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
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
Answers
GPT-4o mini
To address your questions, I will guide you step by step through the process of analyzing the data and answering your questions based on the information you provided.
### 1) Plot a Graph of M Against e
To create a graph of mass (M) against extension (e):
- **Mass (M):**
- For 0g (mnol), M = mnol + 0g
- For 250g, M = mnol + 250g = 300g
- For 450g, M = mnol + 450g = 500g
- For 650g, M = mnol + 650g = 700g
- For 850g, M = mnol + 850g = 800g
- For 1050g, M = mnol + 1050g = 1100g
- Note: Make sure to convert grams into kilograms if necessary for consistency, but for the purpose of plotting, we will use grams.
- **Extension (e)** corresponds to each load. The extensions provided are:
- For 0g: e = 0cm
- For 250g: e = 6cm
- For 450g: e = 11cm
- For 650g: e = 16cm
- For 850g: e = 2cm
- For 1050g: e = 26cm
When plotting, on the x-axis you would place extension (e) and on the y-axis mass (M).
### 2) Measure the Slope and Explain Its Physical Meaning
The slope of the graph corresponds to the ratio of mass to extension, which can be interpreted in terms of stiffness (k) of the material according to Hooke’s Law:
\[
F = k \cdot e
\]
Where:
- \( F = M \cdot g \) (force due to mass, with \( g \) being the acceleration due to gravity, approximately \( 9.81 m/s^2 \)).
- Therefore, \( k = \frac{M \cdot g}{e} \).
**Units:** The unit of the slope \( (k) \) will be grams per centimeter (g/cm) or Newtons per centimeter (N/cm) if converted to the SI unit.
**Error Estimation:** Estimate the uncertainty in the slope based on the errors in your mass measurements (0.05g) and extension measurements (0.05cm). Use standard error calculations for this.
### 3) Did Your Graph Pass Through the Origin?
If your graph does not pass through the origin (0,0), it may be for a few reasons:
- There may be an initial extension in the spring or material before any load is applied (pre-load).
- Measurement errors, particularly if the ruler or compliance is not properly calibrated.
- Non-linearity in the material's elasticity at lower loads.
### 4) Deduce from Your Load (Mass) Extension Graph the Weight \( WA \) of the Given Object \( A \)
To find the weight \( WA \) of the object \( A \) from the extension graph:
1. Identify the corresponding extension \( e \) for mass \( A \).
2. Use the slope (stiffness) to convert that extension back to force.
\[
W_A = k \cdot e
\]
### 5) Estimate the Error in \( WA \)
The error in \( WA \) can be estimated by propagating uncertainties in the slope and extension:
1. For the slope \( k \) (say \( \Delta k \)) and the extension \( e \) (say \( \Delta e \)):
2. The error in the weight \( WA \) can be computed using:
\[
\Delta WA = WA \cdot \left( \frac{\Delta k}{k} + \frac{\Delta e}{e} \right)
\]
This provides you with a thorough analysis of your experiment and answers your questions based on the data you've provided.
### 1) Plot a Graph of M Against e
To create a graph of mass (M) against extension (e):
- **Mass (M):**
- For 0g (mnol), M = mnol + 0g
- For 250g, M = mnol + 250g = 300g
- For 450g, M = mnol + 450g = 500g
- For 650g, M = mnol + 650g = 700g
- For 850g, M = mnol + 850g = 800g
- For 1050g, M = mnol + 1050g = 1100g
- Note: Make sure to convert grams into kilograms if necessary for consistency, but for the purpose of plotting, we will use grams.
- **Extension (e)** corresponds to each load. The extensions provided are:
- For 0g: e = 0cm
- For 250g: e = 6cm
- For 450g: e = 11cm
- For 650g: e = 16cm
- For 850g: e = 2cm
- For 1050g: e = 26cm
When plotting, on the x-axis you would place extension (e) and on the y-axis mass (M).
### 2) Measure the Slope and Explain Its Physical Meaning
The slope of the graph corresponds to the ratio of mass to extension, which can be interpreted in terms of stiffness (k) of the material according to Hooke’s Law:
\[
F = k \cdot e
\]
Where:
- \( F = M \cdot g \) (force due to mass, with \( g \) being the acceleration due to gravity, approximately \( 9.81 m/s^2 \)).
- Therefore, \( k = \frac{M \cdot g}{e} \).
**Units:** The unit of the slope \( (k) \) will be grams per centimeter (g/cm) or Newtons per centimeter (N/cm) if converted to the SI unit.
**Error Estimation:** Estimate the uncertainty in the slope based on the errors in your mass measurements (0.05g) and extension measurements (0.05cm). Use standard error calculations for this.
### 3) Did Your Graph Pass Through the Origin?
If your graph does not pass through the origin (0,0), it may be for a few reasons:
- There may be an initial extension in the spring or material before any load is applied (pre-load).
- Measurement errors, particularly if the ruler or compliance is not properly calibrated.
- Non-linearity in the material's elasticity at lower loads.
### 4) Deduce from Your Load (Mass) Extension Graph the Weight \( WA \) of the Given Object \( A \)
To find the weight \( WA \) of the object \( A \) from the extension graph:
1. Identify the corresponding extension \( e \) for mass \( A \).
2. Use the slope (stiffness) to convert that extension back to force.
\[
W_A = k \cdot e
\]
### 5) Estimate the Error in \( WA \)
The error in \( WA \) can be estimated by propagating uncertainties in the slope and extension:
1. For the slope \( k \) (say \( \Delta k \)) and the extension \( e \) (say \( \Delta e \)):
2. The error in the weight \( WA \) can be computed using:
\[
\Delta WA = WA \cdot \left( \frac{\Delta k}{k} + \frac{\Delta e}{e} \right)
\]
This provides you with a thorough analysis of your experiment and answers your questions based on the data you've provided.