters. There is also a rubber stopper at the end of the string just for safety purposes.

This is the mechanism that we were observing and analyzing.

This is the diagram of what the mechanism looks like
and also what we helped use to solve for the different
variables in order to find a relationship.

Once we got the electric generator to turn on we began watching the top of the mechanism spin allowing the rubber stopper attached to the string to begin spinning, as it reached equilibrium and was stable we began recording the times on our phones and the amount of revolutions (complete circle) done with each different wattage of the generator. In order to find the height from the ground of the spinning rubber stopper, we used a metal rod which had a paper taped at the end. We would higher and lower the height of the metal pole in order to get the best height, we knew how high to put it when the rubber stopper just skimmed the paper. This height was then recorded and represented the distance of h. We repeated this process eight times with eight different heights which we found when the professor began to higher the voltage of the generator allowing the wooden stick with the string and rubber stopper attached to rotate faster. This data will help us to solve mathematically for the angular velocity.

This is the tool used to measure the different heights from the ground of the rubber stopper.

After we worked out the data was used to find the results by using an excel spread sheet. We also used some white boards to show the work that we did too. We first used the first diagram that helped us to solve for the angular velocity (W). By using the force diagram we were able to use trig to find the angle at which the string and rubber stopper swung at (ø) by first finding the distance from the pendulum to the rubber stopper vertically and then using the length of the string to derive a trig equation which we took the cosine inverse of to solve for the angle .

This is the work which solved for the angular velocity that we first started with.

After we solved for theta to find the angular speed we had to convert the answer into radians. We then used the proven physics equation to solve for the angular velocity. We used the physics equation  W= 2(pi)/T and compared it to the equation we derived using our mechanism 

W^2= (gsinø)/cosø(d+lsinø). 

This is the work we used to solve for theta (ø) which is a function we derived.

The equation derived will be compared to a physics law that has been proven which is W(angular speed)= 2(pi)/T. Which is shown above.

Now we plotted an equation for the lows and highs of angular velocity using the derived equation and compared them to the angular velocity of the given W=2(pi)/T. In doing so we found below that the slope of the line is .971 which is the period equation (W= 2(pi)/T).

We found that when we solved for the lows and highs of the equation as well, that the results were almost identical. And the slopes of the lines were almost completely equal. 

This the slope of the high equation meaning that the highest error value it could have had. Which we found the slope to be .9824 

This is the slope of the low equation which is the lowest error value it could have had. We found the slope to be .9696.

The slopes show that we were off by only 2 percent from the from the hypothetical value since the slopes are not completely identical. But as each trial went on the graph shows that the angular velocity did increase. The small error we had could have been from measuring the height of the mechanism as it rotated or when we measured the period which involved several decimals which may have cause the error because of rounding. In conclusion, the experiment was successful in finding a relationship between omega(W-angular speed) and theta(ø-angle). As the angular speed increased the angle also increased showing that the large the angle the higher the angular speed of the object and in this case the rubber stopper. 

25 March 2015: Centripetal Acceleration

Lab 8: Centripetal Acceleration

Purpose: Today in class we used a wheel powered by a small motor which was rotating, known as an accelerometer, which we placed against a a flat disk which then spun in continuous circles which helped us to record the data as a piece of tape passed through a photo gate. By doing so we were able to establish a relationship between angular acceleration and time which lead us to establishing a relationship between centripetal acceleration and angular velocity. Ca=mv^2/r. w=2(pi)/t.

Accelerometer is at the farthest left and the photo gate is the small black object that is attached to the pole under the paper towels.

Procedure: Due to time constraints, professor did the lab for the whole class. He first began by placing the accelerometer next to the disk with the taped photogate. By using the small generator to the far left, (next to the scooter wheel) we were able to power the wheel with different hertz of electricity allowing for the wheel to accelerate at different speeds. By using logger pro the photo gate captures the revolution per second of the wheel spinning. We used this data to record the acceleration, period, and angular acceleration.

Angular acceleration vs. acceleration 

This is the data for acceleration,  rotations, period, and centripetal acceleration. 

Results: As we plot our data in the table we find that our results are true and the radius of the disk spinning is nearly identical to the slope of the line, which we find by comparing our acceleration and omega. We find that the radius is .1386 nearly identical to the spinning wheel, off by only a few hundredths, showing that angular velocity and acceleration do share a relationship.

16 March 2015: Coffee filters

Lab 5: Coffee Filters

Purpose: Today we used coffee filters to determine the relationship between air resistance force, as something is falling, and the objects speed as it free falls. We are using the equation Fresistance = kv^n which should support the relationship we are looking to find.

Procedure: Beginning the experiment, we dropped coffee filters from the second floor of the balcony in the information technology building at Mt.SAC. We used video capture in order to project the fall of the coffee filters by marking every position every 1/10 of a second. We did this five times and each time we increased the number of coffee filters until we did 5 coffee filters. The fall was then analyzed and recorded by logger-pro were we found our data that we were going to test out using excel by analyzing our answers.

This pictures above show our data recording on logger-pro(left-side) and the data taken with the different number of filters(right-side).


Logger-pro is able to make a graph by marking the distances after each time frame that the filter fell. Which is how we came up with the position vs. time graph below. We also used a curve fit to find the slope which will help us to eventually find the acceleration since we now know the velocity.

We then used the force and velocity information to make a graph that would help us toy solve for our k and n values. We would have then got several values and compared them with our values of the function we created by solving for our own k and n values in terms of coffee filters. When using excel the values should be similar if not then you most likely had an air in your experiment with measuring the the position of the coffee filters as they fell to the ground. 

Final Thoughts: All in all, the lab had some errors in calculations due to the lack of equipment , if the equipment was better our data could have been more accurate and since the filters were white they were very hard to see as they fell closer to the ground. We still were able to find a relationship between the force resistance of air and kv^n but even though our calculated value wasn't exact to the expected value they were very similar. 

11 March 2015: Propagated Uncertainty

Lab 4: Propagated Uncertainty

Purpose: The purpose of this lab is to understand how to mathematically solve for the propagated uncertainty of several masses. Then use this knowledge in order to calculate three masses of different densities and find the uncertainties of this density.

Part 1: We took three metal cylinders that are made of copper, aluminum, and steel.We then used a scale and found the mass of each object and used calipers which helped us to find the dimensions of each object which we would then use to solve for the density of each of the cylinders.

The picture above shows the materials we used, the three masses are steel,copper, and aluminum which are on the bottom left. The scale we used to find the masses is also at the bottom left. And the caliper is the measuring instrument also on the bottom left in the picture. All of our data was recorded and the next steps taken was determining the densities which we found by using the volume of a cylinder (pi*d^2h/4). 

This is how we solved for the densities of each of the cylinders. We can then begin to find the propagated uncertainty by then differentiating the diameter, height, and mass, we were able to add up those uncertainties to figure out the exact propagated uncertainty.

These calculated values will now replace the original uncertainties of .1g,.01cm, and .01cm.

aluminum: 2.61g/cm^3 +/- .03

copper: 8.73g/cm^3 +/- .12

steel: 7.66g/cm^3 +/- .11

Part 2: The second part of the lab was found by choosing a tension diagram made by the professor in class. Where we first found the able of measurement of tensions and then had to solve for the mass of the equation. 

This is the work above on how to solve for the hanging mass by setting up vectors directions for the tensions. I solved for the unknown ganging mass by solving for the mass in both the y and x directions of vectors. 

Final Thoughts: The main purpose of this lab was for us to practice our free body diagrams for part two and also being able to solve for an unknown mass given two tensions and a hanging mass. We learned in part one that most things have propagated uncertainty which we now know how to solve using calculus.

02 March 2015: Non-Constant Acceleration

Lab 3: Non- Constant Acceleration (Rocket/Elephant)

Purpose: The purpose of the lab is to solve an equation with a non-constant acceleration mathematically by using an excel spreadsheet.

Problem: There is a 5000 kg elephant on skates who is on a hill which is frictionless, as he comes to the bottom of the hill. The rocket is then thrusted on and lets out a force of 8000N(newtons) constantly, but it is pointed in the opposite direction of where the elephant is moving. The mas of the rocket is its mass before the rocket is on, 1500kg, minus the fuel which leaves the rocket causing it to lose fuel at 20kgs a second. Meaning mathematically the rockets mass is 1500-20kg/s*t=M(rocket).
The question is asking us to find the distance the elephant has traveled before running out of fuel or coming to rest.

Procedure: In order to solve for the equation we worked it out mathematically and we also used excel to start a spreadsheet in order to input different data and test our results to find a time to then find a distance that the elephant has traveled. Mathematically speaking we labeled all of our known values.
M(rocket)=1500kg-20t
elphant mass=5000kg
v(intitial)= 25m/s
masses together= 5000+1500-20t=6500-20t
We can use the equation of the masses to solve for a force value which gives us our a(t), acceleration function.

The picture above shows the integral of a(t)= Fnet/M(together) which equals F(t)= -8000/6500-20t(all functions integrated from 0 to t) a(t)= -400/325-t. Leading to the equation v(t)=25-400ln(325)+400ln(325-t). After taking the integral of this function we find the position equation x(t)= (25-400ln(325))t+400(t-325)*ln(325-t)-t+325ln(325), which we need to find the time he traveled in order to find the distance that the rocket and elephant traveled. We found that time was equal to 19.69 seconds by solving for velocity at V(0). We use this t value to then solve for the distance the elephant had traveled which approximated to about 248.7. To prove our mathematical work we used excel spreadsheet to calculate the a(t),v(t), and x(t) by setting up several different columns where we input equations with respect to what they represent. 

Each column was labeled,( t) represents time, (a) represents acceleration, (a-avg) represents the average acceleration on a given time interval, (deltaV) change in velocity on a given time interval,(v) represents velocity, (Va) represents the average velocity on a time interval, (deltaX) represents the distance traveled in a time interval, (X) represents the total distance traveled completely. We used a small time interval to guarantee accuracy. A was represented by the acceleration equation we integrated. A-Average was found by a(final)-a(initial)/2. deltaVelocity = acc*time.Velocity = V(initial)+ deltaV(change in v).Velocity (average)= Vf-Vi/2. DeltaX= A-avg*t. X is the position which is solved by plugging in each time for x(t) equation above.

Class data: In class we were asked to have a very large time scale in order to check our work so we had the columns go down an addition two hundred rows. By simply highlighting the row and the equation used to solve for the respective value we got around 230 new columns with new data. We continued to do so with every row until our entire data had each column down to about 230 rows each. 

The second way of solving is by going to the velocity column and checking where time is equal to zero because we want to know the distance that the rocket and elephant traveled after coming to rest when the rocket runs out of fuel. We find that our data matches!!!! The excel sheet calculated a 19.6-19.8 time window since we made our time interval so small were were able to see just what a tenth of a second would change. Our time we found was 19.69 seconds approximately which is between the time window we found on excel. 

Final Thoughts: Sometimes when solving numerically and analytically the resultant is equal making your working easy knowing that you did the correct steps mathematically to solve for the problem in this case a non-constant acceleration problem involving an elephant on skates thrusted by a rocket strapped to his back. The elephant traveled a distance of 248.7 m before coming to rest which we proved in both methods, numerically and analytically. The thing that allowed us to get such accurate data was the fact that our time interval was shortened to a tenth of a second allowing more data and more accurate numbers to be calculated. This lab helped to retouch on skills on excel to analytically solve something by first proving it mathematically. 

02 March 2015: Free Fall Lab and determining gravity(g)

Lab 2- determining free fall (g= 9.81m/s^2)

Purpose: The purpose of the lab is to prove the theory and validate the statement that, "In the absence of all other external forces except gravity, a falling body will accelerate at 9.8m/s^2.

Materials/ Procedure: The apparatus consisted of many parts including a long metal tripod stand which is used as a track for the falling electromagnet. Before beginning the process the wooden cylinder with a ring attached to it must be attached to the electromagnet, acting as a conductor for the electric spark to hit the paper strip. This electromagnet is powered by a spark generator which allows it to mark its distance as it free falls from the top of the tripod to the bottom. There is a long strip of spark paper which is zapped with around 60Hz of electricity leaving dark enough dots to see where it was zapped by the electromagnet. After the apparatus is done recording the dots zapped on the paper the first part is done now the mathematical solving and analyzing comes into play.

part 2 procedure:  Since we were limited to only one apparatus we were given strips that were made from previous classes in order to give us more time to work on the mathematical part of the lab by using excel. Every 1/60th of a second was the time frame used for the electromagnet to shock the paper strip. We determined a starting point of the paper and then measured the distances apart from each of the dots generated by the spark. The direction is shown but can also be found by analyzing the distances from left to right as they get larger. We were then asked to make at least 20 or more data points that would be plugged into an excel sheet and used to solve mathematically.














 









The table to the right is a list of the data we took. The 1st column is representational of time which in our case we took every sixtieth of a second which explains why we got such small decimals for time in column 1. The 2nd column is the distances we recorded between each point of the falling electromagnet the first point we recorded being the origin. The change in distance is represented by the 3rd column which means how much more it traveled from the previous distance. The 4th column is when when we doubled our time frame for the shock to hit the strip which is every hundred and twentieth of a second. Which would be the midpoint between two of the sixtieth of a second intervals. The 5th column was the average velocity of the falling object on a hundred and twentieth of a second time interval. Using this data we set up a time vs distance graph.

Distance vs. Time

We found out by using curve fit that the line is representational of a parabola. This equation is almost identical to the position equation used in physics 2A (x(t)= Xo+Vot+1/2at^2 ). By using this graph we can now have an equation where we can derive the v(t)(velocity vs. time) equation and also derive the a(t)(acceleration vs. time) equation which will give us an acceleration of 9.56m/s^2. This differs from the already given value of g= 9.8m/s^2. This tells us that somewhere along the way we had error or the computer may have miscalculated. 

Velocity vs. Time

We derived the acceleration from here by taking the derivative of the equation given. 

V(t)= 954x+42.529

Giving us a g value of 9.54m/s^2 which isn't the g value of 9.8m/s^2.

Class data: We did an entire graph together as a class and found that the average class values were also off from the actual value of g (9.8m/s^2). The picture on the bottom shows the first column as the solved g values for each group.  The second is the difference between the class average at the bottom of column one and the actual deviations of groups. The third column shows the average of the second column squared and square rooted to find the mean of the deviation from the class average to their solved g value. The class average was about 20.12. 

We were able to find the range of the g value as a class which was between 916 to 996 giving a possible error of about +/- 2. 

Given by the lab packet, we found the percentage of it being between our data was around 95 percent. An error could have been made with measuring accurately causing our numbers to be off by a few tenths. 

Final Thoughts: Although we did not solve an exact value for g we did get fairly close in our data and identified a relationship using excel spreadsheet to solve for the g of our strip paper. Our class average of about 950 and our groups value of 956 differs in the excepted value of 980. Which gives us reason to believe there was a small error in recording the data causing our numbers to easily differ from group to group. We learned that by doing this lab we will run into some trial and error after analyzing our data. We also retouched on our skills with excel spreadsheet and how to input equations to get the results, although we didn't solve for g exactly we did learn about excel and how solving mathematically may not always get you the exact value, but doesn't mean your wrong. 

23 February 2015- Inertia Balance Lab

Lab 1: Inertia Balance

Purpose- In class we used a known mass and c- clamp to measure the period of an inertia balance, as we added weight(kg) to see how it affected the period(t) of the inertia balance. We found that they mathematically share a relationship within the graphs and will use the data from logger pro to prove that the data matches a power law equation which best fit the curve of the graph as we analyzed on logger pro.

Materials: Inertia balance, masses 0-800g, eraser, calculator, c-clamp,LabPro, and Photogate.




Procedure: The first part of the procedure was to set up the c-clamp stand and the inertia balance so we could use a photo-gate as a motion detector. This helps keeping track of the period by picking up light rays as the pendulum goes through its complete oscillation the flashing red light on the photo-gate acts as the sensor  showing the time it takes for it to go back forth( one cycle). We did this eight times adding 100 g masses for every new trail we did. After recording the data from logger pro we decided to also find a curve fit to see how we would go about solving the equation of the inertia balance with increasing masses. We found that a power law equation fit best. We recorded the data of the inertia balances period(seconds) with respect to the mass(0-800g).

There are also two random objects we found and measured there masses. These two masses we ended up solving for there periods as if they were in our graphs by using the power law equation which fit our graph almost perfectly. By finding there periods we can analyze our data and see if they match up.

 This photo above shows all the work we did in order to find the time of calculator and the eraser.
Equation: lnt= nln(mass+Mtray) + lna
T for the period was equal too T= A(low/high)(m+Mtray(low/high))^n(low/high)
The low is representational to the lowest period with the lowest mass and high is the highest period with the heaviest mass. We solved for the slope of the equations first so we could find there periods. After doing so we used the period to estimate the lowest and highest possible masses they could be. by doing this we were able to later analyze our data and find that we had an error.

Data: All in all our data table worked out nice but there was an error in calculations due to the fact the we were probably not being consistent with the amount of strength we used on the inertia balance or a miscalculation done by the computer. By working out the answersby hand of two random masses we were able to compare it with the data from picture 2 (chart) which showed that our masses did not fit into the correct period.

Final Thoughts: We found the relationship between masses and its period depend highly on the amount of mass you use. The more you use the longer the period of the object will take in order to complete a full oscillation. We mathematically solved an equation that roughly calculated the mass of an object with respect to its period.