23 September 2014: Spinning Lab 8

Objectives:
1) Finding the radius of a spinning object using centripetal acceleration and rotational speed
2) Finding the relationship between rotational speed, omega (w), and angle (theta) of a spinning system.

Part 1:
Materials:
1) Lazy Susan
2) Accelerometer
3) Timer
4) Logger Pro

Set Up:

An accelerometer was taped to the lazy susan.










Procedure:
Each group in the class was given a stopwatch. They were asked to time the time it took the lazy susan to turn 2 revolutions. These values were then reported and the professor took an average. This was repeated for 4 other trials in which the lazy susan was turned at varying speeds.

The professor then made a graph with all our values. The graph looked like this:

r=0.1493

From this graph we were able to determine from the slope that the radius=0.1493 m. This is possible because r= a/(w^2). (w^2) was calculated from the periods(T) we reported as w= (2pi)/T. And a was collected using the accelerometer.

So, finally:

r from data=0.1493 m

r in actuality= 0.18 m

So there is a 17% difference between both the radii.

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Part 2:

Materials:

1) Preset Setup

2) Timer

3) Ruler

4) Paper

Set Up:

It is important to note that we had nothing to do with this set up as it was done for us. This is a conical pendulum with a motor that allows for almost uniform rotation.

Procedure:

The first thing to do was to record the height of the system(h1), the length of the stick from which the pendulum hangs(d), and the length of the string from which the pendulum hangs(l).

h1= 210 cm

d= 62 cm

l= 177 cm

We then let the motor run and recorded data for 8 different trials. We were specifically recording at what height(h2) a rubber object tied to the end of the string hit a sheet of paper. This can be seen below:

The data that resulted can be seen below:

We were then required to derive an equation that would relate the angle between the stick and the string (theta) and angular velocity. Our equation can be seen below:

Using this data we plugged the following data into Logger Pro:

Lastly we created a graph of Omega vs F(theta). The graph looks like this:

Conclusions:

We took care in our derivation of our formula. We were also careful when plugging in numbers to create values. When creating our graph the graph is very close to 1 except for one dot. This shows that we have gathered good results.

18 September 2014: Friction Forces Lab 7

Objectives:
1) To use our understanding of friction to calculate either the static friction or the kinetic friction of a system

Part 1:
Materials:
1) 1 wooden block with felt on one side
2) String
3) Frictionless pulley wheel
4) 1 Styrofoam cup
5) water

Set up:

As you can see from this picture, we have clamped the pulley wheel to the table. We then placed the block with the felt side against the table and tied a string from the block, over the pulley wheel, to a cup which lies hanging over the table. We will then slowly add water to the cup.














Procedure:
First we must measure the mass of the cup without water in it. Our cup had a mass of .030 kg

Then we had to measure the mass of the block. Next we had to patiently add water to the cup until the block started to move. Once the cup had moved we had to record the mass of the water and cup. Then we repeated the experiment except we added more and more blocks to the first and recorded the mass of the added blocks.

3 blocks

2 blocks

We continued adding blocks until we had 4 blocks in total tied to the cup. The data that resulted from each trial can be seen below:

table 1

As seen from the table, we were able to calculate the normal force between the block and the table (column 3 of table 1). We did this by multiplying the total mass of the blocks x "g". We were also able to calculate the max static friction force between the block and table (column 4 of table 1). We were able to do this by multiplying the mass of the water and cup x "g". In both these calculations "g" = 9.81 m/(s^2)

We then wanted to plot our data. So we created a graph of the normal force between the block and the table (column 3 of table 1) and the maximum static force between the block and the table (column 4 of table 2). The resultant table could be seen below:

As you can see, we have added a trend line to the data. The slope that occurred from the graph was 0.3957. This was our coefficient of static friction.

So our coefficient of static friction was 0.3957.

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Part 2:

Materials:
1) 1 Block with felt on one side
2) String
3) Force Sensor
4) Logger Pro

Set up:

We have placed the block with the felt side against the table and have tied a string to the force sensor. Someone was then supposed to slowly pull the force sensor horizontally.

















Procedure:
First we had to calibrate the force sensor using a known mass. Then we recorded the mass of the wooden block. We then held the force sensor horizontally, started collecting data on Logger Pro, and started pulling the force sensor slowly moving the block at a constant speed. We then recorded the mass of additional blocks and repeated the procedure until we had a total of 4 blocks. The graphs produced by Logger Pro of these trials can be seen below:

From these graphs, we could collect the mean Force from the most flat part of the graph. The mean force represents the force at which the block was moving at a constant speed.

The data acquired from this lab was organized into the following table:

Table 2

The normal force between the blocks and the table (column 3 of Table 2) was calculated by multiplying the total mass of the blocks (columne 2 of Table 2) x "g". For our calculations "g"= 9.81 m/(s^2).

We again decided to create a graph from our data by plotting Normal Force (column 3 of Table 2) vs Kinetic Friction Force (column 4 of Table 2). The resultant table looked like this:

We again added a trend line to the data The slope that occurred from the graph was 0.3763. This was our coefficient of kinetic friction.

So our coefficient of kinetic friction= 0.3763

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Part 3:
Materials:
1) Ring stand
2) Clamps
3) A smooth track
4) Block
5) Device to measure angles
6)Logger Pro

Set up:

The track has been propped up with help of the ring stand to the point where the block is just about to slide. The bottom part of the track is held in place with a clamp.





Procedure:
First we had to measure the angle made with the track and the table. The angle was found to be 30.5 degrees.

We were then asked to calculate the coefficient of static friction. The calculations can be seen below:

u_s=0.589

For the next part we were asked to find the coefficient of kinetic friction. To do this, we raised the track so that the block slides down the track. We measured the angle at which the block slides down the ramp and found it to be 36.5 degrees. Then using the motion detection and Logger Pro, we recorded the acceleration of the black as it traveled down the track. With this data and the mass of our block, we were able to solve for the coefficient of kinetic friction. The calculations can be seen below:

u_k = 0.852

Conclusions:
From this lab we learned how to calculate static friction and kinetic friction. Unfortunately for part 1 the mass of the cup has a large range of error as we were not very patient. This might have had negative results for our data.

16 September 2014: Propagated Uncertainties and Unknown Masses Lab 6 Part 2

Objectives:
1) Finding the mass of an unknown object using tensions and angles
2) Using the skills acquired in part 1 of this lab to propagate the uncertainty in our calculations

Materials:
1) A preset setup
2) An unknown mass

Setup:
Our setup looked like this: (It is important to note that we had nothing to do with the setup as it was already made for us)

From this picture you can see the set up consists of 2 10 N spring scales placed at some unknown angle that are holding up an unknown mass object.

Procedure:

First we collected the data needed to calculate the mass of this object. We needed the Force exerted by both spring scales and the angles at which both were placed. The data gathered can be seen below:

From this data we were able to calculate the mass of the unknown object. The calculations can be seen below:

m=.501 kg

Now that we had the mass of the unknown object, we could propagate uncertainty. The calculations for the propagated uncertainty can be seen below:

dm=0.0605 kg

Conclusions:

Our final data showed that

m=0.501 kg

dm=0.0605 kg 

So it can be said that m= 0.501 kg + 0.0605 kg

Sources of uncertainty most likely lie in the angle measurement. It was hard to get this measurement without disturbing the preset set up.

16 September 2014: Propagated Uncertainties and Densities Lab 6 Part 1

Objectives:
1) Learning how to calculate propagated uncertainty in our density measurements.

Materials:
1) Caliper
2) Scale
3) 3 Metal cylinders  (in our experiment we chose copper, steel, and aluminum)

Procedure:
First we measured the mass of our metal cylinders using our little scales:

Steel

Aluminum

Copper

 Using the calipers, we found the height and diameter of each cylinder. The height, diameter, and mass was organized in the following table:

table of values

After we had these values we could calculate the densities along with the propagated uncertainties. The calculations for these values can be seen below:

Aluminum calculations

Steel Calculations

Copper Calculations

Conclusions:

Our calculated densities along with their propagated errors were found to be:

Aluminum:  2.47 + 0.1653

Steel: 6.56 + 0.60

Copper: 7.428 + 0.2856

11 Septmeber 2014: Trajectories Lab 5

Objectives:
1) Study the components of motion in projectile motion.
2) Use our understanding of projectile motion to predict the landing point of a ball on an inclined board

Materials:
1) 2 Aluminum v-channels
2) 1 steel ball
3) Board
4) 1 Ring stand and clamp
5) Tape
6) Carbon Paper
7) 2 Blocks of wood to hold one of the aluminum v-channels
8) Plumb bob (weight and string) (optional)
9) Ruler

Set up:

As you can see the horizontal v-channel is held in place by the 2 blocks of wood. The v-channel at an angle is held in place by the ring stand and clamp. Also, it is held in place by tape at the bottom so it does not slide while the ball is rolling down it.

As you can see from this pictures, in addition to the set up on the table top, we also taped carbon paper to the floor to document where exactly the ball lands. We also hung a plumb bob (weight and string) from the edge of the table.











Procedure:
Part 1: Now that our setup was complete, we launched the ball from the v-channel at an angle. We marked the exact spot from which we launched it the first time and launched it from the same spot another 5 times. Remarkably, the ball landed in exactly the same spot all 5 times:

I added a red circle to clearly identify the mark of the landing point of the ball

We were then asked to determine the height from the table to the floor and the distance from the table's edge to the mark. We used a ruler to determine these measurements and can be seen below:

horizontal v-channel=31.5 cm, height= 93 cm, distance from table to mark=47.2 cm

We were then asked to determine the launch speed of the ball when it leaves the table. The calculations can be seen below:

v0= 1.08 m/s

Part 2: We were then asked to imagine attaching an inclined board at the edge of the table in a set up that looked something like this:

We were then asked to derive an expression that would allow us to determine the value of d. Our calculations of this expression looked like this:

We then actually ran the experiment by placing the board against the table and taping the carbon paper to the board. We then measured the angle that the board created with the floor and found the angle to be 45 degrees. We then launched the ball five times from the exact same spot. Our experimental d was found to be 0.31 m.

We then determined our theoretical d. Our calculations can be seen below:

d=0.33 m

We were able to determine our v0 from part one and plug into our calculations for part 2. Overall, these were our values:

experimental d= 0.31 m

theoretical d= 0.33 m

Conclusions:
If we were to create a percent error with our results it would look like this:

Percent Error= 0.31-0.33   x 100% = 6%
                                0.33

Our percent error was 6%. There were sources of error that could account for this percent error. Sources of error might lie in the rotational friction from the track that was not accounted for. It was also difficult to exactly measure distances from the table edge.

9 September 2014: Air Resistance Lab 4

Objective:
1) Observe how air resistance affects terminal velocity of a falling object
2) Discover the behavior of air resistance with regards to change in mass

Materials:
1) Tape
2) Ruler
3) Coffee Filters
4) A high indoors place from which to drop the coffee filters
5) Video
6) Video analysis program such as Logger Pro

Set Up:

As you can see from this photo, we have taped the ruler as a reference point for later video analysis. We are filming far away enough to capture the whole fall of the coffee filter. We are dropping coffee filters; first 1 at a time then 2 at a time and so on until we have dropped 5 at a time. Lastly all this has happened in an area where wind cannot affect our results and also a place that is high enough for proper data collection.








Procedure:
Before we started our data collection we made some predictions about the whole experiment. First we made a free body diagram of all the forces acting on the coffee filter as it fell with air resistance.

Then we sketched a graph of what we predicted for acceleration vs time and velocity vs time:

We predicted the acceleration vs time graph would look like this because the acceleration would decrease until the object reached terminal velocity regardless of the amount of mass. We also predicted the velocity vs time graph would look like this because the coffee filter(s) would gain speed as they are falling until it reaches terminal velocity and stays constant.

Lastly in class, it was discussed that acceleration would look something like this:

Then we traveled over to a different part on campus to collect our data.
We first dropped one coffee filter, then 2 at a time, then 3 at a time, then 4 at a time, and lastly 5 at a time. We then analyzed our data on Logger Pro and created a position vs time graph and took a linear fit and recorded the slope of the linear fit of this graph.

Position vs time graph of 1 coffee filter 

We looked at the slope of the the position vs time graph because the slope of this graph represents the velocity. More importantly the slope of this graph represents the terminal velocity of the coffee filter.

Here are the graphs of the other trials:

2 coffee filters

3 coffee filters

4 coffee filters

5 coffee filters

Then we applied the knowledge that the Force of air resistance=k(terminal velocity)^n
We then calculated the forces and created a graph of Force of air resistance vs terminal velocity. We used the terminal velocity gathered from the slopes of the graphs of position vs time for each trial. The resultant graph looked like this:

To this graph we added a trend line. As mentioned before Force of air resistance=k(terminal velocity)^n. With this graph we were able to discover our "k" and "n" which were modeled by the graphs "A" and "B" respectively. 

So our k= 12.22 + 1.190

and our n = 1.60 + 0.1358

Lastly we tested the accuracy of our results by numerically solving for terminal velocity using excel. Here is a picture of how we set it up. As you can see, we used 1/30 of a second increments.

Then we created a position vs time graph with our excel data:

We then compared this graph to the one produced by our video and Logger Pro

Conclusions:

There was a big discrepancy between the excel graph and our graph created by our data. There could be many sources of error. Firstly, the background was white as were the coffee filters so when analyzing the video it was hard to identify the coffee filters. I believe our model data created in excel is much more accurate than our experimental data because of our inexperience in gathering data using our videos.