Roshni,+Ariel,+Evan,+Ryan

=Lab: Frequency, Harmonic Number, and Wavelength= toc Date: May 23rd, 2011 Group: Dylan, Roshni, Ariel, Evan, Ryan Class: Honors Physics Period 2

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?

MATERIALS String, sine wave generator/oscillator, clamp, masses hanging on string, pulley & clamp assembly, weight holder

HYPOTHESIS Frequency and harmonic number are directly related, while frequency and wavelength are indirectly related. Frequency and tension are directly related. Rationale: We formulated our hypotheses based on the equations and

PROCEDURE 1. Set up the string with the oscillator at one end of the pulley, the table clamp, and with the weights hanging off of the other end of the table. 2. Picked frequency with dial. (repeat for each trial) 3. Measured each wave length. (repeat for each trial) 4. Pick a wave length and harmonic number to use consistently. 5. Add a mass to one end of the string. (repeat for each trial) 6. Change the dial until we got our consistent harmonic number and wave length. (repeat for each trial)

DATA 

This is a graph of frequency vs. wavelength. The equation for our trend line is "y=94.112x^-1.0405". The slope, which is 94.112, is velocity of the wave. This velocity is also equal to the square root of tension over mass per length. The x is raised to a power of -1.0405, which is very close to -1. Because x also represents wavelength, the equation can also be read as y equals velocity times wavelength to the negative one. With a 500 gram mass, the slope should be around 60, but this was not the case because we used a mass of 1000 grams. This is a graph of frequency vs. tension. The equation reads "y=35.881x^.5023". Because x is raised to a power of about .5, the equation also means that x is under a square root. The slope of this graph is 35.881. In reality, that number represents 1/(lambda*sqrt(m/L)).

This is a graph of frequency vs. harmonic number. For the trend line, we got an equation of "y=27.336x". The slope is the natural frequency of the string we used; natural frequency is also known as the first harmonic and the fundamental. This slope should be around 20, but we did not get that because we used a different mass on the end of our string. instead, we had a slope of 27, which is the frequency of our first harmonic number.

CALCULATIONS 

DISCUSSION QUESTIONS 1. Calculate the tension T that would be required to produce the n = 1 standing wave for the red braided string.  -evan

<span style="color: #000105; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">2. What would be the effect if the string stretched significantly as the tension increased? How would that have affected the data? <span style="color: #000105; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;"> - as the tension would increase, the string would get tighter and tighter, and the elastic property of the string would increase. This makes the velocity get larger as well. A large velocity requires a large frequency in order to get the same number of anodes as we had originally. <span style="color: #000105; font-family: 'Palatino Linotype','Book Antiqua',Palatino,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. <span style="font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;"> - Strings with higher elasticity have higher velocities, and therefore higher frequencies as well. Higher frequencies imply more antinodes. The more elastic the string is, the less mass is needed to create a constant number of nodes. <span style="color: #000105; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">4. What is the effect of changing frequency on the number of nodes? <span style="color: #000105; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif; font-size: large; line-height: 27px;">- The number of nodes (the harmonic number) is directly related to the frequency. Therefore, when frequency is raised, so is the harmonic number, and vice versa. <span style="color: #000105; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">5. What factors affect the number of nodes in a standing wave? <span style="font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif; font-size: large;">- Frequency, tension, mass, and length of the string are all factors that affect the number of nodes in a standing wave, due to the equation (derived from the equations  and )

<span style="color: #000105; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">CONCLUSION/ERROR ANALYSIS <span style="color: #000105; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">Frequency and wavelength have an indirect relationship. This is because wavelength times frequency equals velocity. As wavelength increases, frequency decreases**.** This relationship is displayed in the power graph. The r^2 value is equal to .99, meaning the values are very precise. <span style="color: #000105; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">Frequency and harmonic number are directly related. This is because the length of the string equals one half of the harmonic number times wavelength. Therefore, as wavelength increases, harmonic number does as well. Our data supports this, as the frequency vs. wavelength graph shows a linear fit with a positive slope. The y value of the equation represents frequency, the coefficient represents natural frequency, or fundamental harmonic, and x represents the harmonic number. The data is extremely precise as the r^2 value is 0.9922. Frequency and tension are also directly related. This is because wavelength times frequency equals the square root of elastic, tension, divided by inertial, m/L. Although the r^2 value of .9597 is not precise, our power graph slopes upward and illustrates the general relationship between frequency and tension.
 * <span style="color: #000105; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">Ariel **

There are a few sources of error in this lab that mostly involve human error and preciseness of the machinery. The percent differences betweent he measured and calculated data is all pretty low, and less than 20%. Percent difference between calculated frequency and measured frequency for frequency vs. harmonic number trial: This percent difference can be explained by errors in experimentation. The machine measuring the frequency may not have been adjusted correctly.

Percent difference between calculated frequency and measured frequency for frequency vs. wavelength: This percent difference can be explained by human error. This is because the length of the string was measured. Length was used to determine wavelength. The measurement of the length of the string may have been off or not perfectly precise, leading to error in the calculations.

Percent difference between calculated frequency and measured frequency for frequency vs. tension:

=Lab: Spring Mass vs. Period of Oscillation= Date: May 16th, 2011 Group: Roshni, Ariel, Evan, Ryan Class: Honors Physics Period 2

OBJECTIVE To directly determine the spring constant k of a spring by measuring the elongation of the spring for specific applied forces, to indirectly determine the spring constant k from measurements of the variation of the period T of oscillation for different values of mass on the end of the spring, and to compare the two values of spring constant k.

MATERIALS spring, tape, clamp and rod, masses, balance, timers, meterstick

HYPOTHESES The mass m hanging on the spring is directly related to the period T of oscillation of the spring squared. On a Force-Distance graph, the slope should match up with the spring constant k.

PROCEDURE

media type="file" key="arielMOVIE OF LABBBB.mov" width="300" height="300"

1. Hang a mass from the spring 2. Measure the "x" value by seeing how far the spring stretches 3. Measure the period by oscillating the spring 4. Plug the values into excel while populating the graphs 5. Repeat with different masses 6. Solve for k - the spring force constant using Hookes law and period equation

DATA

CALCULATIONS

Pre-lab Derivation:

Percent Difference: Note: K values used were averages from the four trials in the above spreadsheet.

CONCLUSION Based on our results, we can conclude that both of our hypotheses are correct. Based on our Period squared vs. Mass graph, there is a linear relationship between the two variables with a slope of .3931. When that value is multiplied by 4 x pi squared, we receive the value of k, or the spring force constant. We found this value to be 15.01, which is very close to our slope of the Distance vs. Force graph. Since the graph of distance and force is linear, it can be concluded that k is the slope of the graph. We found this value to be 15. 038. Since both k values calculated were found using the same masses and spring, the two should theoretically be equal. While they weren't completely the same, they were within decimal places of each other, showing that our results worked out very well. Human error accounted for part of our 9.308% difference. Also, the spring did not oscillate in a perfectly vertical motion. It bumped into the stand a few times, creating some error.

DISCUSSION 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, the spring constant is indeed constant. As seen in our data, although the four trials did not result in exactly the same K value, they all were within one to two newton meters and, therefore, very similar.

2. Why is the time for more than one period measured?
 * Because one period occurs too fast for us to measure. We measured the time for 10 periods and divided the time we found by 10 to get the time of one period.

3. Discuss the agreement between the k values derived from the two graphs. Which is more accurate? The k values are very close to each other using both methods. When calculated k using Hookes law, k = 15.038. When using the relationship between period and k, k = 15.01 because the slope of the graph equals .3931. This equals k/4pi^2. The results were really close to each other, and there was a percent difference between the two of .1864%. The value of k calculated using Hooke's law at equilibrium is probably more accurate because there is less of a chance of human error when figuring out x than there is when figuring out the period for using the period equation to calculate k. Since we used a stop watch, the timer may have stopped the time earlier or later than necessary. Additionally, the r^2 value when using Hooke's law was closer to 1, meaning that the graph was more precise.

4. Using results from Trial 4, generate the equations and the corresponding graphs for:
 * position with respect to time
 * [[image:x(t)_graph_ryan_listro.png]]
 * [[image:x(t)_graph_ryan_listro1.png width="720" height="469"]]
 * velocity with respect to time
 * [[image:v(t)_graph_ryan_listro.png]]
 * [[image:v(t)_graph_ryan_listro_1.png width="720" height="469"]]
 * acceleration with respect to time
 * [[image:a(t)_graph_ryan_listro.png]]
 * [[image:a(t)_graph_ryan_listro_1.png width="720" height="468"]]

5. A massless spring has a spring constant of k=7.85 N/m. If the spring is displaced -0.150m 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=0.425kg 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 M+1/3Ms (where M is the hanging mass and Ms is the mass of the spring)? Redo graph #2 using sqrt(M+1/3Ms), and explain these results.

When the mass used is changed, the general trend of the graph remains the same. There is a positive slope. However, when taking into account the mass of the spring, the results are probably more accurate because the "m" variable in the equation is more realistic.