- Calculate total times \(t_1+t_2\) and periods \(T = \frac{t_1+t_2}{40}\) for each height to determine \(T^2\).
- Perform linear regression of \(h\) against \(T^2\) to find the slope \(\approx -24.08 \, \text{m/s}^2\) and y-intercept \(\approx 148.70 \, \text{m}\).
- Implement precautions: keep oscillation angles small and use precise timing techniques.
- A spherical bob is chosen for its defined center of mass and low drag. Examples of SHM include a mass-spring system and guitar string vibrations.
- Final values: Slope \(\boxed{-24.08 \, \text{m/s}^2}\), y-intercept \(\boxed{148.70 \, \text{m}}\).
Explanation
- Data Analysis and Calculation of Period squared
The problem requires analyzing data from a simple pendulum experiment to determine its oscillatory properties. The data given includes the height \(h\) and two time measurements \(t_1\) and \(t_2\) for a set number of oscillations (\(N=20\)). First, we calculate the total time for each height as the sum of the two trials, \(t_1 + t_2\). Then, we determine the period \(T\) for one oscillation, which is the average total time divided by twice the number of oscillations, \(T = \frac{t_1 + t_2}{2N}\). Finally, we compute the square of the period, \(T^2\), to investigate the relationship between height and time.
The calculated values are as follows:
- For \(h = 20.00\) m: \(t_1 + t_2 = 91.40\) s, \(T = 2.285\) s, \(T^2 \approx 5.221\) s\(^2\).
- For \(h = 40.00\) m: \(t_1 + t_2 = 86.80\) s, \(T = 2.170\) s, \(T^2 \approx 4.709\) s\(^2\).
- For \(h = 60.00\) m: \(t_1 + t_2 = 76.20\) s, \(T = 1.905\) s, \(T^2 \approx 3.629\) s\(^2\).
- For \(h = 80.00\) m: \(t_1 + t_2 = 66.80\) s, \(T = 1.670\) s, \(T^2 \approx 2.789\) s\(^2\).
- For \(h = 100.00\) m: \(t_1 + t_2 = 57.50\) s, \(T = 1.4375\) s, \(T^2 \approx 2.066\) s\(^2\).
- Graphical Analysis and Precautions
To determine the relationship, we plot \(h\) on the vertical axis against \(T^2\) on the horizontal axis. Using linear regression on these data points, we find the line of best fit in the form \(h = m T^2 + c\).
The slope \(m\) is approximately \(-24.08\) m/s\(^2\), and the y-intercept \(c\) is approximately \(148.70\) m. This indicates a negative correlation between the height and the square of the period in this specific experiment.
Experimental precautions include:
- Ensuring the oscillations are kept at small angles to maintain the simple harmonic motion approximation.
- Starting and stopping the stopwatch precisely at the equilibrium position to minimize measurement errors in time.
- Additional Questions and Final Answers
(i) A spherical bob is preferred because it has a well-defined center of mass, which makes the effective length of the pendulum easier to measure. Additionally, its streamlined shape minimizes air resistance (drag) during oscillation, ensuring that the motion remains as close to simple harmonic motion as possible for a longer duration.
(ii) Two other examples of simple harmonic motion include:
- A mass attached to a spring oscillating vertically or horizontally.
- The vibrations of a guitar string or any vibrating tuning fork.
The final slope is \(\boxed{-24.08 \, \text{m/s}^2}\) and the y-intercept is \(\boxed{148.70 \, \text{m}}\).
Examples
Understanding the relationship between physical parameters like length and period is fundamental to mechanical engineering and clock design. For example, by analyzing how a pendulum's period changes with height, engineers can calibrate mechanical clocks to maintain accurate time across different gravitational environments or adjust the length of components to achieve specific resonant frequencies in structures, ensuring safety and performance in dynamic systems.