Question
mnol=50g
xnol=19cm
M=(m+mnol) plus or minus 0.05cm 250, 450, 650,850,1050
loading =25,30,35,40,45,45
unloading =25,30,35,40,45
Mean,pointer (x plus or minus 0.05cm)= 25,30,35,40,45
Extension e=(x-xnol) plus or minus 0.05cm 6cm , 11cm, 16cm, 2cm, 26cm
1) plot a graph of M against e
2) measure the slope and explain its physical meaning. state its unit and estimate the error
3)did your graph pass through the origin? if not, give reasons
4) deduce from your load ( mass) extension, graph the weight Wa of the given object, A .
5) estimate the error in Wa
xnol=19cm
M=(m+mnol) plus or minus 0.05cm 250, 450, 650,850,1050
loading =25,30,35,40,45,45
unloading =25,30,35,40,45
Mean,pointer (x plus or minus 0.05cm)= 25,30,35,40,45
Extension e=(x-xnol) plus or minus 0.05cm 6cm , 11cm, 16cm, 2cm, 26cm
1) plot a graph of M against e
2) measure the slope and explain its physical meaning. state its unit and estimate the error
3)did your graph pass through the origin? if not, give reasons
4) deduce from your load ( mass) extension, graph the weight Wa of the given object, A .
5) estimate the error in Wa
Answers
GPT-4o mini
Let's break down the tasks you've provided step by step.
### 1) Plot a graph of M against e
To plot this graph, you will use the values of M (the total mass, calculated as \(M = m + m_{nol}\)) and the extension \(e\). The provided values for M and e are:
- For M: 250g, 450g, 650g, 850g, 1050g
- For e: 6cm, 11cm, 16cm, 21cm, 26cm
Plot M (on the y-axis) against e (on the x-axis). Ensure to properly label the axes and plot the corresponding points to visualize your data.
### 2) Measure the slope and explain its physical meaning. State its unit and estimate the error.
To find the slope \(S\) of the graph:
1. Calculate the change in mass (\(ΔM\)) and change in extension (\(Δe\)) between two points. For instance, you can take the first and last points for maximum change:
\[
ΔM = M_{final} - M_{initial} = 1050g - 250g = 800g
\]
\[
Δe = e_{final} - e_{initial} = 26cm - 6cm = 20cm
\]
2. Slope (S) can be calculated as:
\[
S = \frac{ΔM}{Δe} = \frac{800g}{20cm} = 40 \frac{g}{cm}
\]
**Physical Meaning**: The slope represents the stiffness or spring constant (k) of the material. It shows how much mass needs to be added (in grams) to produce a unit extension (in cm).
**Units**: The unit of slope is \(\frac{g}{cm}\).
**Estimate of the Error**: If you've measured M and e with uncertainties (±0.05 cm), propagate these uncertainties to calculate the error for the slope using the following propagation formula:
\[
\text{Error in S} = S \cdot \sqrt{\left(\frac{\text{Error in } M}{ΔM}\right)^2 + \left(\frac{\text{Error in } e}{Δe}\right)^2}
\]
Assuming your errors in M and e are negligible in comparison to the values, this gives a good base for your error estimation.
### 3) Did your graph pass through the origin? If not, give reasons.
If the graph does not pass through the origin, it might indicate a fixed length (the initial extension) when no mass is applied (i.e., when \(M = 0\)). Factors that could prevent the graph from passing through the origin include:
- There may be inherent elasticity in the material leading to a non-zero extension before any load.
- Measurement errors might have shifted the graph away from the origin.
### 4) Deduce from your load (mass) extension graph the weight \(W_a\) of the given object A.
The weight \(W_a\) can be calculated by multiplying the total mass \(M\) by the acceleration due to gravity \(g\) (approximately \(9.81 m/s^2\)).
Taking the last mass (M = 1050g):
\[
W_a = M \cdot g = 1050g \cdot \frac{9.81m/s^2}{1000g} = 10.3N
\]
### 5) Estimate the error in \(W_a\)
To estimate the error in \(W_a\):
1. Use the mass M with its error propagation.
2. The weight's error can be calculated using:
\[
\text{Error in } W_a = W_a \cdot \frac{\text{Error in } M}{M}
\]
Assuming the error in mass is ±0.05g:
\[
\text{Error in } W_a \approx 10.3N \cdot \frac{0.05g}{1050g} \approx 0.00049N
\]
So, the total weight with uncertainty would be \(W_a = 10.3 \pm 0.00049 N\).
### Summary
- Plot your graph and carry out the calculations.
- Use proper labeling, and remember that each of the calculations, especially slope and errors, can depend on many factors such as height of the graph paper measurement and understanding of measurement errors. Adjust accordingly based on actual experimental observations.
### 1) Plot a graph of M against e
To plot this graph, you will use the values of M (the total mass, calculated as \(M = m + m_{nol}\)) and the extension \(e\). The provided values for M and e are:
- For M: 250g, 450g, 650g, 850g, 1050g
- For e: 6cm, 11cm, 16cm, 21cm, 26cm
Plot M (on the y-axis) against e (on the x-axis). Ensure to properly label the axes and plot the corresponding points to visualize your data.
### 2) Measure the slope and explain its physical meaning. State its unit and estimate the error.
To find the slope \(S\) of the graph:
1. Calculate the change in mass (\(ΔM\)) and change in extension (\(Δe\)) between two points. For instance, you can take the first and last points for maximum change:
\[
ΔM = M_{final} - M_{initial} = 1050g - 250g = 800g
\]
\[
Δe = e_{final} - e_{initial} = 26cm - 6cm = 20cm
\]
2. Slope (S) can be calculated as:
\[
S = \frac{ΔM}{Δe} = \frac{800g}{20cm} = 40 \frac{g}{cm}
\]
**Physical Meaning**: The slope represents the stiffness or spring constant (k) of the material. It shows how much mass needs to be added (in grams) to produce a unit extension (in cm).
**Units**: The unit of slope is \(\frac{g}{cm}\).
**Estimate of the Error**: If you've measured M and e with uncertainties (±0.05 cm), propagate these uncertainties to calculate the error for the slope using the following propagation formula:
\[
\text{Error in S} = S \cdot \sqrt{\left(\frac{\text{Error in } M}{ΔM}\right)^2 + \left(\frac{\text{Error in } e}{Δe}\right)^2}
\]
Assuming your errors in M and e are negligible in comparison to the values, this gives a good base for your error estimation.
### 3) Did your graph pass through the origin? If not, give reasons.
If the graph does not pass through the origin, it might indicate a fixed length (the initial extension) when no mass is applied (i.e., when \(M = 0\)). Factors that could prevent the graph from passing through the origin include:
- There may be inherent elasticity in the material leading to a non-zero extension before any load.
- Measurement errors might have shifted the graph away from the origin.
### 4) Deduce from your load (mass) extension graph the weight \(W_a\) of the given object A.
The weight \(W_a\) can be calculated by multiplying the total mass \(M\) by the acceleration due to gravity \(g\) (approximately \(9.81 m/s^2\)).
Taking the last mass (M = 1050g):
\[
W_a = M \cdot g = 1050g \cdot \frac{9.81m/s^2}{1000g} = 10.3N
\]
### 5) Estimate the error in \(W_a\)
To estimate the error in \(W_a\):
1. Use the mass M with its error propagation.
2. The weight's error can be calculated using:
\[
\text{Error in } W_a = W_a \cdot \frac{\text{Error in } M}{M}
\]
Assuming the error in mass is ±0.05g:
\[
\text{Error in } W_a \approx 10.3N \cdot \frac{0.05g}{1050g} \approx 0.00049N
\]
So, the total weight with uncertainty would be \(W_a = 10.3 \pm 0.00049 N\).
### Summary
- Plot your graph and carry out the calculations.
- Use proper labeling, and remember that each of the calculations, especially slope and errors, can depend on many factors such as height of the graph paper measurement and understanding of measurement errors. Adjust accordingly based on actual experimental observations.