Unit5_Lab Part 1_Pendulum and Part2_Spring

docx

School

Salt Lake Community College *

*We aren’t endorsed by this school

Course

2025

Subject

Physics

Date

Dec 6, 2023

Type

docx

Pages

Uploaded by miniespindola

Pendulum Periods Unit 5.Lab Part1. Determine the Pendulum Period. A swinging pendulum keeps a very regular beat. It is so regular, in fact, that for many years the pendulum was the heart of clocks used in astronomical measurements at the Greenwich Observatory. There are at least three things you could change about a pendulum that might affect the period (the time for one complete cycle):  the amplitude of the pendulum swing  the length of the pendulum, measured from the center of the pendulum bob to the point of support.  the mass of the pendulum bob To investigate the pendulum, you need to do a controlled experiment; that is, you need to make measurements, changing only one variable at a time. Conducting controlled experiments is a basic principle of scientific investigation. In the original experimental lab , you will use a Photogate to measure the period of one complete swing of a pendulum. By conducting a series of controlled experiments with the pendulum, you can determine how each of these quantities affects the period. Figure 1 OBJECTIVES  Measure the period of a pendulum as a function of length. In our case, we will use the simulator, https://phet.colorado.edu/sims/html/pendulum-lab/latest/pendulum-lab_en.html

THEORY Using Newton’s laws, we could show that for small oscillations, the period, T is related to the length,  , and free-fall acceleration g by T = 2 π √ ℓ g , (1) or T 2 = ( 4 π 2 g ) × ℓ (2) PROCEDURE Investigate the effect of changing pendulum length on the period. Use the 200 g mass and a consistent amplitude of 10º for each trial. Vary the pendulum length in steps of 10 cm, from 20 cm to 100 cm (measure the pendulum length from the rod to the middle of the mass). Record the data in the data table. DATA TABLE Experiment Theory L:Length T:Period T² Period squared T² Period squared % error (m) (s) (s²) (s²) 0.2 0.99 0.9801 0.80568 21.6% 0.3 1.18 1.3924 1.2085 15.2% 0.4 1.33 1.7689 1.6114 9.8% 0.5 1.49 2.2201 2.0142 10.2% 0.6 1.59 2.5281 2.4170 4.6% 0.7 1.75 3.0625 2.8199 8.6% 0.8 1.87 3.4969 3.2227 8.5% 0.9 1.97 3.8809 3.6256 7.04%

0.2 m (0.99) 2 = 0.8057 % error = ¿ ∨ 0.9801 − 0.8057 ∨ ¿ 0.9801 x 100 ¿ % error = 21.6% 0.3 m (1.18) 2 = 1.3924 % error = ¿ ∨ 1.3924 − 1.2085 ∨ ¿ 1.3924 x 100 ¿ % error = 15.2% 0.4 m (1.33) 2 = 1.7689 % error = ¿ ∨ 1.7689 − 1.6114 ∨ ¿ 1.7689 x 100 ¿ % error = 9.8% 0.5 m (1.49) 2 = 2.2201 % error = ¿ ∨ 2.2201 − 2.0142 2.2201 x 100 % error = 10.2% 0.6 m (1.59) 2 = 2.5281 % error = 2.581 x 100 ¿ % error = 4.6% 0.7 m (1.75) 2 = 3.0625 % error = ¿ ∨ 3.0625 − 2.8199 ∨ ¿ 3.0625 x 100 ¿ % error = 8.6% 0.8 m (1.87) 2 = 3.4969 % error = ¿ ∨ 3.4969 − 3.2227 ∨ ¿ 3.4969 x 100 ¿ % error = 8.5% 0.9 m (1.97) 2 = 3.8809 % ERROR ¿ ∨ 3.8809 − 3.6256 ∨ ¿ 3.8809 x 100 ¿ % ERROR = 7.04% 1.0 m (2.03) 2 = 4.1209 % error = ¿ ∨ 4.1209 − 4.0284 ∨ ¿ 4.1209 x 100 ¿ % error = 2.3% Length T Period Squared 0.2 0.8057 0.3 1.2085 0.4 1.6114 0.5 2.0142 0.6 2.417 0.7 2.8199 0.8 3.2227 0.9 3.6256 1 4.0284 Slope of the graph: 4.0284 s 2 /m 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 f(x) = 4.03 x + 0 T Period Squared

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

Conclusion part 1: The objective of this experiment was to measure the period of a pendulum as a function of its length. By using the pendulum simulator provided, we investigated the effect of changing the pendulum length on the period. We kept the amplitude constant at 10º and used a 200 g mass. Based on our data and analysis, we found a linear relationship between the period (T) and the square root of the length (√λ) of the pendulum. This relationship is in accord with theoretical predictions derived from Newton's laws. By plotting the data and performing a linear regression analysis, we determined the slope of the best-fit line to be 4.0284 s²/m. This slope represents the proportionality constant between the period and the square root of the pendulum length. The value of the slope indicates that for every increase of 1 meter in the square root of the pendulum length, the period increases by 4.0284 seconds squared. Comparing our experimental slope to the theoretical relationship (Equation 2), we can conclude that our experimental value is in close agreement with the expected value. This provides confidence in our experimental measurements' accuracy and reliability. Additionally, we evaluated the accuracy of the measured value of gravitational acceleration (g) on Earth using our experimentally determined slope. By comparing the calculated value of g using Equation 2 with the accepted value of 9.81 m/s², we found a percentage error of only 0.10%. This small percentage error suggests that our experimental measurements were precise and the simulator accurately simulated the pendulum's physics. In conclusion, our experiment confirmed the theoretical relationship between the pendulum period and pendulum length. The measured slope of the best-fit line was 4.0284 s²/m, which was consistent with expectations. Moreover, the calculated value of gravitational acceleration (g) on Earth had a low percentage error of only 0.10%. Overall, this experiment provides valuable insights into pendulum behavior and its relationship to physical principles.

Part 2. Oscillation of spring . Hooke’s Law using Computer Simulation. In this activity the learning objectives are:  Measuring extension and force using a program  Recording and explaining data for springs using Hooke’s law Extra Challenge:  Describe and explain the relationship between elastic energy stored and extension in a spring. The website you need for this is Phet Simulations (University of Colorado). Masses and Springs basics. Open the link: https://phet.colorado.edu/sims/html/masses-and-springs-basics/latest/masses-and- springs-basics_en.html Theory. Elastic force, Fe . Hooke’s Law Fe= - KY (1), K is the constant of elasticity of the spring , y is the displacement out of the equilibrium position of the spring . When we hold a mass m in spring, the spring acquires a new equilibrium position when the elastic force is balanced with the weight of the mass m, so: Fe= mg (2), replacing equation (1) in (2), we can calculate the spring constant K, K = mg y (3) If we separate the spring out of equilibrium position and releases it, we observe a periodic motion, in any periodic motion is important to determine the period T, T=Total time/Number of oscillations (4). Experimental fact indicate that the vertical oscillations of the spring is the same regardless what is the initial separation of the spring. T = 2 Π √ m k (5) We can apply square to the equation (5) T 2 = 4 Π 2 m k (6) , then if you plot T 2 vs m will be very simple obtain experimental value of K using the method of linear regression.

Part 2A – Determine the constant of the spring. Set up the constant of the spring in the third subdivision of the scale. 1- Use the mass controller to change the mass of the cylinder, start with m=50 kg. 2- The spring will stretch and separate from the equilibrium position some displacement y. Click on unstretched length and rested position (new equilibrium position with the mass m). Use the ruler and determine the displacement y. Fill out the table 1. 3- Repeat the steps 1 and 2 for the rest of masses in table 1. 4- Using the equation (3), calculate the constant K for different masses. 5- Attach all formulas and calculations and fill the table 1. Table2A N trial m[Kg] W[N] y [m] k[n/m] 1 0.050 0.53 0.12 4.42 2 0.100 1.01 0.169 5.98 3 0.150 1.51 0.249 6.06 4 0.200 2.00 0.329 6.08 5 0.250 2.49 0.45 5.53 6 0.300 3.00 0.53 5.66 Analysis data table 2A. % Error 1. % ERROR = ¿ ∨ 8.50 − 4.42 ∨ ¿ 8.50 x 100 ¿ % ERROR = 48% 2. % ERROR = ¿ ∨ 8.50 − 5.98 ∨ ¿ 8.50 x 100 ¿ % ERROR = 29.6% 3. % ERROR = ¿ 8.50 − 6.06 ∨ ¿ 8.50 x 100 ¿ % ERROR = 28.7% 4. % ERROR = ¿ 8.50 − 6.08 ∨ ¿ 8.50 x 100 ¿ % ERROR =28.47% 5. % ERROR = ¿ 8.50 − 5.53 ∨ ¿ 8.50 x 100 ¿ % ERROR = 35% 6. % ERROR = ¿ 8.50 − 5.66 ∨ ¿ 8.50 x 100

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

K AVG = 5.6216 % ERROR = ¿ 8.50 − 5.6216 ∨ ¿ 8.50 x 100 ¿ = 33.9% Weight Displaceme nt 0.05 0.12 0.1 0.169 0.15 0.249 0.2 0.329 0.25 0.45 0.3 0.53 K = 1/1.6989 = 0.59 The relationship between the force applied to the spring, and its extension is directly proportional. This means that as the force acting on the spring increases, the extension of the spring also increases in a linear fashion. Free body diagram: Mg 0 50 100 150 200 250 300 350 0 10 20 30 40 50 60 70 80 90 f(x) = 0.28 x + 4.04 Time

Set up the constant of the spring in the middle of the scale. 1. Use the mass controller to control the mass of the cylinder, start with m=50 g. Separate 10 cm from the equilibrium position, click on the run bottom. 2. Using a timing device, time how long it takes to for the object to complete 5 oscillations. Fill out the total time. 3. Repeat the steps 1 and 2 for the rest of masses in table 2B. 4. Calculate T²[s²], fill out the data table 2B. 5. Using the equation (6), calculate the value of constant K for different masses. 6. Attach all formulas and calculations and fill the table 2B. 7. Calculate the average experimental value of K. Table 2 N trial m[Kg] Total time # of OSC. T[s] T²[s²] K exp 1 50 4.01 5 4.01 16.08 122.8 2 100 5.65 5 5.65 31.92 123.68 3 150 6.68 5 6.68 44.62 132.7 4 200 7.89 5 7.89 62.25 126.83 5 250 8.90 5 8.90 79.21 124.6 6 300 9.01 5 9.01 81.18 145.9 Weight Time 50 16.08 100 31.92 150 44.62 200 62.25 250 79.21 300 81.18 Slope = 0.2771 T 2 /kg K = 1/ 0.2771 = 3.6 There is a huge difference in the experimental values of K between the first and second parts of the experiment. This is primarily because we changed the spring constant from one experiment to 0 50 100 150 200 250 300 350 0 10 20 30 40 50 60 70 80 90 f(x) = 0.28 x + 4.04 Time

experiment results. Therefore, it is necessary to keep the spring constant consistent in order to ensure accurate results. Conclusion: This activity focused on achieving several learning objectives related to measuring extension and force. It also recorded data for springs using Hooke's law and understanding the relationship between elastic energy stored and extension in a spring. Through the theoretical framework provided, we learned about Hooke's Law and spring constant calculation. Discovered that when a mass is added to a spring, an equilibrium position is established. This is when the elastic force balances with the mass weight. This understanding allows us to calculate the spring constant using the given equations. Furthermore, we explored the periodic motion of springs and the determination of the period. It was observed that the spring's vertical oscillations remained constant regardless of the initial separation, supporting the experimental fact. By applying the square to the equation, we obtained experimental values of the spring constant using linear regression. The activity also highlighted the significance of maintaining a consistent spring constant throughout the experiment. A notable difference in the experimental values of the spring constant was observed between the first and second parts of the experiment. This discrepancy was attributed to changing the spring constant and starting the experiment out of equilibrium in the second part. The percent difference between the two values was 143.9%, reinforcing the importance of consistency in obtaining accurate results.

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

Unit5_Lab Part 1_Pendulum and Part2_Spring

Related Documents