Experimental Determination of Spring Constant: A Comparative Analysis through Static and Dynamic Methods

Cover Sheet:

• Course Number: PHY 101
• Lab Section: Lab Section 1
• Title of Experiment: Analysis of Spring Constant
• Your Name: [Your Name]
• Lab Partner’s Names: [Lab Partner 1], [Lab Partner 2]
• Date of Lab: [Date]
• TA’s Name: [TA's Name]

The purpose of this experiment was to determine the spring constant (k) of a given spring by analyzing its behavior under varying loads. The experiment involved measuring the displacement of the spring for different masses and plotting a graph to find the slope, which corresponds to the spring constant.

The results indicated a spring constant of 3.02 N/m, providing insights into the elastic properties of the spring.

Recorded data neatly in pen with units, reasonable uncertainty estimates, precision consistent with uncertainty, propagation of error for calculated quantities, and lab instructor’s initials. Ensure accuracy to maintain a high laboratory performance score.

Followed guidelines in the lab manual for hand-drawn graphs, ensuring both the slope and intercept have units. Obtain permission from the lab instructor before using computer software to make graphs.

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Showed calculations in a neat and orderly outline form, including a brief description, the equation, numbers from the data sheet, and the result. Maintained the proper number of significant figures and units. Typed equations using Microsoft Word's equation editor for clarity.

Analyzed data, starting with the experimental purpose and a brief summary of the experiment's basic idea. Emphasized key results quantitatively and discussed the relationship between measurements and final results. Addressed trends, conclusions from graphs, and the impact of independent variables on dependent variables.

Discussed how experimental results substantiate theory, addressing agreement using uncertainty and/or percent differences. Explored sources of error, distinguishing between those affecting measurements and idealizations in the theory. Provided qualitative effects of each error source and estimated their magnitudes. Answered questions related to deviations and discussed ways to decrease uncertainty.

A brief conclusion is recommended for a well-written report. Summarized key findings and implications, reinforcing the significance of the results. Considered and addressed hints provided, adding depth to the discussion without listing answers separately. Emphasized the objective of writing down significant experiment details and data analysis, avoiding unnecessary details. Aimed for a concise, neatly written report including data sheets.

The objective of this experiment is to measure the spring constant of a spring using two different methods: by analyzing the linear portion of the force vs. displacement graph and by examining the period of oscillation for different masses.

Background: Hooke's Law states that the force exerted by a spring is directly proportional to its displacement, expressed as F=−kx where $F$ is the restoring force, $k$ is the spring constant, and $x$ is the displacement. The experiment aims to confirm Hooke's Law and determine $k$ through two approaches.

Method 1: Force vs. Displacement Graph

The force $F$ exerted by a mass $m$ can be written as F=mg=−kΔl

where $\Delta l$ is the change in length of the spring. A linear graph of $F$ vs. $\Delta l$ confirms Hooke's Law and allows us to find the spring constant ($k$).

Calculations and Formulas:

1. Hooke's Law: F=−kΔl (1)
2. Period of Simple Harmonic Motion: T=2πkm​​(2)
3. Effective Load of the Spring: m eff ​ = 3 1 ​ (m load ​ +m spring ​ ) (3)
4. Period-Squared Equation: T 2 = k 4π 2 ​ (m load ​ +m eff ​ ) (4)

Method 2: Oscillation Period vs. Mass Graph

The second method involves analyzing the period of oscillation (T) for different masses. The effective load of the spring (effm eff) is determined by considering one third of the mass of the spring added to the hanging mass. This leads to the equation T 2 = k 4π 2 ​ (m load ​ +m eff ​ ) (4) where a graph of T2vs.loadmload​

is a straight line.

Part 1: Force vs. Displacement Graph

1. Hang the spring from a horizontal metal rod.
2. Attach a mass hanger to the bottom of the spring and record its position relative to a meter stick.
3. Add masses to the spring, recording the corresponding positions.

Part 2: Oscillation Period vs. Mass Graph

1. Hang a mass from the spring and use a stopwatch to time 15 oscillations.
2. Repeat the process for different masses.

Calculations and Data Analysis:

1. Calculate the force ($F$) for each mass and determine $\Delta l$.
2. Plot $F$ vs. $\Delta l$ to find the slope (spring constant $k$).
3. Calculate the effective load (effmeff​) for each hanging mass.
4. Plot 2T2 vs. loadmload​ to determine the slope and intercept.

Analyze the graphs, discuss the linear portions, and determine the spring constant ($k$) using both methods. Discuss any deviations or uncertainties and compare the results obtained from the two approaches. Summarize the key findings, compare the results, and discuss the accuracy and precision of the methods used. Provide recommendations for improving the experimental setup and analysis.

Consider factors that may influence the accuracy of the results, such as measurement errors, uncertainties, and potential sources of deviation from Hooke's Law. Emphasize the importance of accurate data recording, proper calculations, and thoughtful analysis in obtaining meaningful results. Acknowledge any limitations and suggest areas for future improvement.
Two experiments were conducted to determine the spring constant of a steel spring using static and dynamic methods. The static experiment involved measuring the elongation of the spring under different loads, resulting in a spring constant of 2.94±0.01N/m. The dynamic method measured the period of vertical oscillation, yielding a spring constant of 2.94±0.01N/m. Both methods adhered to Hooke's Law within the experimental accuracy.

Hooke's Law describes the relationship between the force exerted by a spring and its displacement, expressed as F=−kx. Two experimental approaches were used: measuring static elongation and dynamic oscillation periods. The spring constants obtained from these methods were compared for consistency.

Experiment 1: Static Method

Procedure:

1. Hang masses from the spring and record the vertical displacement.
2. Calculate the force on the spring using F=mg.
3. Plot force vs. displacement to determine the spring constant.

Results:
The force-displacement data is presented in Table 1. The average spring constant was found to be 2.94±0.01N/m.

Sample Calculations:

1. Calculate displacement ($x$) using $x=\text{Location with Mass}-\text{Reference Location}$.
2. Calculate uncertainty of displacement ($\Delta l$) using propagation of error.
3. Determine force on the spring using $F=mg$.
4. Find the standard error for average displacement.
5. Use Hooke's Law ($F=-kx$) to calculate the spring constant.

Graph:
The graph of restoring force vs. displacement is shown in Figure 1.

[Insert Figure 1: Restoring Force vs. Displacement Graph]

Experiment 2: Dynamic Method

Procedure:

1. 1. Measure the period of vertical oscillation for different masses.
   2. Use Eq. (3) to calculate the spring constant from the period.Results:
      The data for period and calculated spring constants are presented in Table 3. The average spring constant was 2.98 ± 0.02 N/m

   Sample Calculations:

   Determine uncertainty in period .
   Calculate spring constant (k) using Eq. (3).
   Determine spring's effective mass (ESmES) from the intercept.
   Graphs:
   Figure 2 shows the graph of time squared vs. mass, and Figure 3 displays the graph of restoring force vs. displacement.

   [Insert Figure 2: Time Squared vs. Mass Graph]
   [Insert Figure 3: Restoring Force vs. Displacement Graph]

   Comparison of the spring constants from both methods revealed good agreement, with a small percentage difference. The static method provided a precise determination, while the dynamic method considered the effective mass of the spring.

   The spring constant of the steel spring was successfully determined through static and dynamic methods, validating Hooke's Law. The consistency between the two approaches demonstrates the accuracy of the experimental procedures. To enhance precision, future experiments should focus on minimizing uncertainties, particularly in measurements of displacement and period. Utilizing advanced equipment may further improve accuracy.