30- Sept - 2014 Work and KE Lab

Work and KE Lab

Objective:

 Show that the change in work equals to the value of Kinetic Energy at that point in position.

Set- Up: 

Figure 1a. Our apparatus set up at a top view

 We had a set up with a motion sensor and a force sensor plugged into labber pro for this lab. We also had a cart. aluminum track, a spring, a block for mass, a block for leveling the spring and rods with clamps to hold the force sensor in place as seen in (fig 1a- 1b).

Data Collection:

 After we had set up our apparatus, we reset both sensors to zero when the spring was at a relaxed state. Then, we reversed the motion sensor setting to toward the sensor is the positive direction. We then stretched the spring between 20 cm to 30 cm. When data collection began, the cart was released and the spring pulled the cart away from the sensor. 

Our readings were shown initially as a velocity graph and a force graph. We took the force graph and changed the time axis into position. Using 

KE= (1/2) mass* (velocity)^2

Figure 1b. apparatus set up at a side view

we inserted a kinetic energy graph with a x-axis of position along with the force vs. position graph. Now as seen in (fig 2.)

We integrated our Force graph to find a change in work and compare our integration with our KE values to see if they matched.

Theoretical Value= .413 N*m

Experimental Value= .378 N*m   

.378 * 100% = 91.523%

.413                                

 Apparently the target value for our integral is higher than our KE value at that point in position by approximately 8.477%. 

Conclusion:

 What may be the main source of our 8.477% error is that the spring may not have had a spring constant that applied to the entire spring since some parts are more tightly coiled than other places in a relaxed state, or the spring was not ideal. With that in mind, our results were close enough to show that the concept holds water and completes our lab objective.

Figure 2. Our resulting integral of the change in work in an interval between different points in position.

Figure 3. Our row where our target value of KE is found
unfortunately it does not match our work integral.

23- Sept.- 2014 Spinning Lab

Spinning Lab

Objective: 
1)Find the radius using centripetal acceleration and rotational speed

2) Find the relationship between omega (w) and angle (theta) for a particular apparatus.

Figure 1. the spinning apparatus with an
accelerometer taped on the edge and a
block of wood for reference to count rotations

Set up (2 of them):

1) Using a spinning apparatus and a accelerometer attached to it, you time how many times it makes 2 rotations. There are 5 trials at various accelerations. 

2) Using a specific apparatus with an electric motor, a stand, string and a rubber weight, we must time it in a similar fashion as set up 1 but for 5 rotations.

Part 1)

 Using the first set up, each class group made times of how long the wheel to turn 2 rotations and Prof. Wolf made an average period for each trial. We then graphed the results and made a linear fit to find the slope of omega vs. rotational acceleration (r) seen in (fig 2.):

Where a is acceleration and T is average period:

a= r*w^2

r= w^2

    a

since, w= 2(π)

               T

Then, r = a

                 T  

Figure 2. The resulting scatterplot with a linear fit of r= .1493 m

If we compare our theoretical radius r to the actual, we find that:

rtheo = 15cm

ract = 18 cm

17% more than theoretical

Part 2)

Figure 3. the rotating apparatus for this
lab.

 We now move on to the second set up in the lab. We use a spinning apparatus where the angle (theta) from the vertical effects the radius r (the faster it spins, the higher theta gets approaching 1.57 radians. The length of the string and the length in which the wood sticks out all affect r as well as the height h that the rubber end is from the ground. After putting all this in perspective, we derived the equation: where r= (Lstr + Larm)                     

w = [g*tan(theta) ] ^1/2      

[  (Ls + Larm)  ]       

After, we then recorded the trials and labeled our results on an excel spreadsheet. The biggest difference is between omega's both experimental and theoretical. We used our derived equation for omega experimental and for omega theoretical we used:

wtheo= 2(π)

         T

Figure 4. Our recorded data, very similar to most of our classmates' results since we all participated in each trial.

We then made a omega experimental vs. omega theoretical graph. Idealy their slope should come out to r = . The graph is not an ideal one but it is very close at a slope of r= .9926m . This comes out to less than a percent error for good results. What may have caused it was the stick flexing and friction in the system from air and the motor.

Figure 5. the resulting graph from our spreadsheet.
The ideal slope was supposed to be r=1

Figure 6. All recorded values for both part 1 and 2 an results for each trial and derivations of the equation for omega in part 2 

16- Sept. - 2014 Friction Lab

Friction Lab

Objective: 
 Use the lab set ups to find both  s  and k  between wooden block(s) and the surface it is on.

Set- up (4 of them):

1) We used 4 wooden blocks (one is piled on top of the previous per trial with the first having a specific red fabric underneath), around 1 m of string, 2 cups with on filled with water, weights, dropper, and a pulley as seen in (fig 1a). We used this set up to find the static friction of the block by comparing its total mass to how much water it takes to make the block move.

Figure 1a. The set up with the initial block with a pulley and string with a
 cup with water at the end of it 

2) We attached a force meter to the same initial block and attempted to drag it at a constant speed. Each trial is the number of blocks stacked on top (up to 4).

3) We set up a sloped new surface and tracked the acceleration of the block sliding down with a motion detector set up parallel to the surface as seen is in (fig 3).

4) We attached a pulley to the top of the sloped surface and attached 500g to the end of the string so its weight will overcome the static friction. We set up the block with a motion detector and one end of the ramp to track its acceleration as seen in (fig 5).

Part 1)
 Using the first set up we measure the following spreadsheet as the results of each trial where the Normal Force was the weight of the stack of blocks for each trial and the force of Max Static Friction was the weight of the cup and water that resulted movement from the block.

Figure 1b. The spreadsheet from excel of the different values and measurements we found

Figure 1c. The graph of the Max static force vs. Normal force where the slope is the s

From the graph above, we found our slope (s) to be .3546. 

Part 2)
 For this part of the lab we used the second set up by using the force sensor and pulled at a constant speed to record the average force of kinetic friction after analyzing the data seen in (fig 2a.)

Figure 2a. a graph with all 4 results for the force 
readings and we found a statistical average (mean)
to find our average kinetic friction coefficient k 

Using the known mass for each trial and our calculated values of Normal Force and the Average Kinetic Friction (from the figure 2a), we graphed the Average Kinetic Friction vs. Normal Force and linear fitted it.

Figure 2b. Similar to the spreadsheet in part 1, we use excel to graph the values of Average kinetic friction along the axis of Normal Force.

Figure 2c. The resulting graph with a linear fit where we found our slope (k) to be .2914 and seeing that it is smaller than our value for the coefficient of static friction is reassuring that it may be accurate.

When we used linear fit, we got the slope (k) to be .2914.

Part 3)
 Using a new block with a mass of .273 kg, we then used the third set up and slowly tilted our aluminum flat- track until the block started to slide due to gravity overcoming its static friction.

Figure 3. Our third set up for the lab where 
we have a sloped surface and motion detector 
to track its position and calculate acceleration 
when we change its angle.

We found that the angle in which the block starts to slide is approximately when theta is .2443 rads above the horizontal. As seen in (fig 6.) We derived an equation and solved:

s= tan(theta)

s= tan(.2443) = .2493

Part 4)

 From the last set up we used the same block and flat track. However, we made the track go from .2443 rads to .4363 rads. Using the motion detector (also seen in fig 3.), we recorded its velocity and velocity changes to find the acceleration of the system. To do so, we had to make a linear fit for the graph and read its slope.

Figure 4. Our resulting graph from the part 3 set up where
our average slope in the linear fit is m= .8548

After reading its slope for average acceleration to be 0.8548 m/(s^2), we used it in (fig. 6) to find the system coefficient of kinetic friction which came to be a very interesting value of k= .370. At first it was a surprise that the coefficient was higher than our previously found static coefficient, but then we realized that this was a different angle so there are different coefficients. To check, we calculated s= .4663 in this case. With that reassurance, we moved on to part 5.

Part 5)

Figure 5. Our final set up with a pulley and weight to
 drag the block along a slanted surface.

Before we performed this lab, we had to predict the system's acceleration in which we derived the following:

asys= g(mwt -mbl (sin(theta) + cos(theta)k) 

(mbl + mwt)

Theoretical value:

asys= 2.810 m/(s^2)

For this lab, we used the fifth set up and hung a mass of .4 kg and recorded the results.Unfortunately our actual graph was lost, however we have recorded the actual acceleration in this scenario as aact = 2.695 m/(s^2) up the ramp. Comparing the two values shows that there was a small amount of error:

atheo= 2.810 m/(s^2)

aact= 2.695 m/(s^2)

approximately -4% than theoretical

Figure 6. Lab notes and work for parts 3 to 5 showing derivation of equations and our calculations

11- Sept. - 2014 Unknown Mass Lab

Unknown Mass Lab

Objective:

 To practice our ability to determine uncertainty through finding an unknown density and unknown mass respectively. (Lab Partners: Kevin and Ivan)

Set up (2 of them):

1) We used calipers and were given 3 metal cylinders to determine density and propagate uncertainty

as seen from (fig 1 - 3).

2) We were given a set up of an unknown mass in equilibrium in a tension set up with spring scales to label tension forces as seen in (fig 4). To find angle we used a piece of string, a small weight and a protractor.

Part 1)
 This part of the lab is to apply what we learned about propagating uncertainty through finding the volume of three metal cylinders and their mass to determine the object's density. Density is the amount of mass (kg) per cubic meter (m^3). The next three figures were our work for each cylinder to find their densities and propagated error:

Figure 1. One of the metal cylinders with work on calculating its density.

Figure 2. The second metal cylinder with work on finding its density.

Figure 3. The third cylinder and work on finding its density.

The calculated densities with their propagated error for each figure are:

fig. 1= 7.49 ± .26 g/(cm^3) 

3.0% uncertainty 

fig. 2= 8.005 ± .714 g/(cm^3)

8.9% uncertainty

fig. 3= 17.574 ± .404 g/(cm^3)

2.3% uncertainty

Part 2)

During this part of the lab, we had to find the mass of an unknown on two pieces of string connected to spring scales at unknown angles. To find the angles we used a protractor, a piece of string and a small weight. We tied the weight to the string  and tied the other end on the hook (we were sure to only use enough weight to keep our string straight, but not too much to bend the string too much. We then used the protractor along the string connected to the unknown mass and read the angle between the two strings.

Figure 4. An unknown mass hanging on string attatched to spring scales each at an angle.

This is our work and set up to find the equation that could help us find our mass along with uncertainties in the find measurements to use for propagating our uncertainty for the mass.

Figure 5. The process in which we solved for our mass function in terms of:
 gravity, angle 1,  angle  2, Tension 1, and Tension 2.

We then plugged in our values into our new found equation and got:

m= (2.3 N)(cos(.4363) - sin(.4363)) + (3.9 N)(sin(.6458) + cos(.6458)) = .670 kg

(9.81 m/(s^2))

From there we had to find the partial of each independent variable (except gravity) and find our uncertainty from this example as shown below in (fig 6).

Figure 6. Calculations in which to find uncertainty of our unknown mass by finding the partial in respect to each independent variable then square rooting the dot product of the derivative of  our mass function (treating it as a vector) to find dm or uncertainty of mass. 

Through that we find that our mass with our uncertainty is:

m= .670 ± .034 kg

5.1% uncertainty

Source of uncertainty lie mostly with the reading of angles because we don't know if there was enough change in angle that could effect the results.                                                                                       

9- Sept- 2014 Projectile motion

Projectile Motion Lab

Objective: 

 To use what we know from projectile motion to predict the impact point of a ball on an inclined board.

Set- up:

 We used a simple apparatus that involved aluminum tracks (one at an incline and the other at a horizontal on top of a table/ desk) seen in (fig 1), a wooden plank longer than the height from the floor to the top of the rail, a steel ball, carbon paper attached to a regular sheet of paper that's taped to the ground where it is estimated for the ball to hit. From there you will need an angle finder for the plank, and a 2 meter stick to do other necessary measurements like the height from the end of the track relative to the ground.

Figure 1. the aluminum tracks assembled
to roll the ball off the table and onto the floor
or the inclined surface

Part I:

During the first part of the experiment, you need to have you apparatus set up and the papers on the relative area where the ball will impact the ground. After getting 5 consistent tries and your point of ball release is marked on your railing, you must get the 2 meter stick and measure your mark from the carbon paper from the base of the end of the track as well as its height relative to the ground, see in (fig 2). From there, you are asked to calculate you vo :

y= 1/2* (g)* t^2

Since our y= .942m,

(.942)= .5* (-9.81) * (t^2)

t= .483s

Now, it is assumed that the acceleration for x is zero and our initial velocity in the y direction is zero as well, with our given time we find our initial velocity to be:

vo=  (x)     

(.483s)

We found our x =.663± .002 m because we were confident in our time and y measurement, but the place where the ramp ended was a small yet noticeably off from the edge of the table where we used as reference for the measurement, in this case:

vo= (.663m) = 1.514 ± .006 m/s

(.483s)                      

Part 2:

 Our next step was to predict that if we put an inclined plank from the edge of the ramp to the floor at an angle (alpha) where would the ball land. Assuming that the apparatus was not altered in any way, the initial velocity will still be the same. Below in (fig 2.) Are diagrams and work showing the the derivation of the equations for each part of the lab.

Figure 2. The derivation of each part of the
 lab and final calculations for the vo and distance d on the inclined surface 

The equation from above that we were able to derive for our angle (alpha) and vo would be:

alpha = .8378± .0175 rads

vo = 1.514 ± .006 m/s

d= 2* tan(alpha)* (vo)^2 = 2* (1.1106) *(1.514^2) = .776 ± .022m

      g                                    (9.81)                        

We've also included our uncertainties that we found in the figure below.

Figure 3. Derivations of both equations used in this lab and their uncertainties.

If we compare the experimental results with the theoretical results:

Theoretical: .774 ± .022 m

3% uncertainty

Experimental: .794 m

+2% more than theoretical

Sources for uncertainty lie in mostly in the rotational friction from the track that was not put in account. Error may have been from an inconsistent placement of the steel ball when beginning a trial, and small measurement errors.

25- Sept. - 2014 Work and Power Activity

Work and Power Activity

Objective:

 Calculate the work and power for each task and compare results of each.

Set- up (2 of them):

 1) Walk a flight of stairs and time yourself until you reach the top and for the second trial you run. We measured the height of one step and counted steps to get an approximate height as seen in (fig 1).

2) With a long rope, weighted bag with a given mass and a plant with a pulley on the top of the balcony with the same height, time how long it takes to hoist the bag to that height as seen in (fig 3).

Figure 1. Classmates measuring the steps while one is timing
 themselves on the stairs. Set-up for task 2 in background

Part 1:

 Following the procedures in Set up 1, we found the following:

Height of one step= 17cm 

# of steps= 26

h= (26)*(17cm)(1m) = 4.42m 

(100 cm)                    

I. Walk time: 14.2s 

II.  Run time: 8.3 s 

 My weight (w)= 377.91 N 

Below in (fig 2.) is the calculations to find the work and Power for each Trail (I. and II.) in this task.

Figure 2. Calculations using what was known and what was recorded in the trials.
Notice Work is the same for both but Power varies depending how fast we did the work.

Part 2:

 In this task, we had to perform the set up 2 procedures. We have found the following:

Figure 3. Me starting my task to hoist the bag.

h= 4.42 m

mass of weighted bag= 6 kg

Lift time: 16.14 s 

Below in (fig 2) is the calculations to find the Work and Power for the task.

Figure 4. The calculations using what is known and what was recorded in the trial.
Once again, no matter how many trials I do, as long as change in height h is the same,
 work will be the same but Power will vary.

 Conclusion:
 Through observation and calculations, if the force applied and distance of a task are constant (in respect to h) then no matter how fast one may perform it, the Work is the same. However, Power is purely dependent on time. So even with the same amount of work in numerous trials, Power will vary as shown in Task 1. Task 2 has also shown that if an individual's weight was being hoisted the same height as the stairs, the work in both tasks would be the same.

9- Sept. - 2014 Air Resistance Lab

Air Resistance Lab

Objective:

 1) Determine the relationship between an object's terminal velocity to its mass.

 2) Model a falling object with air resistance.

Set- up:

 To set up for this experiment we needed a computer with logger pro, a video recording via smartphone or FireWire, an area with very little draft or wind (room seen in fig. 1a), a stack of coffee filters (fig 2.), and a scale reference (meter stick). We are to record the coffee filters falling with each trial having a slight more mass than the previous trial up to 5 trails (5 filters stacked).

Figure 1a. The area in  which we drop our coffee filters as well as
the frame of perspective we were advised to film our trials.

Data Collection (Part I):
 Before starting this experiment, we first , predicted what our relationship between force of air resistance Fres.  to be related to a mass's terminal velocity vter. by a power rule:

Fres. = k*vterm. ^n

Where n and k are constants.

Using the set up we have, we attempted to fulfill our experiment, but our last trial with 5 filters was lost      

and so our Data will only have 4 trials listed hence forth. Now since the room    

Figure 2. Coffee filters of different colors

to make them easier to chart movement with

 a white background.  

is indoors with limited draft can be assumed negligible. We filmed our four trials and charted its movements. If you notice, that graph was linear fitted at the most linear interval where it can be determined as the terminal velocity in the trial as seen in (fig 5). The reason being, just like in the initial prediction, we were trying to find the terminal velocity by finding the slope of the mass's position graph when the slope is linear representing m= vterm for that particular mass.

After graphing each trial and finding their terminal velocities, we used the total mass of each trial to graph a Fair vs. vter graph because at the terminal velocity the force of air resistance is equal to the mass's weight w. Our ordered pairs of this new graph are shown in (fig 3a.) numbered by the trial they represent. Lastly, we were told to create a power fit for this new scatter
plot to find our n and k values for our initial formula:

Fair.= k*(vter.)^n

Figure 3a. Found vter. for each trial and
F is the total mass of the filter stacks
respective to trial number.

So after we created our power fit for our 4 points (trials), we looked for which points fit closest to our power fit is on trial number 4 seemed to fit the graph very well seen in (fig 3b) which is helpful in choosing our best reference interval for part 2. 

Figure 3b. The resulting scatterplot of the previous figure of force vs. calculated terminal velocity. There is also a power fit where the value of y is Fres , A is the value of k, and B is the value of n.

With this we have also identified our n and k constants:

n= 2.162 ±  0.1645 

k= 0.0109 ± 0.0007 

This are our experimental values for our values in this power law we made for air resistance.

Data Collection (part 2):
 We then tested our accuracy of our results by finding terminal velocity numerically for one of our trial in which we previously mentioned to be trial 4 seen in (fig 4). We graphed both results in increments of (1/30th) of a second. With our current values for n and k, the two graphs are very similar so there wasn't a reason to alter them.

Figure 4. The movie capture graph from our 4th trial (4 filters)
with a linear fit of v is approximately 1.942 m/s while the model
with a close fit has a slope of 1.8836 m/s or 1.884 m/s.

Figure 5. The lay out for the numerical calculation on excel for trial 4. Velocity seemed
 to approach 1.8836 m/s and can be determined as the terminal velocity for this trial.

Figure 6. The work on deriving different sections of the excel lay out. There is also the
propagated uncertainty of the force of air resistance due to the uncertainty for n and k as well as the error between theoretical and experimental values of terminal velocity.

Conclusion:

We have concluded that the amount of uncertainty (shown in fig 6. above) for the force of air resistance due to the uncertainty which are:

dk= .0007

dn= .1.645

We have propagated an uncertainty for our Force of air resistance for trial 4 as:

Fair. theoretical= .0429 ± .008 N

or  ± 18.6% uncertainty
Fair experimental = .04586 N
+6% more than theoretical

Seeing that the level of uncertainty is poor, our model would not be very useful in calculating the force of air resistance. It may be due to the difficulty of tracking the filter's movements with a similar color background making the filters hard to track. The inaccuracies in the tracking may have caused this large amount of         uncertainty.