15 April 2015: Impulse-Momentum Activity

Purpose:

The purpose of this experiment was to prove that the impulse-momentum theorem applies to systems that involve both elastic and inelastic collisions  (such as the one included in this experiment). The impulse-momentum theorem states that the change in momentum of an object is equal to the impulse applied on an object.

Apparatus:

Figure 1: Apparatus used in the experiment

The apparatus of this experiment consisted of a metal track with negligible friction, which was used to allow a cart to move in one direction on a level surface. The cart had attached to it a force senor, which was used to measure the force applied on the cart during the collisions. A motion sensor was also applied to the side of the track to measure the velocity of the cart. A computer with Logger Pro software was needed to collect and record the data from the force sensor and motion detector. To simulate a nearly perfect elastic collision, another cart with a springing piece was fixed on the other side so that upon collision with the force sensor on the moving cart, the force would be recorded and the cart would be bounced off to simulate an elastic collision. The cart was fixed onto a vertical rod that was held to the table by a C-clamp. Lastly, to simulate an inelastic collision, a wooden block with clay attached to its side was used instead of the springy cart, and this block was fixed to the same side of the track. In addition, a nail was attached to the force sensor via a rubber stopper to ensure that the cart sticks to the surface it collided with.

Procedure/Discussion (Part 1):

Once the apparatus with the springy cart was set up, the mass of the moving cart (with the force sensor attached) was measured and recorded to be 0.683 kg. Then, the force sensor was zeroed and a 500 g mass was hung on it to ensure that it is accurate in reading the actual force (for the calibration, a force of about 4.9 N). Then, the fixed cart was set up so that the hook on the force sensor directly touches the springy part during the collision. Then, a test run was made to ensure that everything is working normally, which turned out to be so. 

Then, on Logger Pro, the positive direction reading on the motion detector was reversed so that the cart's initial velocity, moving away from the detector, is read as negative. Next, a specific experimental file was opened on Logger Pro so that the motion detector would record data points at a faster rate, so that the results of the experiment would turn out more accurate due to the collision being a very small amount time. 

Next, the actual experiment was run. Before starting it, the force probe was once again zeroed. Then, the cart was given initial push and right after the cart started moving the data from the force sensor and the motion detector was begun to be recorded. Data was recorded until a few seconds after the collision between the two carts ended.  The graphs of the velocity of the cart and the force applied by the cart are shown below:

Figure 2: Force / velocity vs time graphs of the elastic collision of the cart

As can be seen on the force vs time as well as the velocity vs. time graph, since the friction of the track is negligible and it is level, the net force exerted on the cart right before and after the collision is zero (velocity is constant). Then, the magnitude of the force on the cart is maximum when the springy piece of the fixed cart is fully compressed and is right above to go back to its natural state, which is seen as the top of the peak in the force vs time graph. To prove this, looking at Hooke's Law (F = -kx), the force of the spring is the largest when x (the distance the spring is stretched or compressed) is the largest, which is why the peak resembles max compression of the spring. Lastly, looking at the force vs time graph, the duration of the collision (the amount the peak stretches for) is about 0.18 seconds.

Then, using the force vs. time graph, the area under the peak (the time interval where the collision occurred) was found using Logger Pro. The area under that curve provides the net impulse applied on the cart. Then, using the measured mass of the cart and the velocity data obtained from the motion detector, the momentum of the cart was obtained throughout the experiment. The momentum is the mass of the cart multiplied by the velocity of the cart at each specific point in the experiment. 

Figure 3: Collected and calculated data for this experiment

For example,  to calculate the momentum of the cart at 0.66 s, p = (0.683 kg) (0.328 m/s) = 0.224 kg m/s. Then, the change in the momentum of the cart was compared to the net impulse applied on the cart (area under the force vs. time curve during the collision). To get the change in momentum, the momentum after the collision and before the collision had to be found. The graphs used to find them are shown below:

 Figure 4: Graph of momentum & force vs time displaying impulse and momentum right after collision

Figure 5: Graph of momentum & force vs time displaying impulse and momentum right before collision

The total change in momentum of the cart during the collision can then be found by subtracting the momentum after the collision by the momentum before the collision. Doing so, the obtained change in momentum during the collision  is 0.224 - (-0.264) = 0.488 kg m/s. The calculated impulse is 0.5170 N s (N s = kg m/s). As can be seen, the impulse and the change in momentum are not equal but, ruling out the error in the data due to uncertainties present throughout the experiment, are close enough (5.8% difference) to prove that the impulse-momentum theorem applies to this system and all other elastic collisions.

Conclusion (Part 1):

Looking at the velocity vs time graph shown in figure 2, the velocity is not exactly constant before and after the collision. This is due to error in the measurement of velocity cause by uncertainty in the motion detector. In addition, the fact that there is friction in the axles of the cart was never taken into account. Also, air resistance was never taken into account, which does cause the velocity of the cart to decrease slowly over time. 

Also, looking at the same velocity vs time graph, the speed of the cart before the collision is not equal to the speed of the cart after the collision, showing that the collision was not completely elastic. The causes for this include the fact that the spring on the fixed cart is not ideal, therefore resulting in some energy lost to vibrations and heat in the spring. In addition, there is friction in the springy piece on the fixed cart, also resulting in loss of energy from the system, known as non-conservative work.

Looking at the force data, the force read is not zero when the cart is not colliding. This is due to the fact that the cart is experiencing air resistance, which was assumed to not be present. In addition, there is also uncertainty in the force sensor.

Lastly, it was found that there was small difference between the impulse applied on and change in momentum of the cart. This is caused by uncertainties in the measurements and the equipment used as well as the fact that the collision is not ideal and not perfectly elastic. Overall though, although the uncertainties do have some effect on the results, they are not large. They have a noticeable effect on the results because the values of the results are fairly small, so any uncertainty would have a noticeable effect on them. To lower the uncertainty in this experiment, more accurate and precise equipment would be needed as well as the use of larger measurements to result in larger results that are not as sensitive to any uncertainty.

Procedure/Discussion (Part 2):

The same procedure in part 1 was performed again for part 2, with the exception that the cart was made heavier by adding to it a 500 gram mass. The graphs displaying the velocity and force data obtained are shown below:

Figure 6: Force / velocity vs time graphs of the elastic collision of the more massive cart

As seen in the previous graphs for part 1, the net force on the cart is zero before and after the collision, and the peak represents the collision. Then, the momentum of the cart at each point was calculated and is shown below:

Figure 7: Collected and calculated data for this experiment

Then, the area under the peak of the force vs time graph was found (impulse) and the change in momentum of the cart was found using the same process (subtracting momentum after collision by momentum before collision).

 Figure 8: Graph of momentum & force vs time displaying impulse and momentum right after collision

Figure 9: Graph of momentum & force vs time displaying impulse and momentum right before collision

The impulse applied on the cart was found to be 0.5820 N s and the change momentum of the cart was calculated to be 0.583 N. They are almost exactly equal (0.17 % difference), strongly proving that the impulse-momentum theorem still applies to an elastic collision using heavier objects.

Conclusion (Part 2):

As stated before, the velocity is not exactly constant before and after the collision because of error in the measurement of velocity cause by uncertainty in the motion detector, as well as the presence of friction in the axles of the cart and air resistance. Also, the speed of the cart before the collision is not equal (although very close) to the speed of the cart after the collision, showing that the collision was not completely elastic, due to  the spring  not being ideal and friction in the springy piece on the fixed cart, resulting in loss of energy from the system.

Again, the force read is not zero when the cart is not colliding because of air resistance and uncertainty in the force sensor. Lastly,  there was a very small difference between the impulse applied on and change in momentum of the car, which is caused by uncertainties in the measurements and the equipment used as well as the fact that the collision is not ideal and not perfectly elastic. The uncertainty is very small, though, proving that this model is very accurate to simulate the impulse-momentum theorem. Stated as an improvement for part 1,  the fact that higher measurement and result values was done in part 2 by increasing the mass of the cart greatly decreased the effect of uncertainty on the results.

Procedure/Discussion (Part 3):

The same procedure for part 3 was done in part 2 except the fact that the wooden block with clay attached was used instead of the cart with the springy device, and that a nail was attached to the force sensor using a rubber stopper. This was done to model an inelastic collision, where the moving object collides and sticks onto the surface it collided with, and to prove that the impulse-momentum theorem also applies to inelastic collisions. The added 500 g mass was left on the cart to give the cart a mass of 1.183 kg. The impulse of an inelastic collision would be smaller due to the fact that the collision time is smaller and that the object is stopped and not given force to move opposite its initial direction. The graphs of force vs. time as well as velocity vs time are shown below:

Figure 10: Force / velocity vs time graphs of the inelastic collision of the more massive cart

The net force is also zero before and after the collision and the peak is the collision. Because this collision was inelastic, the final velocity is zero. The momentum was also calculated and is shown below:

Figure 11: Collected and calculated data

Then, as before, the impulse was found by taking the area under the force v time graph and the change momentum during the collision was found (momentum after minus momentum before collision). The graph before the collision is displayed below:

Figure 12: Graph of momentum & force vs time displaying impulse and momentum right before collision

The reason that the graph showing momentum after the collision is not shown is simply because the momentum after the inelastic collision is zero (since the velocity is zero). The curve of the graph is different from the curve of elastic collisions since a negative force is needed to stop the moving cart. The impulse was found to be 0.3891 N s and the momentum was calculated to be 0.341 kg m/s. As expected, the impulse/momentum change of an inelastic collision is less than an elastic one. The difference between the two is somewhat large (13%) and is caused by the uncertainties and errors in the experiment, also proving that the impulse-momentum theorem applies to inelastic collisions

Conclusion (Part 3):

In this part, the velocity of the cart came out to be fairly constant before the collision (and obviously after since the velocity was zero). However, the force rad was not zero before and after the collision, which is due to uncertainty in the force sensor and the fact that the rubber stopper on the hook of the force sensor is applying some force on the hook (the mistake is that the force sensor was not zeroed after placing the nail and the rubber stopper on the force sensor). Lastly, as stated before, the uncertainty in the measurements and the equipment was the cause of the error in the difference between impulse and momentum. The uncertainty is small, but because the data and results also have small values, even the small uncertainty has some effect on the results, resulting in error. In addition, this part of the experiment was performed by only one person, making it difficult to collect data and do the experiment at the same time, resulting in larger human error. Overall, though, it can be concluded that the model for this experiment was a strong portrayal of the impulse-momentum theorem in an inelastic collision as well as an elastic collision for the previous parts. To lower the error for this part, the experiment should be done with more than one person and better equipment should be used.  

15 April 2015: Magnetic Potential Energy

Purpose:

The purpose of this experiment was to derive an equation that models magnetic potential energy by relating potential energy to force. This model was then used to prove that the conservation of energy principle applies to the system used in the experiment.

Apparatus:

 Figure 1: Apparatus of this experiment

The apparatus of this experiment consisted of a metal track shaped as a wide triangle that is hollow inside and contains many holes in it. These holes are to allow air to run through the track and out of it so as to minimize almost all friction forces between the track and a gliding object. It is also shaped as a triangle to keep a gliding object from going off course. A small magnet was fixed in place onto the track. An air track glider containing a magnet and a metal sheet was also used in order to derive the magnetic potential energy function and determine if conservation of energy applies to this system. An air pump was also used to run air through the air track. Textbooks were used to change the angle of the air track and a digital protractor was used to measure that angle. Lastly, a motion detector and a computer with Logger Pro were needed to provide data for the calculations of the energies of the system.

Procedure (Part 1):

Once the apparatus was set up, the air track was set at an angle and the track glider was allowed to come close to the fixed magnetic and come to rest while the air track was on. Then, the relationship between the magnetic potential energy and force was identified in this system by noticing that the force applied by the glider's weight parallel to the track is equal to the magnetic force applied on the glider, or mgsin(theta) = Fmag. Then, using this force relationship, a graph of magnetic force vs. the separation distance between the magnet on the cart and the one fixed on the air track was created. The data used to plot this graph was found by changing the angle the air track is sloped on, since, looking at mgsin(theta), the only value that can be changed is theta. Once the slope of the track was changed, the angle was measured via the digital protractor and the distance between the faces of the magnets was measured through the use of a ruler. The mass of the glider also had to be measured, which was found to be 0.343 kg. The data gathered  along with the graph created by that data is shown below:

Angle (degrees)	Force (N)
1.30	0.0763
2.80	0.164
5.90	0.346
10.5	0.613

Figure 2: Graph of force vs separation distance of magnets (r)

Discussion (Part 1):

A sample calculation for the force based on the angle is shown. Using theta = 10.5 degrees, the force was found to be F = (0.343 kg) (9.80 m/s/s) (sin(10.5)) = 0.613 N.

The last two data points had to be crossed out since they resulted in a large deviation from the power fit. The force as a function of r equation was assumed to be a power law based on the slope of the curve created by the data points. The force of the magnets was assumed to take the form of F = Ar^n, where A and n are obtained from the power fit. The value of A was measured to be 8.464 x 10^-7 +/- 6.325 x 10^-7. The value for n was found to be -3.213 +/- 0.1808. The uncertainty in B (5.6%) is reasonable compared to the errors and uncertainties in this system. These uncertainties/errors include assuming that no friction is present between the track and glider when in fact there was (just not so large), uncertainty in the measurement of the angles, and uncertainty in the measurement of the separation distances between the magnets.  On the other hand, as can be seen, the uncertainty in the value for A (74.7%) is very large. This is primarily caused by the fact that A is a very small value (10^-7!). It is also caused by simply the use of inconsistent data, which is why the last two data points had to be crossed out. Taking the rest of the errors/uncertainties into account, these resulted in a large uncertainty in the value of A.

The equation for the magnetic force was found to be F(r) = (8.464 x 10^-7) r^-3.213. Taking the negative integral of the force function with respect to r, the potential energy function between the magnets can be found. The potential energy function was derived to be U(r) = (3.825 x 10^-7) r^-2.213.   

Procedure (Part 2):

Next, the track was made so that it is flat (0 degrees) and the glider does not slide when left alone. Then, a motion detector was applied to the side of the track with the fixed magnet to be able to measure the position and speed of the glider. With the air track off, the glider was placed close enough to the fixed magnet so that the relationship between the separation distance between the magnets and the distance between the glider and motion detector were found. The distance between the motion detector and glider was measured to be 0.702 m and the separation distance of the magnets was found to be 0.149 m. Subtracting the two, it was found that the relationship was 0.553 m. 

Then, the glider was moved to the far end of the track and given a small push. The motion detector was run, and the position and velocity of the glider were measured and collected. Then, the KE of the cart was found using the measured velocity and the magnetic potential energy was found by using the derived function, where r equals the position data from the motion detector minus the relationship between the separation distance of the magnets and the separation between the motion detector and glider (0.553 m). For example, at 1.20 seconds, the kinetic energy of the cart is 0.5 (0.343 kg) (-0.259 m/s)^2 = 0.0115 J. The magnetic potential energy is (3.825 x 10^-7) (0.791-0.553 m)^-2.313 = 0.00 J. The data recorded and calculated as well as the graph are shown below:

Discussion (Part 2):

Figure 3: Recorded and calculated data of the system 

Figure 4: KE, U magnetic, and total energy of the system

The kinetic energy and magnetic potential energy graphs were expected. At the point of "collision" the kinetic energy would drop down to zero and then come back up. In addition, it was expected that U would be almost zero, increase up dramatically during the "collision", then fall back down to nearly zero, as seen in the graph. However, what was not expected was that the total energy would decrease greatly in such a small amount of time (from 0.012 to 0.008 J). Again, this is probably caused by the fact that the energy is so small and since there is some friction and air resistance, some of the energy is lost over time. In addition, the magnets are not ideal, resulting in not all of the energy in the magnets being conserved and then thrown back into kinetic energy of the glider. However, due to the fact that there was a large uncertainty in A, it was surprising to find that the energy graph was a fairly accurate representation of the conservation of energy principle.  

Conclusion: 

As was seen, although there was high uncertainty for the value of A for the potential energy function, the magnetic potential energy function turned out to be a fairly accurate model of the system. In addition, it was proved that the conservation of energy principle also applies to this system. The largest reason for the uncertainty and error in this experiment is not that the uncertainty and error was large, but the fact that the values were very small, so that any uncertainty would have a large effect on the error in the values. These errors include the assumption that the track is frictionless and that there is no air resistance, when in fact there is both of some degree. In addition, there was also uncertainty in the measurement of the angle when the track was sloped and uncertainty in the measurement of the separation distance between the magnets. In addition, the assumption that the magnets are ideal in that all kinetic energy becomes potential magnetic energy and then all back to kinetic energy is not a reasonable assumption, resulting in error in the energy values. However, if non-conservative work was taken into account, it would be expected that the conservation of energy principle would be better proven and the total energy would remain constant.

13 April 2015: Conservation of Energy-Mass Spring System

Purpose:

The purpose of this experiment was to prove that the conservation of energy principle applies to a vertically oscillating spring that does not have a negligible mass.

Apparatus:

Figure 1: Mass-Spring Apparatus Setup

The apparatus for this experiment consisted of a long vertical rod held pointing upwards by a C-clamp. A horizontal rod was also attached to the vertical rod, which was used for holding up the spring in the vertical direction, as shown in figure 1 above. Some masses and a mass "holder" were also attached to the spring such that it allows the spring to oscillate continuously. An index card was also applied to the bottom side of the mass "holder" to allow the motion detector under the mass-spring system to detect the presence of the oscillating spring and mass. A meter stick was also used to measure the dimensions of the system. Lastly, a computer with Logger Pro software was also needed to allow the collection and analysis of data obtained from the motion detector.

Abstract:

The three types of energies involved in this system include kinetic energy, gravitational potential energy, and elastic energy. The hanging mass contains kinetic and gravitational potential energy while the spring has elastic potential energy. However, since the spring in this experiment does not have negligible mass compared to the rest of the system, it must be taken into account that the spring also has kinetic and gravitational potential, however not as much as the mass does. In addition, because the potential energy and kinetic energy of the spring varies throughout its mass, calculus must be used to determine functions that represent it gravitational potential energy and kinetic energy. To do so, a part of the mass (dm) along with its corresponding length (dy) must be observed, and the ratio between the two (dm/dy) must equal the ratio of the mass and length of the spring as a whole. Then, dm must be solved for in terms of y and must be substituted into the energy equation for that piece (in this case, dGPE and dKE). Then, through integration the sum of the energies of all the pieces of the spring must be found. Finding these functions of energy for the spring allows the use of the spring as a point particle, making it easier to measure its energies as it oscillates. The derivations of these two energy functions are shown below:

Figure 2: Derivation of the gravitational potential energy (GPE) of the spring

In this derivation, H is the height from the ground to the top of the mass-spring  system and y "naught" is the height from the ground to the bottom of the mass-spring system. Therefore, the length of the spring can be found at any point during the oscillation by subtracting the height H from the ground to the top of the spring (which is constant) by the height y "naught" from the ground to the bottom of the mass (which constantly changes throughout the oscillation. Because H is constant, MgH/2 is a constant in the equation for GPE of the spring. The value of y "naught" at each point was obtained from the position data of the oscillating mass gathered by the motion detector. The derived formula allows us to treat the spring as a single mass located at the center of the spring, but only when using this formula.

The same derivation, shown below, is done for the kinetic energy of the spring, with the exception of different data values and a different base formula (0.5mv^2).

 Figure 3: Derivation of the kinetic energy (KE) of the spring

In this derivation, the length y of the spring is the length L of the relaxed spring. The length of the relaxed spring is the needed length because it is the length at which the mass of all the pieces of the spring (dm) are proportional. In addition, the velocity of the piece of spring is equal to the velocity of the end times y/L. This is because the velocity of the spring at its different points varies, and the only velocity that can be collected and recorded is the velocity of the end of the spring.The velocity of the piece is proportional to the velocity of the end times y/L because, for example, at the top of the spring  y=0, so the velocity of the top of the spring is zero, which is seen. At the bottom of the spring, the length L is equal to y, which results in the velocity of the bottom to be equal to the velocity of the end, which also makes sense. Therefore, based off of this, the velocity of the center of the spring must be half the velocity of the end, and so on. This derived formula allows us to treat the spring as a mass at the end of the spring, but only when using this formula.

Therefore, there are now five different energies found in this mass-spring system: the kinetic energy of the mass and the spring, the elastic potential energy of the spring, and the gravitational potential energy of the mass and the spring,

Procedure/Discussion:

First, the dimensions of the mass-spring system were measured. The length L of the relaxed spring was measured as 47.3 cm. In addition, the mass of the spring was measured to be 85 g. Also, the height H from the ground to the top of the sprig was recorded as 145.7 cm. The distance from the top of the motion detector to the bottom of the unstretched spring was measured and recorded to be 85.7 cm. Lastly, the height y "naught" from the ground to the bottom of the mass holder when the spring was relaxed was recorded to be 98.5 cm.

Then, the spring constant of the spring was calculated by hanging various masses onto the spring and recording the change in the length of the spring. Then, using Hooke's Law (F = kx), where F is the weight of the added mass and x is the change in the length of the spring, the spring constant k was calculated. Using a 200 g (F = 1.96 N) mass, the spring stretched a distance  0.216 m. Then, the spring constant k was calculated to be 9.07 N/m (k = F/x = 1.96 N/0.216 m). The same procedure was done with 300 g and 400 g masses, whose k values were calculated as 8.80 N/m and 8.87 N/m, respectively. Taking the average of the k values, the actual experimental spring constant k was found to be 8.92 N/m. 

Next, a 250 g mass was attached to the spring and the spring was allowed to oscillate vertically by pulling down on the mass slightly and letting go. Then, using the motion detector, the position and velocity data of the system was collected. The experimental position and velocity graphs are shown below.

Figure 4: Position and velocity graphs of the oscillating mass spring system

These graphs correspond with the predicted because, theoretically, the velocity of the mass is greatest when the spring is unstressed and is zero when the string is at its maximum stretch and compression, which is seen in the graphs above. As can be seen, the  maxima and minima of the position graph translate to zero velocity in the velocity graph and the maxima and minima of the velocity graph are caused from the center between the maxima and minima of the position graph, as was expected.

Next, the graphs of KE, EPE, and GPE were predicted based on the position and velocity graphs. After the prediction, the experimental graphs of the same energies were found with respect to time. New columns of EPE, GPE (including both mass and spring), and KE (including both mass and spring as well) were created using the derived formulas as well as 0.5kx^2 for EPE, 0,5mv^2 for KE of the mass, and mgh for the GPE of the mass. the value of h for the GPE of the mass was calculated by subtracting the unstretched position by the position data gathered. These graphs are shown below:

Figure 5: KE, EPE, GPE and Total E vs time graphs of the mass-spring system

These graphs also corresponded with the predicted graphs. As predicted, the kinetic energy graph must correspond to the velocity graph; the maxima and minima of the velocity graph should result in peaks in the KE graph, as seen. This is because the velocity is greatest at those time intervals, resulting in the most kinetic energy. In addition, as predicted and seen, the kinetic energy graph must read zero at the maxima and minima of the position graph. In addition, as predicted, the maxima and minima of the GPE graph must correspond directly with the maxima and minima of the position graph, as seen. In addition, as predicted, the GPE graph is never zero since the mass never reaches the motion detector (the defined zero GPE point). Also, the maxima  of the elastic potential energy graph are  proportional to the minima of the gravitational potential energy graph, as predicted since the elastic potential energy is greatest when the spring is stretched the greatest (lowest GPE). Lastly, the total energy is nearly a straight horizontal line, as predicted. This proves that the conservation of energy principle is true and applies to situations that involve energy, such as the oscillation of a spring system seen in this experiment.

Then, the following experimental plots were found using Logger Pro: KE vs. position, KE vs. velocity, EPE vs. position, EPE vs. velocity, GPE vs. position, and GPE vs. velocity, The KE graphs are shown below:

Figure 6: KE vs position and KE vs velocity graphs of the mass-spring system

The KE graphs shown in Figure 6 are as expected. For the position graph, as the position of the mass-spring system reaches the unstretched position of the spring, the KE is the greatest (max velocity), and as the position of the system reaches the maximum stretch or compression of the spring, the KE is zero (zero velocity), which was expected and witnessed in the oscillation of the graph. Looking at the velocity graph, the kinetic energy is proportional to the velocity squared, as expected and seen from the result of a parabolic graph. When the velocity is zero, the KE is zero, and when the velocity is highest (in both directions), the KE is the highest, as expected. Next, the GPE graphs are shown below:

 Figure 7: GPE vs position and GPE vs velocity graphs of the mass-spring system

The GPE vs. position graph shown in Figure 7 was also expected. As the position (distance from the motion detector to the mass) increases, the GPE increases. It was also expected that the graph of GPE and position is linear because, looking at the equation for GPE (mgh), the change in "h" results in a proportional change in GPE. The GPE vs velocity graph was not expected to be an ellipse, but after some observing it made sense to be an ellipse. In the oscillating mass spring system, the velocity constantly changes from zero to +max, then back to zero and then to -max, then to zero back again. This results in a constant rotation in the slope of the graph, causing an elliptical arc. In addition, the points of minimum and maximum GPE being at zero velocity also made sense because, when the velocity is zero, the spring is either at its maximum stretch (lowest GPE of the mass) or at its maximum compression (highest GPE of the mass). Lastly, the EPE graphs are shown below:

Figure 8: EPE vs position and EPE vs velocity graphs 

The EPE vs. position graph seen in Figure 8 was expected and made sense. As the mass-spring system reached the unstretched position of the spring (0.857 m), the EPE of the spring would decrease and would be transferred to KE of both the spring and mass, as witnessed in the experiment and seen in the graph. In addition, the graph takes the shape of half a parabolic arc, which makes sense because  x is equal to the unstretched position minus the position obtained from the data collected by the motion detector as well as squared. Interpreting this, x can never be larger than 0.857 and x^2 can never be negative, resulting in EPE always being positive and taking the shape of a parabola with the "slope" of the parabola equal to 0.5k, where k is the spring constant. Lastly, the EPE vs velocity graph was not expected but upon further observation made some sense. The EPE would be greatest at a velocity of 0 since that is when the spring is stretched or compressed at its maximum. In addition, for the same reason the GPE vs velocity graph is circular, it is too. However, one anomaly is that the lowest EPE is also shown by the graph at a velocity of 0 m/s. It is not known why, but one guess would be that it is affected by GPE, since at those points of minimum EPE it is the highest energy of the three.

Putting all of the above graphs together, the following two graphs are obtained:

Figure 9: Energy vs position and energy vs time of the mass-spring system

Looking at the total energy, it is nearly constant with around a 3.4% change throughout the graph, but, looking at the uncertainties present in the experiment and the fact that the system and the world is not ideal, it shows that the system was a very accurate model for proving that the conservation of energy principle is in fact true. Therefore, it can be said that the energy in this system was conserved.

Conclusion:

There were some sources of error and uncertainty in this experiment, but they did not have a very large effect on the results of the experiment, with those results already being expected. For example, the 3.4% error in the total energy not being constant results from the fact that an ideal spring was not used yet it was still assumed that it was. Assuming so does not take into account that the spring does not symmetrically stretch and compress, resulting in different areas of the spring being in different conditions (variation in dm and elastic energy, making 0.5x^2 an inaccurate potential function for EPE). In addition, the stretching and compressing of a spring that is not ideal results in the loss of energy through non-conservative ways, such as the formation of heat arising from friction in the spring and the loss of energy due to vibration in the spring.

Also,  the uncertainty in the measurement of the unstretched position is the cause of the elastic potential energy not being exactly zero at the unstretched position. The uncertainty in the unstretched position is caused by both uncertainty in the equipment and the fact that it is difficult for a vertical spring that has mass to be kept in the unstretched position.

In addition, the uncertainty in the value of the spring constant had an effect on the error in this experiment. In order to lower the error from measuring the k value, better equipment could have been used to find k, such as the availability of an accurate force sensor, a better version of LoggerPro that can better monitor and connect with the equipment that collects data. Lastly, more time for the experiment could have been given to allow for the running of more trials and collection of more data, lowering the error in the experiment. Overall, though, the uncertainty/error in the experiment was very low and had little effect on the success of proving that the system is an accurate model for the conservation of energy.

6 April 2015: Work-Kinetic Energy Theorem Activity

Purpose:

The purpose of this experiment was to prove the work-kinetic energy theorem holds to be true. The work-kinetic energy theorem states that the total amount of work done on a system is equal to the change in kinetic energy of the system.

Apparatus:

Figure 1: Apparatus of this experiment

Figure 2: Experimental setup used to obtain data

The apparatus in this experiment consisted of a computer with the LoggerPro software, which was used to obtain data needed for this experiment. The data needed was the position of the cart, which was obtained with a motion detector, and the force applied by the spring, which was obtained with a force sensor. As just stated, a cart and a spring were also used. Some PostIt notes were also pasted to the cart to allow the motion detector to more effectively sense the presence of the cart. Two 250 g weights and a "weight hook" (used to hold the weights) were also used to calibrate the force sensor. Although weights are shown on the cart, they were actually removed off the cart later on in the experiment since they affected the data obtained in a slightly negative way. A calculator was also used to hold the spring up and keep it in its "natural" position (unstressed). Lastly, a C-clamp was used to hold a metal stick, which was used to hold the force sensor in place. 

Procedure (Part 1):

First, the motion detector and force sensor were set up to the LoggerPro software. The force sensor was then zeroed, and calibrated with a 500 g mass on LoggerPro. The rest of the apparatus was then set up, as shown in Figures 1 and 2. One end of the spring was attached to the force sensor, and the other end was attached to the cart. A force (N) vs position (m) graph was then opened on LoggerPro. The force sensor was again zeroed with the unstressed spring attached. The cart was then pulled toward the motion detector so that the spring was stretched about 0.4 m, with the cart at rest (the force applied by the hand equals the force applied by the spring). Collecting data with the motion detector and force probe, the cart was then let go so that the only force acting on it was the force of the spring until it reached the zero position (where the force sensor is located. The force vs position graph was then formed with the collected data.

Discussion (Part 1):

Figure 3: Force vs position graph of the cart using raw data

As seen in the graph above, the farther the position of the cart, the larger the force of the spring (consistent with Hooke's Law: F=kx). The rest of the graph after the origin (in the negative x axis) is just bad useless data resulting from the collision of the cart with the force sensor. Removing this data and forcing the graph to go through the origin (since the spring does not apply a force when its in its natural state) by subtracting each point by the amount the point on the y axis is above the origin, the actual graph of the force applied by the spring and the amount the spring is stretched (the position of the cart) for this experiment was obtained.

Figure 4: Actual force vs position graph of the spring

Since  the graph goes through the origin, a proportional fit was created. Relating the slope-intercept form equation of a line (y = mx + b, where b = 0 since the graph's y-intercept is the origin) and Hooke's law (F = kx), it can be seen that the spring constant k can be found from this graph by taking the slope of the graph. The spring constant k of the sued spring was found to be 1.758 +/- 0.01568 N/m. 

Conclusion (Part 1):

The uncertainty in the slope of the graph is very small, proving that the method to obtain the spring constant is effective. The uncertainty in "k" results from the fact that an ideal spring was not used, since an ideal spring is unattainable and not realistic, while Hooke's law assumes that an ideal spring is used. 

Procedure (Part 2):

The mass of the cart was measured to be 0.505 kg. Using the same data from part 1, a new column for kinetic energy was created. The kinetic energy at each point was calculated using the kinetic energy formula: KE = (1/2)mv^2, where the velocity was obtained by the motion detector and the mass of the cart measured was used. The force vs position curve and the kinetic energy vs position curve were both displayed on the same graph, and, using the integration feature in LoggerPro, the work done by the spring on the cart and the kinetic energy of the cart were compared by taking the area under the curves. 

Discussion (Part 2):

The work done was found from the area under the force vs. position curve and the kinetic energy of the cart was found from the area under the kinetic energy vs position curve. They were both compared at three different positions to prove the validity of the work-kinetic energy theorem. These positions are shown below:

   Figure 5: Comparing the work done by the spring and the KE of the cart at the first position

Figure 6: Comparing the work done by the spring and the KE of the cart at the 2nd position

Figure 7: Comparing the work done by the spring and the KE of the cart at the 3rd position

As seen in Figures 5 through 7, the work done by the spring and the kinetic energy of the cart at the three different positions were very close (only off by an average of around 0.001), proving that the work-kinetic energy theorem is applicable and holds to be true. 

Conclusion (Part 2):

The small error (difference) between the work and kinetic energy is mostly caused by the fact that an ideal spring was not used, which Hooke's Law assumes. It is also caused by the uncertainty in the measurement of the velocity of the cart by the motion detector. 

The work-kinetic energy principle for this system can be stated as follows: The work done by the spring on the cart is equal to the change in kinetic energy of the cart; therefore, energy is conserved in the spring and cart system.

Procedure (Part 3):

Using LoggerPro, a previously made video that also proves the work-kinetic energy theorem was investigated. The professor in the video uses a machine to stretch a large rubber band as much as possible. The force exerted is recorded by a force transducer onto a graph. The stretched rubber band is then placed onto a cart, and the cart is released. It travels through two photogates of a known distance apart, and the time it takes to travel through the two photogates is measured.

The graph created from the force transducer was replicated onto the graph provided, with the y axis containing the force applied by the rubber band (in N) and the x axis consisting of the amount of the stretch of the rubber band (in m). The amount of work done by the rubber band can then be calculated by finding the area under the curve of the graph. The curve of the graph was simplified so that different shapes can be constructed under the curve, making it easier to calculate the area.

The kinetic energy of the cart was then found by first finding the velocity of the cart, which was done by dividing the total distance between the photogates by the total time it took for the cart to travel through the photogates. Then, the kinetic energy of the cart was found by plugging in the known mass and velocity into the kinetic energy fomula.

Discussion (Part 3):

The graph of the force of the rubber band vs its stretch is shown below:

Figure 8: The graph of force exerted by the rubber band vs its stretch

The area under the curve was split up into 6 usual shapes whose areas are easy to find. They include a large triangle (A1), two rectangles (A2 and A3), a thin rectangle (A5), a small thin triangle (A4), and a trapezoid (A6). Summing the areas of these shapes, it was found that the work exerted by the rubber band was 26.6 J. 

To compare the work exerted by the rubber band to the kinetic energy of the cart, the previously proven work-kinetic energy theorem was used. The kinetic energy of the cart was found by first finding the velocity of the cart in between the two photogates. Using the distance it traveled (0.15 m) and the time it took to travel that distance (0.045 s), the velocity of the cart was found to be 3.33 m/s. Then, using the velocity and the given mass of the cart (4.3 kg), the kinetic energy of the cart was found to be 23.9 J. 

As can be seen, there is a relatively small difference between the work exerted by the rubber band and the kinetic energy of the cart (2.7 J). 

Conclusion (Part 3):

The error (difference between the KE of the cart and work done by the rubber band) is caused by many things. First, the uncertainty (simplification) in the replicated curve of the graph is one of the major contributions to the difference between the two. This simplification resulted in the addition of extra area under the curve, therefore causing extra calculated work of the spring that the spring actually did not do. In addition, other sources of error include the fact that the professor in the video tried to move the graph at constant speed, which causes some uncertainty in the curve of the graph. Friction between the graph and the marker as well as between the graph and the sliding holders of the graph are some sources that resulted in the uncertainty of the graph moving at constant speed. Lastly, the uncertainty in the force transducer and the timer that measured the time it took for the cart to go through the two photogates also caused the difference between the kinetic energy of the cart and the work done by the rubber band. However, this difference is very small, also proving that the work-kinetic energy theorem is legitimate and is an excellent model that is applicable to almost all systems.

1 April 2015: Centripetal force with a motor

Purpose:

The purpose of this experiment was to find a model that relates the angle a hanging mass tied to string makes with the vertical and the angular speed of the hanging mass as it rotates around a central axis.

Apparatus:

The apparatus for this experiment consisted of an electric motor that spins at a certain speed depending on the voltage that runs through it. The motor is mounted on a surveying tripod. It also consists of a vertical rod attached to the motor that also spins with the motor, with a horizontal rod that is attached to the vertical one. A long string is attached to the end of the horizontal rod, with a rubber stopper attached to it. This apparatus was used to find the relationship between the angle and angular speed by performing different trials with running increasing voltages through the motor, resulting in a faster angular speed. The angular speed was measured by measuring the time it took for the hanging mass to go through 10 revolutions, then finding the period by dividing the time by 10. Then, angular speed is found by dividing 2pi by the period. The experimental angular speed was used to measure the accuracy of the model. 

In order to find the angle, the height from the top of the horizontal rod to the mass when it was rotating had to be measured. In order to be possible to find that height, the height of the whole apparatus was measured and it was subtracted by the height of mass from the ground as it was rotating. The height of the hanging mass was found by using a ring stand with a horizontal piece of paper; the height of the paper was modified until the hanging mass barely scraped the paper, and the height of the paper from the floor was measured.

Procedure:

First, the dimensions of the apparatus were measured. This includes the height of the apparatus (200.0 cm +/- 0.1 cm), the length of the string (166.4 cm +/- 0.1 cm), and the distance from the center of the horizontal rod to the edge where the string is (87.0 cm +/- 0.1 cm).

Using the apparatus explained above, six trials were performed, with the voltage running through the motor increased between each trial. Each trial, the time it took for 10 revolutions was measured and the height of the mass from the ground as it was in motion was measured. The period and angular speed of the mass for each trial was calculated. As expected, the period was smaller and the angular speed larger every increasing trial. The data is found tabulated below:

Figure 1: Data table for experimental values of w and height

The uncertainty in the height h of the first 5 trials is 0.5 cm, and the uncertainty in last trial is 1 cm. The uncertainty in the time for 10 revolutions is about 0.2 s (average reaction time), so the uncertainty in the period is 0.02 s.

Once the experimental angular speed for each trial was found, the relationship between the angle the mass makes with the vertical and the angular speed was calculated. The derivation is shown below in the results section.

Results:

Figure 2: Free body diagram of hanging mass

The relationship between theta and omega was found through analyzing the forces acting on the hanging mass in motion through the use of the free body diagram above. The two forces acting on the mass are its weight and tension of the string, with the centripetal acceleration pointing inward to the center of rotation parallel to the horizontal direction. Summing forces in the x and y direction, we get the equations shown in the figure below:

Figure 3: Derivation of the model that relates theta and omega

In this experiment, centripetal acceleration was substituted with r times omega squared to get theta and omega in the same formula. Theta can be found by looking at the triangle formed when the mass is spinning, seen in Figure 2. The vertical leg is the height of the whole apparatus (H) subtracted by the height of the mass relative to the ground (h). The horizontal leg is the distance r from the mass to the end of the horizontal rod. The hypotenuse is the length of the string (L). Since r cannot be measured, using cos(theta) is the best alternative. Then, theta = acos((H-h)/L). Using the sum of froces in the x direction, T can be solved for in terms of theta, which is what is needed for this model. The distance r, since it cannot be measured, can be substituted for Tsin(theta), where T is mg/cos(theta). Plugging in the expressions for T, r, and theta, an expression for angular speed that can be used with the limitations of this experiment is found. The model derived that relates both theta and angular speed is:

.

The model that uses the dimensions measured instead of T, theta, and r is:

.

Using the dimensions of the apparatus and the heights of each trial, the expected angular speed for each trial was calculated, and are tabulated below.

Figure 4: Calculated angular speed for each trial using the derived model

To compare the calculated angular speed and the experimental one, a graph was made that plots the calculated angular speed on the y axis and the experimental on the x axis. Perfect proportionality between the two (that is, all of the values for all six trials are exactly equal to each other) results in a slope of exactly 1. The closer the slope is to 1, the better the derived model is. The graph is shown below:

Figure 5: Graph of calculated w vs experimental w

  The slope of the above graph is 0.9943 +/- 0.003639. As expected, the slope of the graph cannot possibly equal to 1 because of all the uncertainties in the measurements in this experiment. However, the slope, along with its uncertainty, is fairly close to 1, indicating that the model derived in this experiment is good enough to relate theta and omega.

Conclusion:

To see if the error/propagated uncertainty in this experiment is only based on the uncertainties in the measurements, dw could be found by taking partial derivatives with respect to T, H, h, L, and R. However, taking partial derivatives of such a model proves to be difficult and is not the best alternative to use since most of the uncertainties in the measurements are not large enough to have a great impact on the uncertainty/error in this experiment. The measurements that have the highest uncertainty is the height h of the mass relative to the floor and T, making them have the biggest impact on the uncertainty/error. 

The best alternative to finding if the error is caused all by the uncertainty in this experiment, one of the values of h from one of the trials can be used. Using the uncertainty of that value of h, the change in the value of w for that trial can be seen upon addition and subtraction of the uncertainty from the value of h. Then, the change in the value of w upon changing the value of h with respect to its uncertainty can be compared with the error of the slope. If the error in the slope of the graph is less than that of the error from the uncertainty in w upon changing the value of h, then the propagated uncertainty in this experiment is all caused by the uncertainties in the measurements of the dimensions of the apparatus and time. 

Using the value of h from trial 4 (118.5 +/- 0.5 cm), the original value of angular speed is 2.742 rad/s. Using the value of h with the uncertainty added (119.0 cm), the value of omega obtained is:

The difference between that value and the actual value of omega (2.742) is + 0.009. Doing the same with the value of h obtained by subtracting the uncertainty in h (118.0 cm), the value of omega obtained is:

The difference is -0.010. Therefore, the average uncertainty in omega is +/- 0.0095, which is 0.35% of the actual value of angular speed for that trial (2.742 rad/s). Doing the same with T for trial 4, the average uncertainty in omega is +/- 0.0237, which is 0.87% of the experimental value of angular speed for trial 4 (2.731 rad/s). Subtracting the two uncertainties, since w - the derived equation (sqrt(...)) should equal zero, the uncertainty obtained is 0.52%.

The uncertainty in the slope of the graph is +/- 0.003639, which is 0.37% of the slope (0.9943). Since the uncertainty in the graph is smaller than the uncertainty in the angular speed, it is safe to say that the uncertainty in the slope is caused by the uncertainty in the measurements of the dimensions and period.