December- 4 2014 Semicircle Physical Pendulum

Semicircle Physical Pendulum 

Objective:
  Find the period of a half circle mass acting as a physical pendulum at two different orientations.

Set up:
  As seen in the photo to the right, we (each group) had to cut semicircles our of a light material and use calipers to measure its dimensions and weight its mass with scales. We then attached light rings to the edge of orientations and record its motion at a small angle.

Data Collection:
  Firstly we had to find the moment of inertia of the center of mass of the object, but in order to do that we had to find the moment of inertia about the flat side of the semicircle and the location of the Center of Mass to apply the parallel axis theorem:
Iflat= ICOM + m(d^2)
ICOM= Iflat  - m(d^2)

As seen below, we found the center of mass first which is at the middle of the x axis parallel to the flat side and (4R/3pi) to the y direction toward the curved side. Then using what we found our moment of inertia to be about the flat side and subtracting according to parallel axis theorem we found the moment of inertia about its center of mass. We used this to then find our moment of inertia about the middle of the curved edge of the semicircle. We found that about the flat side has less inertia than id the semicircle were rotating about the curved side.

From there we set up our scenario as a dynamics problem of unequal non- constant torque to meet the form of simple harmonic motion assuming a small theta or angle. We found that our theoretical calculation for period if it were turning on its flat side is less than 1% off as is expected for using a taylor series on the sine value to assume it to be just theta. However in the actual for the curved side rotating, the period is exactly the same as if it were to be about its flat side which the theoretical calculations did not suggest however we were only 1.9% off.

Conclusion:
It seems friction or damping of the system may have altered the results. Another possibility is that we may have had a too large of a theta during our test for the curved side of the circle trial otherwise the sensor may not have read such a high average for its period.

Mass Spring Oscillations Lab Nov.- 20- 2014

Mass Spring Oscillations Lab

Objective: Find the relationship between period and spring constant as well as with the amount of mass that is oscillating.

Set up: There were a total of four groups each with a different spring, the appropriate mass so that way each group would have the same amount of oscillating mass determined by the equation (mosc= mhang+ mspring/3). We used a motion sensor and lab Pro to graph our results and place an appropriate fit.

Data Collection: The first step for our group to do is to find the spring mass of our given spring. To do so we weighed the spring for its mass, We then hung the spring to measure its length at a relaxed state, then once more with a .1 kg mass hanging at the end without oscillation:

Fnet= 0N= mg - kxstretch               xstretch= x- xo = .067 m

                                                   m= 26 g

mg= k(x -xo) => k= mg              k= 14.64 N/m

                              (x- xo)

We then went on to derive our equation for omega and, eventually, for our period using Newton's second Law as seen in (fig. 1).

Figure 1. All my derivations and notes from the set up as well as comparisons to our actual
unfortunately there is a mistake in the percentage comparison on theoretical period to actual.

Figure 2. Results of our 4 trials to find the average period for the set up.

We now had our theoretical equation for period of our set up and we tested with 4 trials for consistency. We counted the periods with in a 5 second trial and found that our theoretical calculation was almost 100% accurate to our actual. When we graphed our results compared to others and graphed them with an appropriate curve fit we find something interesting. First, how we analytically found the curve fit to be a power fit was this observation:

Our period came to a form of:

T= 2pi*(m/k)^.5

It seems to follow the power fit seeing as there is a constant followed by a value with a variable (either k or m) multiplied to a certain power. Units -wise is simple, since the constant is unitless or in radians so all that remains is to show that what's in the parenthesis needs to be s^2:

T= C*[(kg )]^.5 => [(kg*m) ]^.5        =>  [1 * (s^2)]^.5 = s

          [N/m ]            [kg*m/(s^2)]             [1       1   ]

Seeing this sort of relationship with mass and spring constant we can predict as k=> infinity with a constant mosc T would approach a period of zero at an curve similar to y= (1/x)^.5 seeing as k being in the denominator. However, in that respect, if mosc => infinity with a constant k then T will also approach infinity at an inverse rate compared to the predicted behavior of T vs. k, y= (x)^.5 , seeing as m being in the numerator. According to the graphs below with appropriate curve fit it seems as though our predictions were correct.

Figure 2A. Our graph of T vs. oscillating mass with appropriate curve fit.

Figure 2B. Our T vs. k graph with appropriate powerfit.

Lastly, there was the regard to see how much a 5% spring constant error would affect a calculated period in which for our set up we found the error to be around only 3% difference compared to both actual and correctly calculated spring constant as seen in (fig 1.).

Conclusion:
We found the our source of any uncertainty would lie in the calculations of other groups as well as any inconsistencies in the spring seeing as it swung side to side shortly after the 5 seconds of our trials.

Conservation of Linear and Angular Momentum Nov. -13- 2014

Conservation of Linear and Angular Momentum

Objective:

   Show that the linear and rotational momentum of the system is conserved during this rotational collision.

Set Up:

   There are two set ups for this lab:

1) With a small downward ramp and a steel ball on top of a table, we will use the measurements of its final position to calculate velocity.

2)From there we have a rotating apparatus with a catch for the ball so we may use the found velocity to find the resulting rotation velocity.

Calculations:

We used what we know about the ball's moment of inertia and measurements of the ramp to the table to calculate the ball's theoretical velocity down the ramp without slip. From there we used the measurements of the ball's final location and solved the actual velocity of the ball which when compared to theoretical is smaller suggesting some slip. With this information we moved on to our second set up, but before doing so we have to calculate the inertia of the rotating apparatus in order to find the final momentum of the inelastic collision. To do so we used a hanging mass as a known source of torque then found the average value of alpha to determine the inertia of the apparatus. From there we placed the ball in a similar height on the ramp to have a consistent initial velocity and had it collide to the apparatus. With our calculations we found our final omega to be 1.87 rads/ s with a theoretical value of 2.43 rad/s.

Figure 1. Calculation for both set ups in the lab.

Conclusion:
  It appears that since there is slip of the ball on the ramp greatly affects the value of the actual compared to the theoretical.

Ruler Collision Activity Nov.- 18- 2014

Ruler Collision Activity

Objective:

   Derive an expression for finding the final height for an end of a swinging meter stick after an inelastic collision.

Set up:

We used a meter stick with a hole drilled in at the .2 m mark that can freely swing. We then have a clay piece with tape placed where it may collide with the meter stick and will get stuck to it.

Data Collection:

As a change in usual procedure, we performed the lab prior to doing the theoretical calculations. As seen in (fig. 1), we found our maximum height to be .657 m. With that data we measured the height of our zero mark relative to the ground which came to be .5 m, so according to this information the maximum height from our zero in  height is .157 m.

Figure 1. Our video capture set up and one data point at the maximum height after the collision.

To theoretically calculate our maximum height we split this problem into three phases:

I) Energy from GPE to KErot.

       This step was to find our value for omega just before collision.

II) Rotational Collision Lbefore = Lafter 

      This step assumes that the rotational momentum is conserved to find the resulting omega for both the clay and bottom of the meter stick. Since there is a fixed pivot point we do not need to find the new COM and perform the parallel axis theorem.

III) Energy from KErot to GPE since the final rotational velocity is zero

      At this final phase we determine what the final height of the meter stick will be.

In the figure below (fig 2.) we see that our calculated results came to be .131 m which is lower than our actual height by 16.5%.

Figure 2. This is my calculation work of the three phases and overall answer with the actual on the bottom.

Conclusion:
  Our calculated value was significantly lower than the actual which means there must have been a mistake in scaling during the video capture thus causing the data to read higher numbers than it should.

Moment of Inertia of a Uniform Triangle about its Center of Mass Nov.- 13- 2014

Moment of Inertia of a Uniform Triangle about its Center of Mass

Objective:

Figure 1. The Pasco set up with a triangular mass attached
whose inertia we have to find in different orientations.

Use the apparatus to calculate the inertia of the triangle at different orientations using the average angular acceleration from a hanging mass rotating the apparatus.                             Set up:                                                                    Using the apparatus similar to the set up of the previous angular acceleration lab, the only changes is a triangle and its attachment to a Pasco rotational sensor (as seen in fig 1.).

Data Collection:

 Initially we had to derive the moment of inertia for a triangle about its base center with the help of the parallel axis theorem resulting to:

I= M*b
      18
From there our theoretical values for our .455 kg (.15 m X .098 m) right triangle in both orientations are:

about COM with base b=.15 m
I= .184 kg*m^2

about COM with base b= .098 m
I= 2.428 * 10^(-4) kg*m^2

As we see, the smaller the length of the base of the triangle, the smaller its moment of inertia which coincides with our conceptual understanding of moment of inertia.

As seen in the figure above, we also used what we know about our given set up to derive how we would calculate our inertia for the triangle. Unfortunately, due to time we were only able to test for one of the two orientations of the triangle using a .4 kg weight. Using the known weight we used the mass as a source of a known torque to find our total inertia then subtract the known inertia for the disk apparatus using the value of the difference as our found triangle value of orientation about COM with b= .15 m with an I=.1843 kg*m^2

Conclusion:
With our results only ,1% off, it may be safe to assume that our derived formula may produce values very close to our theoretical values. Only sources of uncertainty may lie in friction in either pulley and the fact we did not check consistency with a second trial.

Moment of Inertia Nov. 4- 2014

Moment of Inertia Lab

Objective:

Calculate the object's mass and predict the rotational acceleration if a cart was attached to it on a string going down a frictionless slope.

Set up:

First we are given a large metal disk on a central shaft(it has a known overall mass) and a caliper to measure its volume. Second, we have a track and a cart on a light string tied to the shaft of the apparatus.

The Approach:

  We approached this problem assuming that the disk has a uniform distribution of density, and that the shaft has negligible friction. From there find the volume of the apparatus by separating it into the sections (two small cylinders that stick out of a large disk) as seen in the figure below. Once we find their volumes, we find what ratio the large disk is compared to the small cylinders since density is evenly distributed by the same ratio.

M = V               M,V= values for large disk

m     v                m,v= values for small cylinders (each)

Since V= 13.18v, then compared to 2v, V is 6.59 times larger in comparison to the sum of the cylinders. Thus, the disk's mass is 6.59 times larger resulting to mass of 3.915 kg from the total 4.615 kg.

 Now that we have the mass values, we calculate each of their inertia and add them together for a system inertia of:

(Big) + 2(small)= (.02 kg*m^2)+ 2(3.99*10^-5 kg*m^2)= 2.008*10^-2 kg*m^2

Figures 1A and B. A picture of the rolling disk apparatus and Lab work for each part including a small diagram of the
second set up for this experiment.

 With the total inertia known, there was a need to derive the predicted angular acceleration  for a cart of a mass (m) moving down a frictionless track while attached to the shaft of the rolling disk. We start off as treating this problem as a dynamic problem or unequal torque is the tension of the string times the force on the cart parallel to the track which turns out to be the sine value of the force of gravity on the cart. From there we had the tangential acceleration become terms of angular acceleration and radius of the turn shaft. The result is the following:

α= mgsinθ*r       Where m is the mass of the cart, r is radius of the shaft, θ is the angle of the slope,
  (m(r^2)-Itot)         and Itot is the total inertia of the apparatus.

We set our surface at 25 degrees with a .4 kg cart attached to the shaft. Before beginning the trial we predicted with out derived formula that the α of the cart would be -1.253 rads/s^2 and our actual came to be -1.266 rads/s^2, only 1.04% off. 
Conclusion:
It seems our derived equation is accurate enough to predict the angular acceleration of our cart set up. Any apparent source of error would source from the friction of the disk, friction of the surface and friction of the axle. 

28- Oct. - 2014 Rotation Lab

Rotational Acceleration  Lab

Objective:

Figure 1. Lab apparatus for this set up.

  To see what factors affect angular acceleration α.

Set- Up:

 As seen in (fig 1.) we have a Pasco rotational sensor with a hanging mass, 2 steel disks and an aluminum one, two torque pulleys and a Lab Pro. We exchange different parts of this set up to change different variables to see what affects angular acceleration.

Data Collection:

Figure 2. The derivation of finding the inertia of the apparatus
as well as what affects angular acceleration. 

First we derived an equation for angular acceleration (α). We approached this problem as a dynamics problem with :

unequal net torque= (inertia of the system) *α

Since our torque will be known as well as the radius of the torque pulley (r), and inertia of our system the only unknown we would have is the average angular acceleration αavg.

We found this to be:

αavg = αup + αdown = MB*(g-asys.)r

                2                  Isys.

With this we find that if we get the sum of the absolute value of each (αup,  αdown) and average it, then we get the average angular acceleration. This is useful to compare resulting α's when we altar the main variables found in the other equation that can alter angular acceleration:

• Hanging Mass
• Radius of the torque pulley
• Inertia of the apparatus

With each of these in mind we performed 6 trials each changing one of these factors (seen in fig 2.). The first three trials were purely change in mass with a single steel disk as the system's inertia. The fourth trial altered only the radius of the torque pulley. The fifth altered both radius and system's inertia by using a lighter aluminum disk of same dimensions. The last altered the radius and doubled the inertia of a steel disk by having the hanging mass overcome 2 steel disks stacked and stuck together.

Figure 2. Our results for the 6 trials of our lab with recorded angular accelerations and αavg.

Conclusion:
From this we see that the apparent that the factors we found to affect angular acceleration all seem to have interesting relationships. When one looks at the results, doubling or even tripling the mass or radius has the same result as having half or a third of the inertia showing just like in our derived equation that they are inversely related. Any source of uncertainty comes down to the apparatuses air distribution and cleanliness of the disks as well as small imperfections of reading the rotations by the machine.

9- Oct.- 2014 Momentum/ Impulse Lab

Momentum Lab

Partner: Adam Moro

Objective: 

  Show that in each scenario (both elastic and inelastic collisions) that momentum is conserved .

Set Up:
There are three set ups:

Figure 1. A set up very similar the first set up (except the extra mass)

1) We use a track and a cart with a force sensor attached to the top of it, another cart that is stationary with a springy bit extended. A motion sensor is on the opposite end of the track reversing the settings in the motion sensor so it read motion toward it as positive. You must collide the movable cart with the stationary one and record your data.

2)Repeat the first experiment, but add a 500 g weight to the movable cart.

Figure 2. Our second set up has an extra half a kg
to increase the amount of impulse in the system

3) Replace the stationary cart with a platform with molding clay on it. Attach a nail to the end of the force sensor and collide.

Data Collection:
  With each experiment with its own objectives each had a very similar approach. For each we had to find the impulse or change in momentum (p) which means there are things we have to consider:

• Since friction in this case is negligible and the surface is level, the amount of net force just before collision in each case is 0N
• The point when the cart is experiencing maximum linear force is during impact at instantaneous rest
• There is no net force after impact acting on the cart
• As we will see later, since m*vf - m*vo=  F*t where t is the how long a collision took place, it varies in different scenarios

As seen below in (fig 4) we use data such as initial and

final velocity from the collision, collision time interval

Figure 3. Our cart now has a nail at the front meant to stick
into a wad of modeling clay for an inelastic collision.

and maximum force experienced during the collision. We then compared the theoretical calculations to the actual results with calculated error for each in percent. Where we got the numbers we used for these calculations will be on the next few figures.

Figure 4. Lab Calculations for each experiment with theoretical answers compared to
actual using known mass and given velocities from our results.

7- Oct.- 2014 Magnetic Conservation of Energy lab

Magnetic Conservation of Energy Lab

By: Jordan Fuentes

Partners: Adam Moro, Jonathan Cole, Jorge Gonzales, and Andrew Tek

Objective:

  Show that energy is conserved in this set up at differing angles.

Set up:

  We use a frictionless cart with a strong magnet on one end approaches a fixed magnet of the same polarity. The track it is on would need to be leveled first then tilted at small increasing angles and a motion detector with its direction reversed so towards it is positive. 

Figure 1. The overall set up of the lab.

Data Collection:
Using the set up as described above we tried to find if the system's energy is conserved. To do so, we had to identify what sources of energy are going into the system:

• Kinetic Energy
• Gravity Potential
• Magnetic Potential

The first half of the Lab is to derive a formula for magnetic potential. For every form of potential energy there tends to be a value of displacement interacting with a force which we found to be the sine value of gravity. To find the approximate behavior of our magnetic potential we used the set up to see changes in angle of the track result to what sort of change in the distance between the cart and the magnet as the cart reaches an instantaneous stop. We then recorded each result and graphed it (as seen in fig 3.) as well as used a power fit of Ax^B due to the assumption that the graph had that sort of relationship. Our values of A and B show to be (.0002133± .0002803) and (-1.903 ± .3145) respectively. We then assume that the initial position of the cart is far enough to not be affected by the magnet and that change in gravitational potential energy is negligible. We then integrate our found power fit from infinity to the value of our separation distance (or when the value of MPE is at its maximum). We chose to integrate because of the non- constant force from the magnet. However, as shown in (fig 2.), we see that after integrating we have a Magnetic potential formula of:

A*(position of the cart - (initial position = 0))^(B+1)
                                    [B+1]

Figure 3. Our force vs. distance the carts stops (dr) with our curve fit where trial 6 (6th point) follows the power rule best and is used to test our derived formula for MPE.

Conservation Test:

Figure 4A. This is a graph of a trial at a random small angle
and each values of energy is graphed, as kinetic energy
approaches 0, magnetic potential reaches its max value.

  Our next step was to test our derived formula by using the same set up, but place at an unknown angle and graphing the energy results to see if it is conserved. After our trial, we set calculated columns for kinetic energy and slightly altered our derived equation by having its value be positive at all times to check for consistency in total energy and also put into account the separation from our motion sensor to the end of the magnet (.405m apart).

(.0002133)*(position of the cart - .405)^(-.903)
                   absolute value [-.903]

Figure 4B. Numerical evidence of what was described in
figure 3A.

 Magnetic Potential peaked at the exact point where kinetic was found to be zero at the 1.25s mark which is a sign that the system is conservative. However, the total energy in the system seemed to increase then decrease during repulsion of the cart thus suggesting a non-conservative system.

Conclusion/ source of Error:
Seeing as though the potential for gravity was ignored, it may have greatly affected the results to show that this system is in fact conserved. Behavior- wise, the graphs of kinetic and magnetic potential energy did show to be consistant to expected. Uneveness of air flow in the track and a possibly unbalanced cart may have cause friction in some parts more than most.

14- Oct.- 2014 2D Collision Lab

2-D Collision Lab

Objective: Show that momentum is conserved for a 2-D collision.

Set Up:

Figure 1. A screenshot of motion tracking
 one of our trials.

With a camera, smooth surface, and a camera set up above looking down (as seen in fig 1)we take 3 round balls 2 of which are made of steel and are similar masses while the third mass is made of aluminum with a smaller amount of mass than the steel balls but relatively the same size. We do two trials, one with two steel and another trial with one steel and the other aluminum.

Data Collection:
As we finished video recording the collisions, we tracked their path in each frame as it was graphed (as seen in Fig 2A. and 3A.) From there we created linear fits for each mass before and after collision to get an average velocity in both ends of the trial. We then made calculated columns for momentum of the system in the x and y axis at any given time and then another calculated column for kinetic energy (seen  in fig 2B, and 3B).:

1) We found that the momentum in the x and y directions:

before collision: (mass 1)*(V1 in x or y)+ (mass 2)*(V2)=
after collision:     (Vf in x or y)*(sum of both masses)

2) We also found momentum of the system to calculate average overall velocity for each mass for any given time during and after collision:

p^2 = px^2 + py^2
m*v^2 = m*vx^2 + m*vy^2

3) Then used the found velocity to find total Kinetic Energy at any given time:

KE= .5*mass 1*(v^2) + .5*mass 2*(V^2) = 5*mass 1*(vx^2 + vy^2) + .5*mass 2*(Vx^2 + Vy^2)

Our first trial with the two steel balls show that resulting velocities after collision are close to parallel since they are the same mass (seen in Fig 2A.). However, possibly due to spin on the ball, it does not seem to show much consistency or conservation in the Kinetic energy column. Nevertheless, it does show consistency in in each momentum column (as seen in Fig 2B.). In trial two since the aluminum ball was initially in motion in this collision, it makes little change to the velocity of the steel ball it collides with and does not seem to have a great change in velocity (seen in 3A.). Numerically, it seems to show much more conservation in kinetic energy compared to trial one while still maintaining consistency in momentum of the system (seen in Fig 3B.).

Figure 2A. Trial one positions before and after collision with linear fit graphs to
find each of the average velocities for calculating momentum in both x and y directions.

Figure 2A. Our numerical results for our first trial for momentum in both the x and y directions as well as our calculated kinetic energy at any given time.

Figure 3A. Trial two positions before and after collision with linear fit graphs to 
find each of the average velocities for calculating momentum in both x and y directions.

Figure 2B. Our numerical results for our second trial for momentum in both the x and y directions as well as our calculated kinetic energy at any given time

Conclusion:

This lab for the most part shows that in both systems there is a conservation in momentum, but conservation in kinetic energy is only found in trial two. Initially it was suspected it was due to misreadings of masses, but we remeasured our masses and found them to be the same once again. We believe that any source of uncertainty would be from spin of the ball, imperfections of the ball, imperfections of the surface and approximated motion tracking credited to human error.

2- Oct.- 2014 Total Energy Lab

Total Energy Lab

Objective:

  Show that energy in a system is always conserved when one adds all forms of energy during each point in time and the sum stays consistent.

Set- up:

  In this set up we used a mass of 600g attached to an end of a hanging spring (as seen in fig. 1a- 1b). Underneath the hanging spring we put a motion detector with its settings set to zero when the spring with the mass is at a relaxed position and its readings reversed so toward the sensor is positive.

Data Collection:

Figure 1a. Our mass on the hanging spring

We set the weight at a height where the spring's length was the same as it was at a relaxed state without a weight attached to it. From there we recorded the data for position, velocity and acceleration for a trial of  seconds. From there we used that information to numerically calculate different components of what we thought was what made up the system's energy. As a clsass we observed that there were 5 forms of energy in the system that is not negligible (as seen in fig 2.):

Kinetic Energy of the Mass (m): .5*m*v(mass)^2

Gravity Potential of the Mass (m): g*m*(position)
where mass (m)=  .6 Kg

Elastic Potential of Spring: .5*k*(position- relaxed)^2

we calculated spring constant k to be :
F=k*(stretch)  where stretch= x = 25 cm

Figure 1b. Our motion sensor with appropriate
settings placed directly under the hanging mass
set up. 

Ms*g= k*x where Ms = .1 Kg
k= m*g = 23.52 spring constant
        x
Kinetic Energy of Spring (Ms)= (Ms)*V^2
                                                           6
Gravity Potential of Spring (Ms) = Ms*g*position*(.5)

To prove that energy of this system is conserved, we add up all the forms of energy and we then see if the sum is consistent. Numerically it is apparently close and evidence of this can also be seen in fig 3. where each form of energy is graphed and the orange graph on the top is the sum.

Conclusion:
Energy is in fact conserved in this system. My level of uncertainty is around 3% seeing as not the entire spring was uniform, but behaved consistent enough during the trial to complete the objective.

Figure 2. Each of our calculated values for different forms
of energy found in the set up with an approximated sum
of all the energy.

Figure 3. Graphed values our our calculated energies with the
sum of the energy values for each point in time graphed at the
top in orange with consistency.

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