Double Spring Harmonic Oscillator Lab

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1 Dylan Humenik and Benjamin Daily Double Spring Harmonic Oscillator Lab Objectives: -Experimentally determine harmonic equations for a double spring system using various methods Part 1 Determining k of your two springs from hanging masses a. Using hanging masses, determine the spring constant Note- use multiple mass trials and find the spring constant based on the combination of all data rather than one data point Construct and insert a graph of your data Part 2- Determining theoretical harmonic oscillations as functions of time Note this portion of the lab incorporates no actual experimentations; This is strictly theoretical equations based on your knowledge of harmonic oscillators. b. Once the spring constant is determined from part 1, use it to find the period of oscillation if two springs were used horizontally, with one on each side (Given displacement (.05m) and mass 1.5Kg) c. Calculate angular velocity d. Give your position equation as a +cosine function of time e. Derive equations for velocity functions and acceleration functions as functions of time f. Plot your equations using function grapher below, and include on your lab (grapherhttp:// Part 3 - Determining k from harmonic oscillation data g. Find the spring constant from oscillation data i. Record period raw data as a function of mass. ii. Create a graph based on this data. Considering the appearance and shape of the graph, determine variables you could plot to product a linear graph. iii. Create a linear graph and determine the spring constant using the appropriate physics equation to assist. Part 4- Construct functions from sinusoidal graphs (opposite of part 2 note you should not be using calculus to assist even though you could)

2 h. With the oscillating system still set up from part 3, video a.05meter displacement with a 1.5Kg mass. Video the oscillations plotting position and velocity graphs as functions of time. i. Construct the position graph as a complete +cos graph and overlay the velocity on that graph (make your central point the zero position and have the initial position be the positive axis) Data: Part 1 Spring 1 Spring 2 Mass (kg) Force (N) Displacement (m) Mass (kg) Force (N) Displacement (m) Spring Constant of Single Spring 1 (N/m) Spring Constant of Single Spring 1 (N/m) N/m N/m Spring Constant Average N/m Part 2 Spring constant (system, assume the average spring constant from part 1 is the constant N/m for each individual spring) Period of System 1.20 s Angular Velocity 5.23 rad/s Position Equation x = 0.5cos(5.23 t) Velocity Equation v = -2.62sin(5.23 t) Acceleration Equation a = cos(5.23 t) Part 3 Trial Mass (kg) Period (s) Linear Plot Variables Mass 0.5 (kg) Period (s) Spring constant (system) N/m Spring constant (1 spring) N/m Part 4 Period From Graph s Angular Frequency 4.96 rad/s Position Equation x = cos(4.96t) Velocity Equation v = sin(4.96t)

3 Stretch Distance (m) Stretch Distance (m) Graphs: Part 1: Stretch Distance as a Function of Force (Spring 1) 0.25 y = x Force (N) Stretch Distance as a Function of Force (Spring 2) y = 0.058x Force (N)

4 Period (s) Part 2: *The link provided did not work. Part 3: Period as a Function of Mass Mass (kg)

5 X (m) X velocity (m/s) Period (s) Period as a Function of Mass^0.5 (Linear Graph) y = x Mass^0.5 (kg) Part 4: Position and Velocity as a Function of Time Time (s) X (m) X Velocity (m/s)

6 Calculations: Part 1: Restoring Force of Spring: F = ma F = (0.05 kg)(9.8 m/s 2 ) F = 0.49 N *Similar calculations were performed for the remainder of the data. Spring Constant from Graph Data: k = N/m k = (0.49 N) / (0.023 m) k = 21.3 N/m *Similar calculations were performed for the remainder of the data, and then averaged to find the spring constant from each spring. Spring Constant Average: Avg k = (k spring 1 ) + (k spring 2 ) / 2 Avg k = (20.54 N/m N/m) / 2 Avg k = N/m Part 2: Spring Constant for a Double Spring System: k T = k 1 + k 2

7 k T = N/m N/m k T = N/m Period of Oscillation: T = 2π/ω T = 2π / 5.23 rad/s T = 1.20 s Angular Velocity: ω 2 = k/m ω 2 = (41.00 N/m) / (1.5 kg) ω 2 = 27.3 rad 2 /s 2 ω = 5.23 rad/s Position Equation: x = 0.5cos(5.23 t) Velocity Equation: v = -Aωsin(ωt) dx = d(0.5cos(5.23 t)) v = -2.62sin(5.23 t) Acceleration Equation: a = -Aω 2 cos(ωt) dv = d(-2.62sin(5.23 t))

8 a = cos(5.23 t) Part 3: Spring Constant (System): ω 2 = k/m ω = 2π/T ω = 2π / (0.636 s) ω = 9.88 rad/s rad 2 /s 2 = k / (0.5 kg) k = N/m *Similar calculations were performed for the remainder of the data. Spring Constant (1 spring): k 1 = k T /2 k 1 = N/m / 2 k 1 = N/m Part 4: Angular Velocity: ω = 2π/T ω = 2π / s ω = 4.96 rad/s

9 Discussion: 1. Why is it more appropriate to calculate the spring constant from a data set than from a single data point? Calculating from multiple sources of data, including ones with different masses attached, allows for more accurate results. 2. Without referencing graphs, conceptually explain where maximum and minimum points will be during oscillation for a. Kinetic Energy Max center Min edges b. Force (spring) Max edges Min center 3. If you double the length of a spring made of a certain material the spring constant will (decrease/increase)? Explain The spring constant will decrease because the displacement distance will get larger. 4. Explain the effect of adding a second a spring on to the system, as in part 2, on the opposite side of the first spring? When a spring is added on the opposite side, the spring constant is doubled. 5. Derive an equation for potential energy stored in the spring showing proper calculus. F = -kx F = -kx F = k -x PE = ½ kx 2

10 6. During oscillation, how does the relative distance of your spring connection points affect the period of oscillation? Increasing the distance of the spring connection points will have no effect on the period of oscillation. 7. Explain in detail possible reasoning for the difference between the results for spring constants in part 1 and part 3 (A: the labs were set up differently; the springs were not the same)-these should only be the start of thorough arguments. This will act as your error analysis, so put emphasis on this question. The difference if spring constants in Part 1 and Part 3 can best be explained by the fact that the labs were set up differently. For Part 1, errors can be made if the stretch distance is not measured accurately. If the measurement taken is too high, then a lower spring constant will result. Conversely, if the measurement taken is too low, then a higher spring constant will result. For Part 3, the time was taken for five oscillations, then divided by five to find period. If the time taken was higher then the actual time for five oscillations, then the spring constant will be lower, and vice versa. The reason that the time for five trials was measured was to produce more accurate results. 8. Could you determine harmonic equations from the period data in part 3, explain in detail why/how you could or could not do this task? A harmonic motion equation can be determined. This is possible because of the equation ω 2 = k/m. The equation can be found by setting a sine or cosine graph to the angular velocity (x = sin(ωt). Essentially, the amplitude would not be relevant/needed, but if it were desired, then the stretch distance would also have to be measured. 9. Describe how you constructed your velocity function in part four based on the graph. You may use formulas and calculations to assist in your explanation. The velocity function was created by taking the derivative of the position function. Assume that the derivative of a cosine function is a sine function. The following model was followed: x = A cos (ωt) dx = d(a cos (ωt) v = -Aω sin (ωt)

Physics 2101 S c e t c i cti n o 3 n 3 March 31st Announcements: Quiz today about Ch. 14 Class Website:

Physics 2101 S c e t c i cti n o 3 n 3 March 31st Announcements: Quiz today about Ch. 14 Class Website: Physics 2101 Section 3 March 31 st Announcements: Quiz today about Ch. 14 Class Website: http://www.phys.lsu.edu/classes/spring2010/phys2101 3/ http://www.phys.lsu.edu/~jzhang/teaching.html Simple Harmonic