Report For Experiment 2 Measurements Answers

Report for Experiment 2: Measurements and Analysis

Introduction
In this report, we analyze the measurements obtained during Experiment 2, which focused on determining the acceleration due to gravity (g) using a simple pendulum. The experiment involved measuring the time period of oscillation for varying lengths of the pendulum and applying theoretical principles to calculate g. This report will detail the methodology, results, and critical evaluation of the experimental process, emphasizing the accuracy of measurements and their alignment with theoretical expectations.

Objectives of the Experiment
The primary goals of Experiment 2 were:

1. To measure the time period of a simple pendulum for different lengths.
2. To calculate the acceleration due to gravity (g) using the pendulum formula.
3. To analyze the relationship between the length of the pendulum and its time period.
4. To evaluate the impact of experimental errors on the results.

Methodology
The experiment was conducted using the following apparatus:

• A simple pendulum (string, bob, and support).
• A stopwatch.
• A measuring tape.
• A protractor (to ensure the pendulum was released at a small angle).

Procedure:

1. Setup: The pendulum was assembled with a string of known length (L), and the bob’s mass was measured.
2. Initial Conditions: The pendulum was displaced to a small angle (≤ 15°) to ensure simple harmonic motion.
3. Data Collection:
   • The length of the pendulum was varied in increments of 10 cm (e.g., 40 cm, 50 cm, 60 cm).
   • For each length, 20 oscillations were timed, and the total time was recorded.
   • The time period (T) was calculated by dividing the total time by 20.
4. Calculations:
   • Using the formula for the time period of a simple pendulum:
     $ T = 2\pi \sqrt{\frac{L}{g}} $
     Rearranging to solve for g:
     $ g = \frac{4\pi^2 L}{T^2} $
   • The average value of g was computed from multiple trials.

Results
The raw data and calculated values are summarized below:

Key Observations:

• The time period (T) increased with the length (L), consistent with the theoretical prediction $ T \propto \sqrt{L} $.
• The calculated values of g (9.42–9.68 m/s²) were close to the accepted value of 9.81 m/s².

Scientific Explanation
The experiment relied on the principle that a simple pendulum exhibits simple harmonic motion for small angular displacements. The time period (T) depends on the length of the pendulum (L) and the acceleration due to gravity (g), as described by the formula:
$ T = 2\pi \sqrt{\frac{L}{g}} $
By measuring T for different L values, we could isolate g and compare it to the standard value. The slight discrepancies between the calculated and accepted g values were attributed to experimental errors, as discussed below.

Discussion of Errors
Several factors could have influenced the results:

1. Human Reaction Time: The stopwatch was started and stopped manually, introducing timing errors.
2. Air Resistance: The bob’s motion was not entirely frictionless, slightly altering the time period.
3. Measurement Inaccuracy: The length of the string was measured with a tape, which could have slight parallax errors.
4. Angle of Release: If the pendulum was released at an angle >15°, the motion would no longer be simple harmonic, affecting the results.

To mitigate these errors, future experiments could use automated timing devices and ensure the pendulum is

To furtherimprove the accuracy of the measurement, the pendulum could be equipped with a photogate or an electronic timer that records the passage of the bob without human intervention. Using a finer‐scale ruler or a caliper to determine the effective length—measured from the pivot point to the center of mass of the bob—would reduce parallax and alignment errors. Additionally, keeping the amplitude of oscillation below 5° ensures that the motion remains within the small‑angle approximation, thereby preserving the validity of the governing equation. Conducting multiple trials for each length and discarding outliers before averaging would also tighten the statistical confidence of the derived g values.

In summary, the experiment successfully demonstrated the inverse relationship between the square of the oscillation period and the length of a simple pendulum, yielding an experimental acceleration due to gravity that was within 4 % of the standard value. The minor deviations observed were consistent with the expected contributions of timing inaccuracies, air drag, and slight deviations from the idealized conditions. These findings reaffirm the utility of the simple pendulum as a pedagogical tool for probing fundamental gravitational principles, while also highlighting the importance of meticulous experimental design to approach theoretical precision.