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Sagot :
Sure! Let's go through the questions step-by-step.
### Part (a): Plotting a Graph of [tex]\( L \)[/tex] Against [tex]\( T^2 \)[/tex]
We'll start by extracting the values from the table and plotting the graph.
#### Given Data:
1. [tex]\( L \)[/tex] (cm): [tex]\( [140.00, 120.00, 100.00, 80.00, 60.00, 40.00, 20.00] \)[/tex]
2. [tex]\( T^2 \)[/tex] (seconds[tex]\(^2\)[/tex]): [tex]\( [6.0639, 5.1179, 4.1514, 3.2942, 2.6001, 1.7437, 1.0639] \)[/tex]
To create a plot:
1. Plot [tex]\( T^2 \)[/tex] values along the x-axis.
2. Plot [tex]\( L \)[/tex] values along the y-axis.
Ensure that you label the axes appropriately. The x-axis will be titled "T^2 (seconds[tex]\(^2\)[/tex])" and the y-axis will be titled "L (cm)".
### Part (b): Shape of the Graph
When you plot [tex]\( L \)[/tex] against [tex]\( T^2 \)[/tex], observe the relationship between the two variables. In the context of a simple pendulum, the period [tex]\( T \)[/tex] is related to the length [tex]\( L \)[/tex] of the pendulum by the equation:
[tex]\[ T = 2\pi\sqrt{\frac{L}{g}} \][/tex]
where [tex]\( g \)[/tex] is the acceleration due to gravity. This equation implies a quadratic relationship [tex]\( L \propto T^{2} \)[/tex].
Given this relationship, the graph of [tex]\( L \)[/tex] against [tex]\( T^2 \)[/tex] should be a straight line. This suggests that as [tex]\( T^2 \)[/tex] increases, [tex]\( L \)[/tex] increases linearly, confirming a direct proportional relationship.
### Part (c): Sources of Error
When conducting experiments involving measurements such as time and length, there are common sources of error. Two possible sources of error in this experiment are:
1. Human Reaction Time: There can be inaccuracies in timing measurements because the time intervals are being recorded manually. When starting and stopping a stopwatch, human reaction delay can cause inconsistencies.
2. Length Measurement Accuracy: The lengths could have been measured with a ruler or a measuring tape. Small errors in alignment or reading off the scale can introduce inaccuracies.
In summary:
1. Plot a graph of [tex]\( L \)[/tex] against [tex]\( T^2 \)[/tex] as described.
2. Shape of the graph: The graph should be a straight line, indicating a linear relationship.
3. Sources of error:
- Human reaction time in timing measurements.
- Inaccuracies in length measurements.
Feel free to sketch this graph on graph paper or use graphing software to visualize the data accurately!
### Part (a): Plotting a Graph of [tex]\( L \)[/tex] Against [tex]\( T^2 \)[/tex]
We'll start by extracting the values from the table and plotting the graph.
#### Given Data:
1. [tex]\( L \)[/tex] (cm): [tex]\( [140.00, 120.00, 100.00, 80.00, 60.00, 40.00, 20.00] \)[/tex]
2. [tex]\( T^2 \)[/tex] (seconds[tex]\(^2\)[/tex]): [tex]\( [6.0639, 5.1179, 4.1514, 3.2942, 2.6001, 1.7437, 1.0639] \)[/tex]
To create a plot:
1. Plot [tex]\( T^2 \)[/tex] values along the x-axis.
2. Plot [tex]\( L \)[/tex] values along the y-axis.
Ensure that you label the axes appropriately. The x-axis will be titled "T^2 (seconds[tex]\(^2\)[/tex])" and the y-axis will be titled "L (cm)".
### Part (b): Shape of the Graph
When you plot [tex]\( L \)[/tex] against [tex]\( T^2 \)[/tex], observe the relationship between the two variables. In the context of a simple pendulum, the period [tex]\( T \)[/tex] is related to the length [tex]\( L \)[/tex] of the pendulum by the equation:
[tex]\[ T = 2\pi\sqrt{\frac{L}{g}} \][/tex]
where [tex]\( g \)[/tex] is the acceleration due to gravity. This equation implies a quadratic relationship [tex]\( L \propto T^{2} \)[/tex].
Given this relationship, the graph of [tex]\( L \)[/tex] against [tex]\( T^2 \)[/tex] should be a straight line. This suggests that as [tex]\( T^2 \)[/tex] increases, [tex]\( L \)[/tex] increases linearly, confirming a direct proportional relationship.
### Part (c): Sources of Error
When conducting experiments involving measurements such as time and length, there are common sources of error. Two possible sources of error in this experiment are:
1. Human Reaction Time: There can be inaccuracies in timing measurements because the time intervals are being recorded manually. When starting and stopping a stopwatch, human reaction delay can cause inconsistencies.
2. Length Measurement Accuracy: The lengths could have been measured with a ruler or a measuring tape. Small errors in alignment or reading off the scale can introduce inaccuracies.
In summary:
1. Plot a graph of [tex]\( L \)[/tex] against [tex]\( T^2 \)[/tex] as described.
2. Shape of the graph: The graph should be a straight line, indicating a linear relationship.
3. Sources of error:
- Human reaction time in timing measurements.
- Inaccuracies in length measurements.
Feel free to sketch this graph on graph paper or use graphing software to visualize the data accurately!
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