Phil,+Alyssa,+Deanna,+Niki

Lab: Transverse Standing Waves on a String
Objective: What is the relationship between Frequency and the tension of transverse waves traveling in a stretched string? What is the relationship between frequency and harmonic number? What is the relationship between frequency and wavelength? Hypothesis 1: We believe that the tension and frequency will share a direct square relationship. This is due to the equation. As the velocity increases, the tension increases, which will increase the frequency.

Hypothesis 2: We believe that as the frequency increases, the number of antinodes increases as well. This is because as the frequency increases on the standing wave, the wavelengths decrease at the same time, allowing for the wave to have more antinodes.

Hypothesis 3: We believe that the frequency will be inversely proportional to the wavelength of the standing wave. This is due to the equation:.

1. Electrically-driven oscillator 2. pulley & table clamp assembly 3. weight holder & selection of masses 4. string 5. electronic balance 6. meter stick
 * Materials:**

**Procedure for setting up the device:** 1. Attach the string to the oscillator and put the string over a pulley that is clamped to the end of a table. 2. Connect a hanger mass to the end of the string. 3. Set the sine generator on to its maximum amplitude and turn the device on.

1. Using the set up procedure originally stated, continue to maximize the frequency, until the device stops making a lot of noise. When you are getting close to when the device is at a point of resonance, the device will be making loud sounds, and when you find the exact point, the device will get quiet. 2. Record the number of antinodes, and the frequency that the device is at. 3. Repeat these steps until you complete about 5-7 different trials. 4. Create an excel graph to determine the relationship between the two.
 * Procedure for relationship between Frequency and harmonic number/antinodes:**

1. Using the steps for the relationship between frequency and antinodes, take a meter stick and measure the length of each wavelength at the point of resonance. 2. Record these results and repeat these steps until you complete about 5-7 different trials. 3. create a graph to determine the relationship between frequency and wavelength.
 * Procedure for relationship between Frequency and wavelength:**

1. Take off the light hanger used for the two previous trials, and place a close to 1 kg hanger on the string. 2. Add a 1/2 kg mass, find a frequency of resonance with exactly two antinodes, and record the frequency and the tension. 3. Continue to add masses and find a frequency of two antinodes until you get about 5 results. 4. Repeat the process and create a graph to determine the relationship between frequency and tension.
 * Procedure for the relationship between Frequency and tension:**


 * Graphs:**








 * Analysis:**
 * Your analysis should include
 * 1) 1. Graphs that will enable you to answer the objectives
 * 2) 2. Interpretation of the equations of the curves on each graph.
 * 3) 3. Use of the slope to solve for an experimental value in each case.
 * 4) <span style="background-color: transparent; color: #000000; font-family: 'Century Schoolbook'; font-size: 18.6667px; text-align: start; text-decoration: none; vertical-align: baseline;">4. Comparison of experimental values to known or other measured values.

1. Calculate the tension T that would be required to produces the n=1 standing wave for the red braided string.
 * Discussion Questions:**

2. What would be the effect if the string stretched significantly as the tension increased? How would that have effected the data? If the string stretched significantly as the tension increased, the velocity would increase as well. We know this from the equation, as the velocity is effected when the medium changes. If the tension increased and the velocity increased, our data would have needed an even higher frequency to find the exact period of resonance for the amount of nodes and antinodes we were looking for.

<span style="font-family: Arial,Helvetica,sans-serif;">3. What is the effect of the type of string on the amount of hanging mass needed to create a set number of nodes? Explain this. //Each type of string has a different natural frequency. A thicjer string will have a different frequency than a thinner one, and therefore resonance occurs as different frequenices. So, if you wanted to creat three nodes for example, it would be a different frequency for different types of strings. The amount of mass hanging affects the tension in the string. Different tensions also result in different natural frequencies. More tense strings have different resonance frequencies from less tense strings.//

<span style="font-family: Arial,Helvetica,sans-serif;">4. What is the effect of changing frequency on the number of nodes? //The higher the frequency the higher the number of nodes.//

<span style="font-family: Arial,Helvetica,sans-serif;">5. What factors affect the number of nodes in a standing wave? //The frequency affects the numbers of nodes. The higher the frequency, the more nodes there are.//

For the relationship between the frequency and the number of antinodes, a linear graph was formed. The slope of this line was positive. The slope represents the natural frequency which is the first harmonic that causes an antinode to form. It should have been around 9.3 which is the fundamental frequency. The relationship between frequency and wavelength was a negative power. Lambda should be to the -1 (which we had.. thank you Alyssa for announcing that to the class). The coefficient should be around 60 because that was the speed of the wave. This all makes sense because to find frequency, you divide velocity by lambda (which is technically the same as multiplying to the -1). For the frequency vs. Tension in string graph, we got a positive power graph. The power should be .5 due to the fact that there is a square root relationship within the equation. Because again we were solving for frequency, and the coeff. is equal to.
 * Conclusion:**

Spring Force Constant Lab
Members: Alyssa Berger, Deanna Magda, Phil Litmanov, Niki Kaiden Due Date: Monday, May 16, 2011 **Lab General Objective:** Determine the relationship between the mass on a spring and its period of oscillation.


 * Hypothesis:** In this lab, we believe that if we add and vary the mass on a spring, it will directly affect the period of oscillation, for the differing masses will force the spring to vibrate in different periods.

1. Place a spring on a stand, and attach a hanger mass to that spring. 2. Determine the equilibrium point of the spring with the hanger attached. 3. Add mass and measure the displacement of the hanger. 4. Using the forces and varying mass, directly determine the spring force constant.
 * Materials:** springs, tape, clamps and rods, masses, balance, timers, meter stick.
 * Procedure for Objective 1:**

1. Keep the attached spring and hanger mass on the stand, and add three masses onto the hanger to make the spring oscillate slower (makes it easier for us to calculate each individual cycle). 2. Have one lab member use the timer and time 10 periods on the stop watch. 3. Divide that number by ten to calculate one individual period for the spring. 4. Using the equation, solve for k, as we know the period from previously calculated and the mass. 5. To create and vary the variable, we will continue to change the mass on the spring to prove that the period increases as more mass is added to the spring.
 * Procedure for Objective 2:**

Picture of Setup:


 * Data:**











Calculations:

We got the k value directly from the slope of the first graph above using Hooke's Law.

k indirectly from the period



Percent Error




 * Discussion Questions:**

**1. Do the data for the displacement of the spring versus the applied force indicate that the data for the spring constant is indeed constant for this range of forces?** Yes, they both definitely indicate that the data for the spring constant is constant for this range of forces.

**2. Why is the time for more than one period measured?** In our group, we measured ten periods and then divided by ten to get one precise time for a single cycle of vibration. We do this because human reaction time can never be good enough to determine and stop the watch at the exactly at the time of one period, therefore, this gives us a more exact result.

**3. Discuss the agreement between the k values derived from the two graphs. Which is more accurate?** The The k value from the first part of the lab averaged to 10.64 N*m, and from the second lab we found numbers ranging from 10.75 to 12.31. Our highest percent error equalled 10.3% between the two values. The first trial was more accurate, as the second trial was more dependent on a human's reaction time, which is not very precise.

4. Generate the equations and the corresponding graphs for i. position with respect to time. x(t)= .017cos(13.75t) ii. velocity with respect to time. v(t)= -.017(13.75)sin(13.75t)

iii. acceleration with respect to time. a(t)= -(0.17)(13.75)^2cos(13.75t)

5. A spring constant k = 8.75 N/m. If the spring is displaced -0.150 m from its equilibrium position, what is the force that the spring exerts?

6. A massless spring has a spring constant of k = 7.85 N/m. A mass m = 0.425 kg is placed on the spring, and it is allowed to oscillate. What is the period T of oscillation?

7. We neglected to take into account the mass of the spring itself. Are your results any better when using the more accurate relationship (where m is the hanging mass and ms is the mass of the spring)? Redo graph #2 using, and explain these results.

The goal of this lab was to determine if the mass of the object directly affected the period. Our hypothesis was correct, because we found that a greater mass did result in a longer period. These findings make logical sense, because a heavier mass will vibrate more slowly, due to the fact that it is greater inertia, and therefore resists the change in direction and speed more than a lighter object would. For example, we found that a mass .11kg of had a period of .6386 seconds, and that a lesser mass, .07kg, had a period of .5095 seconds, a shorter period. Our lab results are fairly consistent, however there were some sources of error in this lab. Firstly, human reaction time affected the times we got. We tried to counteract this by timing ten periods, instead of one, in order to reduce the effect of human error. We also always had the same person using the stopwatch, to ensure we had the same amount of error every time, and that way a pattern would still emerge, because we were at least consistent and precise. Also, we calculated the spring force constant (k) experimentally, therefore if that number were inaccurate, then it would throw off our other results, however a pattern would still emerge, which is the most important part of the lab anyway. If the k value had been known instead of being determined experimentally, than our results probably would have been no less precise, but more accurate. This specific lab does not have many real life applications. However, one of the concepts it encompasses, inertia, does have many real life implications. For example, when driving a car, a smaller car will come to a stop much faster than a much larger car, because it has less inertia. As most of us are fairly new drivers, this applied physics knowledge is helpful in the real world. = = =Mobile Project= Members: Alyssa Berger, Deanna Magda, Phil Litmanov, Niki Kaiden Due Date: May 9, 2011
 * Conclusion**


 * Hypothesis:** When the masses are suspended, the forces and torques will be balanced in each level, resulting in a balanced mobile.


 * Materials:**

Wooden Dowels Nylon String Various Holiday Related Items or any other themed objects Super Glue or Wood Glue Ringstand Ruler/Meter Stick Scale Hacksaw


 * Procedure:**

1. Plan out the design (levels, pivots, etc) on paper first. 2. Cut the dowels into small pieces according to your design, making sure to adhere to the required dimensions. 3. Record the mass of each hanging item and each dowel. 4. Work from the last level of the mobile up to the first level. Balance each dowel with masses on the last level. 5. After the last level dowels are balanced, treat the finished dowels as masses for the subsequent levels. 6. Again, balance the next level up with masses, making sure to have centered and off centered pivots. 7. Repeat step 4 until the mobile is finished. 8. Super glue the string as well as the knot to the dowel to prevent any slipping of the string.

Picture of Mobile:
 * Diagram/ Data:**

Mass Data: Labeled Diagram:




 * Calculations:**

Excel Spreadsheet: Sample Calculation:


 * Net Force Equilibrium Calculations Dowel I Example:**

Link to Spreadsheet:

In conclusion, our results show that our mobile device is very close to being at a state of complete equilibrium. As seen in the calculations, the torque clockwise ended up being very close to equal to the torque counterclockwise, though there was a decent amount of error seen in percent difference. This error can be easily explained, for our mobile device consists of a myriad of smaller masses. For example, take the mini easter egg, which only had a mass of .00574 kg, or same with the mask, which only measured to be .0067 kg. Some of the percent differences are close to 20% due to the fact that the numbers may appear to be the same, but are off due to significant figures that are not necessarily fully shown. This does make a big difference in our mobile scenario, for the smaller the masses, the more of a difference one figure off can make. Also, to make the objects stay on the dowels, we not only used string, but we used super glue as well. This glue could have added a mass that we didn't intend on having, and therefore, could have been a force that caused a loss of perfect equilibrium if we did not take those numbers into account. Error was also seen in our net force equations for the rods. For the sample we put up, out net force equaled -.08N, so clearly some error was involved, but not enough to conclude that our mobile was not in equilibrium, just not flawlessequilibrium. As stated earlier, we had very small masses and positions were very important, so it could not have equalled zero if positions were off in relation to the torques and equilibrium of the mobile. For us, to address the error regarding the small mass, it may have been easier to just use bigger masses. It took us longer to build the mobile, as the objects we hung were so delicate. It also could have helped if we did the calculations prior to building the mobile. This could of helped us in that we wouldn't be building it based on facts and eyesight, rather off of calculated mathematics. It also may have been helpful, in relation to the glue, to take into account the mass of the glue and string combined, and see if it was going to make a difference, or use some sort of sticking device that was mainly massless. Though there was inevitable error in our process of building the mobile, ultimately, our device was mainly in a state of equilbrium!
 * Conclusion:**

= = =Lab: Levers-Static Equilibrium= Members: Phil Litmanov, Alyssa Berger, Deanna Magda (Niki Kaiden was absent) Date: 4/15/11


 * Objective**: To verify the Torque equation, we will balance masses, and solve for the one of the masses.


 * Hypothesis**: Since the Torque equation says that torque is equal to the distance times the force, the weight of the mass time the distance from the fulcrum should be equal to the mass times the distance of the second mass.

1. Acquire the materials listed on the lab handout. 2. Balance the meter stick so that the fulcrum is at the center of mass. 3. For the first trial, take two separate masses and balance them on either side of the meter stick until the meter stick is level. 4. Repeat for the second trials, balance again using three different masses. 5. For the fourth trial, move the fulcrum so that it is 20 cm away from the center of mass of the ruler. 6. To determine the mass of the unknown mass for trial 5, first center the fulcrum. Then, take a known mass and place it on one side. Then take the unknown mass and place it on the other side of the fulcrum and adjust until the meter stick is level. Since both distances are know, and one mass is known, we can solve for the other mass. 7. We measured the distance the mass was from the fulcrum, and and angle of the string. 8. Record all data in an excel spreadsheet. Compare the theoretical mass to the experimental mass.
 * Procedure:**

Set up:




 * Data**
 * Drawings of Each Situation:**
 * [[image:magda_sit1.png width="228" height="271"]] || [[image:magda_sit_2.png width="359" height="266"]] ||
 * [[image:magda_sit_3.png width="253" height="283"]] || [[image:magda_sit_4.png width="321" height="276"]] ||
 * [[image:magda_sit_5.png width="272" height="336"]] || [[image:magda_sit_6.png width="345" height="348"]] ||


 * Calculations:**
 * Trial 1**
 * Trial 2**
 * Trial 3**


 * Trial 4**
 * Trial 5**

FORCE OF WALL X: FORCE OF WALL Y:
 * Trial 6**
 * Percent Error Sample calculation**


 * Analysis Questions:**


 * 1. Does it get easier or harder to rotate a stick as a mass gets farther from the pivot point?**

It gets easier to rotate the stick when the mass is farthest from the pivot point. This is because in the equation T = Fr, the force and distance from the fulcrum are inversely proportional. As the distance increases, the force needs to be smaller to achieve the same amount of torque.


 * 2. Does the weight of the mass increase as you move the mass away from the pivot point?**

No. The weight never changes because it is just mass x acceleration due to gravity. Both factors are constant.


 * 3. Why is more mass required to balance the meterstick as you move another mass farther from the pivot?**

It is necessary to counteract the torque of one mass on one side of the meterstick with another mass on the other side, in order for the torques to be equal and for the stick to balance


 * 4. Why must the mass of the hangers and clamps be taken into account in this experiment?**

Because the hangers and clamps have mass and any mass on the meter stick will create torque.


 * 5. If you are playing seesaw with your younger sibling (who weights much less than you), what can you do to balance the seesaw? Mention at least two things.**

One way would be for you to sit closer to the fulcrum and have your sibling be farther from the fulcrum. A second way would be to add rocks or other mass to your sibling's side to help counteract the torque you create.


 * 6. What kept the meterstick in equilibrium in the fourth trial? In other words, what counterbalanced the known mass?**

Putting the mass on the shorter side of the meterstick kept the system in equilibrium. Because the fulcrum was not on the COM, the COM created torque and pulled down on the longer side of the meterstick. Our mass on the other side counterbalanced this.

In this lab, we demonstrated that when a system is in equilibrium, the torque of the counterclockwise forces is equal to the torque of the clockwise forces. For example, in the first situation, there were two different masses and the fulcrum was at the center of mass. Because of this, the weight of the meter stick did not affect the torque, because the distance from the fulcrum was zero. The heavier mass was located closer to the fulcrum, because the shorter distance compensated for the fact that the mass was heavier than the mass on the other side. We solved for the second mass, which we calculated, based on our experimental distances, to be .0635 and the mass was actually .0664. This is about 3% error, which is pretty small. Another example is trial 3, where the center of mass of the ruler was not the fulcrum. In this case, we had to take the weight of the meter stick. This caused the mass on the right to be slightly farther out than it would have been had the fulcrum been placed at the center of mass, because it had to balance out two forces instead of one. In this example, we solved for the second mass, which we determined to be .467.
 * Conclusion:**

This lab was overall pretty simple, but it did have some sources of error. First off, it was difficult to get the meter stick completely level, and since it was not perfect, it was even more difficult to replicate that exact level of unevenness every single time. This could be corrected by a level perhaps, so that way we could make sure that we were maintaining the same state of equilibrium throughout the trials.

Torque can apply to people's everyday lives as well. For example, when children play on a seesaw they experience the effect of torque, as the heavier child (or the parent) has to sit closer to the center in order to balance with the smaller child on the other side. Another example could be balancing items on a tray. It is common sense that you do not put all the heavy objects on one end of the tray, or else it will tip over. However, now we understand the reason that this occurs.