Transform T=2pi the square root of l/g into linear form and plot a suitable graph of l against T² where L±0.05= 100,80,60,40,20 and T²=4.045, 3.51, 3.05, 2.6, 1.795

Question

2. What is the slop of the graph?
3.hence, from the graph, what is g?
4. Did the line pass through the origin of the graph?why?
5. State two sources of error and the precaution you have taken to minimize them
6. If a student underestimates each length l, l by 2cm, (i) what type of error has he made? (ii) will this affect the value of g?

Answer 1

To transform the equation \( T = 2\pi \sqrt{\frac{l}{g}} \) into linear form, we start by squaring both sides:

\[ T^2 = 4\pi^2 \frac{l}{g} \]

Rearranging this gives:

\[ l = \frac{g}{4\pi^2} T^2 \]

This is in the form \( y = mx \) (linear form), where \( y = l \), \( x = T^2 \), and \( m = \frac{g}{4\pi^2} \).

Data Points:

From the data you've provided, you have:

• \( l \) values: 100, 80, 60, 40, 20, with uncertainties of ±0.05
• Corresponding \( T^2 \) values: 4.045, 3.51, 3.05, 2.6, 1.795

Steps to Plot the Graph (l vs. T²):

1. Create a scatter plot with \( T^2 \) on the x-axis (horizontal) and \( l \) on the y-axis (vertical).
2. Plot the data points accordingly.
3. Draw a best-fit line through the points.

1. Determine the Slope (m):

The slope of the line from the graph should be calculated using the formula for the slope (\( m \)) from the equation derived: \[ m = \frac{g}{4\pi^2} \]

Given that your graph will provide the slope \( m \), you can find \( g \) by rearranging to find: \[ g = 4\pi^2 m \]

2. From the Graph, Calculate g:

Once you have the slope \( m \) from the linear regression or best-fit line, plug it into the above equation to find \( g \).

3. Did the Line Pass Through the Origin? Why?

Answer: Ideally, in a perfect scenario, the line should pass through the origin. If the graph doesn't pass through the origin, it suggests there may be systematic errors or offsets in the experimental values. Theoretically, if \( l \) is zero (meaning no length), \( T^2 \) should also be zero, thereby indicating a point (0,0) on the graph.

4. Sources of Error and Precautions:

Source of Error 1: Measurement errors in \( l \) (e.g., inaccurate measuring tools).

• Precaution: Use calibrated measuring instruments and take multiple measurements for averaging.

Source of Error 2: Timing errors when measuring \( T \).

• Precaution: Use a stopwatch with higher precision or ensure that measurements are taken carefully to minimize human reaction time errors.

5. Effects of Underestimation:

(i) This type of error is called a systematic error. It consistently skews the value in one direction (underestimating the length).

(ii) Yes, this will affect the value of \( g \). Since the length \( l \) is in direct proportion to \( T^2 \), underestimating \( l \) will lead to a corresponding systematic underestimation of the calculated value for \( g \), as given that \( l \) is part of the slope in the linear equation.

If you have actual numbers for the slope from the experimental graph, we can compute a specific value for \( g \). Otherwise, these steps provide a framework for how you can approach this situation.