Lab 6: Propagated Uncertainty

In this lab, we are going to measure the error that we get when we measure things.

For this lab, I had three identical cylinders. One was made of bronze, one silver, and one gold.

Because the scale could measure to a minimum of .1 grams, that is the propagated uncertainty for these three items

The density of bronze is supposedly 9.29.

However my calculations make it 9.69 cm^3 + or - .1 grams

 My calculation of the density silver was 3.54 cm^3 + or - .1which was severely wrong

as the density of silver should be 10.49

My calculation of gold is also extremely wrong as i received 10.86 cm^3 + or - .1 compared to 19.3

Using the equation on the bottom part of the image I get 

silver: dP/p = .015%

bronze: dP/p = .016%

gold: dP/p =  .016%

In the second part of the lab we have to find propagated uncertainty of a unknown mass

For this system the equation looks like 

Fy= mg- F1sin(x) - F2sin(y)

Fx=F1sin(x) = F2sin(y)

F1=3.5-4.5 N

F2 = 2.7-3.7 N

x = 47-49 deg

y = 37-39 deg

By using the y side of the equation we can move over mg

mg= F1sin(x) + F2sin(y)

to get m we then divide by g

m = (F1sin(x) - F2sin(y))/g

m = .505 + or - .015 N

For the first part of the experiment I must have accidentally used to wrong equation for silver and gold. I remember that I had to redo my calculations and was still redoing it while class was over. Unfortunately, I could not find the new sheet and just stuck with my old data. Besides the experimental densities being wrong, I think the rest of my lab is correct. 

Lab 8: Work on a spring

We are going to try to find the work that acts on a spring when a hanging mass is attached to the spring.

First we measure the length of the string stretched and the non-stretched. Using this data, we will calculate the constant of the spring. Afterwards we use the motion sensor to get GPE, EPE, KPE, and the total energy. 

the data of all our energies

this is the graph of the position of the spring vs time

This graphs is very unclear but shows all the energies that are in the system.

These are the calculations of how we derived which energies are going to be in the graph.

First we derived all the energies that were going to be in the lab and their respective equations. Then we calculated the spring constant by getting the position of the spring stretched vs non-stretched. We then used a motion sensor and logger pro and inserted all the equations we derived to get EPE, KPE, GPE, total energy. Our lab was for the most part successful as we were able to show all the energies present in the spring. Unfortunately, the total energy was quite a bit off.

lab 10: Centripetal Acceleration

This was a class lab where Professor Wolf wanted us to find the centripetal acceleration of a spinning disk. 

Unfortunately the apparatus is kind of blocked. However, it was basically a disk spinning on a tripod. 

Each group was supposed to find the time it took for the disk to rotate 4 cycles. 

We got the time by taking the average of each groups estimated time it took for four cycles. 

There were a lot of possible errors as people can have a different start and stop location. Everybody probably did not 100% accurately get the exact time it rotated 4 cycles. Although it is kind of blocked, in the first image we were able to get a graph that shows the relationship between angular speed and centripetal acceleration.

Lab 9: Angular Velocity

The purpose of this lab is to find the relationship between the angle at which the string spins at the speed at which it rotates around.

We checked at eight intervals for it to take ten revolutions and below is the result.

We then used excel to create a graph for the lab using the table below. 

This is a graph of omega vs theta. Our slope was .9877 when it was expected to be 1

After the eight intervals, we were able to solve for the equation that relates the angle and the speed. 

omega = (g*tan(theta)/d+l*sin(theta) )^(1/2)

Our experiment was a success as there was approximately a 1.3% error. 

Lab 4: Air Resistance

We went to the architecture and design building where we would drop coffee filters down from the second floor. We would first drop one coffee filter and add one more each at a time until there became five coffee filters all dropped at once.

The resisting force can be found with the equation, 

F = k*v^n

By taking the natural log of both sides, we derive the equation

ln(F)=ln(k) + nln(V)

where we have to find the value of F and the value of v.

List the mass and value of speed/velocity.

excel sheet of terminal velocities

The terminal velocities are listed below
1 filter: .922m/s
2 filters: 2.129 m/s
3 filters: 1.641 m/s
4 filters: 2.129 m/s
5 filters: 2.525 m/s 

Using the equation 

ln(F)=ln(k) + nln(V)

we derive this graph

We then use excel to calculate the terminal velocities once again.

The experimental values were a bit bigger than the actual values. But not by much. 

Overall it can be said this experiment was successful.

Lab 11: Power Lab

The purpose of the lab was to find the power needed to go up the stairs and pull a backpack up with a rope.

We had three bags to choose from. On the left was the heaviest, the middle and the second lightest or heaviest depending on how you want to look at it, and on the right was the lightest backpack.

This is the stair case which we had to walk up and then run up.

========================================================================There are 24 steps to the stairs and each step has a height of .156 m. 

By multiplying 24 to the height, we get the height of the stair case which is 3.24 m.

I walked up there in 13 seconds and I ran up the stairs in 8 seconds. 

Work is equal to force multiplied by the distance. The force is mass times acceleration and the distance was 3.24 m. 

90kg*3.24m*9.8 m/s^2 = 2857.68 Joules

Work divided by time gives power

2857.68 J/13 seconds = 219.82 watts (walk)

2857.68 Joules / 8 seconds = 357.21 watts (run)

I pulled the 6 kg backpack. 

The work for the backpack is 

6kg * 9.8m/s^2 * 3.24m = 190.5 Joules 

The power is 190.5 Joules / 9.65 seconds = 19.74 watts

In the experiment we would just find the work first. After finding the work, we would divide it by the time to get the power it takes to perform the action. I'm not sure if this is entirely accurate as I took two steps at a time when possible to make myself a bit faster. 

Lab 3: An Elephant on Roller Skates with a rocket attached

         As inhabitants of the earth, it is our job to protect it. When an elephant is rolling down a hill, we have to make sure the elephant does not fall off the edge and die in the process.

         We have to find the original speed of the elephant so that it does not fall off a cliff. A 5000-kg elephant on frictionless roller skates starts at 25 m/s going down a hill. When it hits a level plain, the rocket attached to the elephant then starts generating a constant 8000 N thrust against the direction the elephant is going down the hill. The mass of the rocket follows the equation m(t) = 1500kg - 20kg/s(t).

  From the equation F = ma. We derive a = f/m. 

In this case a(t)= 8000kg/((1500kg+5000kg)-20kg/s*t).

This equation will be simplified to:

a(t) = -400/(325-t) m/s^2

From our experiment the value we got was 248.69 meters which is to what you have calculated for us 248.7.

Ultimately the experiment was a success as the results were almost exactly the same.

Lab 14: Impulse Momentum

The goal of this experiment is to try and find the relationship between impulse and momentum.

Equations useful to this experiment are p = m*v and J = F * t.

In the top image the blue kart collides into the red kart and we use it to determine the momentum of the system

In the middle image there should be a weight on top of the blue kart, however unfortunately I must have forgotten to take a picture of it at the time. 

In the last and bottom picture, there is an inelastic collision. Meaning that the system gets suck together and that energy is lost, as the system then gets connected to each other. 

========================================================================

Results of Experiment 1

 Results of Experiment 2 (block)

Results of Experiment 3 (clay)

In experiment one, our impulse was -.2723 N*s

In experiment two, our impulse was -.6706 N*s

In the final experiment, our impulse was -.2351 N*s

This was the experimental calculations that we did by hand.

Our third and first experiment was really close for our actual and expected results.

Unfortunately that success was nonexistent in the second experiment. Where our impulse was -.6706 compared to our calculated implies of -.764 which is quite a bit off. We were mostly successful as two of the three results were within expected range. 

Lab 15:Magnetic Potential Energy

The purpose of this lab is to find an equation for the magnetic potential energy.

Above is an air track with an air glider. A magnet is attached to the air glider an we tilt the system at an angle by adding more books to the bottom. There is also a magnet at the end of the air track, which repels the cart some distance. In an "ideal" situation the track is completely frictionless by letting air go through the miniature holes in the track.

This is the result of of trials. Where we tested eight different angles and received eight different equilibrium points for the distance between the two magnets. 

Using the angles and the distances between the two magnets, we were able to graph a force vs distance graph. By integrating the graph we were able to get the kinetic, potential, and total energy of the graph.

Our data set used to obtain the force vs distance graph.

What our force vs distance graph looked like

Integrating our force vs distance graph to get the energies in the system. 

========================================================================

The picture above shows the kinetic energy in purple, the potential in green, and the goal energy in orange. 

In the picture below, a straight line should be made for the total energy as no energy should be lost. So if it started with 750 joules the system should end with 750 joules. 

Our total energy is not a straight line, which was unexpected. This could be due to human error such as incorrectly determining the equilibrium point and also due to friction. While our graph was not horrible, there was quite a bit of error in the experiment. 

Lab 20: Conservation of Linear and Angular Momentum

This lab was a class lab, where Professor Wolf did the experiment once. We are trying to show that linear and angular momentum is conserved. 

A ball is put on top of the ramp and flies off and hits a spot on the floor.

Using kinematics, we solve for the velocity at which the ball leaves the ramp.

We found that the velocity at which it exited the ramp was 1.4 m/s.

Above is the calculations for the moment of inertia of the ball. 

Using logger pro, we find the angular velocity of the system

Our answer for angular momentum was 1.74 rad/s

Our answer was close to all other groups. The real angular momentum is most likely smaller though, as we did not account for human error, friction, air resistance, and any other anomalies. All in all, the experiment was successful.

Lab 18: Moment of Inertia of a Triangle

The purpose of this lab is to try to find the moment of inertia of a triangle.

Initially we will solve for the moment of inertia of a triangle symbolically. We will then find the angular acceleration so that the moment of the inertia of the system can be solved thereafter. 

When the triangle is placed horizontally, this is the result.

The result of placing the triangle vertically.

Above is the calculations where we used the parallel axis theorem to derive the moment of inertia of the two positions of the triangles.

Our experiment was fairly close with our results. Differences can be due to not taking into account of friction, air resistance, mathematical errors, or rounding.

Lab 17: Angular Acceleration

In this lab we are trying to find if there is a relationship between the factors that will affect the angular acceleration of a system, 

This is what the lab set up looked like.

We attached a string around the pulley and on the other side of the string was a mass.

The results of the experiment from the first run. We plotted the velocity against time, and by taking the slope, we would get the acceleration. 

This is the v(t) graph for the first trial run.

This is v(t) graph when we doubled the weight

This is the v(t) graph with triple the starting amount of weight.

This is what happens when the size of the torque of the pulley is changed.

These are the actual values we received from doing the experiment. 

From the above graphs, clearly there is a correlation between the acceleration of the system and the weight of the mass. As seen above, when the mass is doubled, the acceleration roughy doubled. Same for when it tripled. All in all, I believe this was a successful lab.

Lab 16: Moment of Inertia

In this lab, we tied a string to a cart and a pulley. We were supposed to determine the time it would take for the kart to descend the track,

We first calculated the moment of inertia of the two cylinders holding the disk in place. Then we calculated the moment of inertia of the system by adding those moment of inertia of the disk.

This is the graph we got from our data table and it shows the velocities of the x and y direction. 

By square rooting the squares of the velocities in the x and y direction and then adding them, we get the graph below which shows the tangential velocity.  

========================================================================

These are our calculations. As can be seen in the image on top, the moment of inertia is 1.92x10^-2kgm^2/s^2. 

The image below shows our calculations for which the time it should take for it to drop one meter. 

Our experiment was unsuccessful. We were expecting  a time of around nine seconds yet it was always around thirteen seconds. After looking at it, the professor said our moment of inertia might be too small even though our calculations are right.