When wire is stretched to double of its length?
2 Answers
When a wire is stretched to double its original length, several important changes occur in its physical properties. Let's explore these effects in detail, focusing on length, area, volume, resistance, and how forces and energy are involved.
### 1. **Effect on Length:**
- The original length of the wire is \( L \).
- If the wire is stretched to double its length, the new length \( L' \) will be:
\[
L' = 2L
\]
### 2. **Effect on Cross-sectional Area:**
- As the wire is stretched, the material's volume remains constant (assuming the wire is made of a material that cannot be compressed or expanded significantly, like most metals).
- Volume of the wire, \( V \), before stretching is:
\[
V = A \times L
\]
where \( A \) is the cross-sectional area of the wire.
- After stretching, the volume of the wire remains the same, but the length increases. Since the volume remains constant, the cross-sectional area must decrease.
- Let the new cross-sectional area be \( A' \), and the new length be \( 2L \). The new volume is:
\[
V = A' \times 2L
\]
Since the volume remains constant, we have:
\[
A \times L = A' \times 2L
\]
Solving for \( A' \):
\[
A' = \frac{A}{2}
\]
So, the cross-sectional area of the wire is halved.
### 3. **Effect on Volume:**
- As mentioned earlier, the volume remains constant. This is because stretching a wire doesn’t add or remove any material, so the volume before and after stretching is the same:
\[
V_{\text{initial}} = V_{\text{final}}
\]
### 4. **Effect on Resistance:**
- The electrical resistance \( R \) of a wire is given by the formula:
\[
R = \rho \frac{L}{A}
\]
where:
- \( R \) is the resistance,
- \( \rho \) is the resistivity of the material (a constant),
- \( L \) is the length of the wire,
- \( A \) is the cross-sectional area.
- Before stretching, the resistance is:
\[
R_{\text{initial}} = \rho \frac{L}{A}
\]
- After stretching, the new resistance becomes:
\[
R_{\text{final}} = \rho \frac{2L}{\frac{A}{2}} = \rho \frac{2L \times 2}{A} = 4 \times R_{\text{initial}}
\]
- So, when the wire is stretched to double its original length, its resistance increases by a factor of 4.
### 5. **Effect on Mechanical Stress and Strain:**
- **Strain** is the measure of deformation or elongation in the wire and is defined as:
\[
\text{Strain} = \frac{\Delta L}{L}
\]
In this case, \( \Delta L = L \), so:
\[
\text{Strain} = \frac{L}{L} = 1
\]
This means the strain in the wire is 1 (or 100%), indicating the wire has been stretched by an amount equal to its original length.
- **Stress** is defined as the force applied per unit area and is calculated as:
\[
\text{Stress} = \frac{F}{A}
\]
Where \( F \) is the force applied, and \( A \) is the cross-sectional area.
Since the cross-sectional area \( A' \) has halved after stretching, the stress increases. If the same force is applied to the stretched wire, the stress doubles because the area is halved.
### 6. **Effect on Young's Modulus:**
- **Young's modulus (Y)** is a measure of the stiffness of a material, and it relates stress to strain by the formula:
\[
Y = \frac{\text{Stress}}{\text{Strain}}
\]
- Since the strain has increased and the area has decreased, the stress will increase. However, Young's modulus is a material constant and does not change because it only depends on the material itself, not on the dimensions.
### 7. **Energy Considerations:**
- When a wire is stretched, energy is stored in it as **elastic potential energy**. The work done to stretch the wire is stored as this potential energy.
- The amount of elastic potential energy stored in the wire is:
\[
U = \frac{1}{2} \times \text{Stress} \times \text{Strain} \times V
\]
As the wire is stretched, the strain and stress both increase, meaning more energy is stored in the wire.
### Summary of Changes When Wire is Stretched:
- **Length:** Doubles (\( L' = 2L \)).
- **Cross-sectional area:** Halves (\( A' = \frac{A}{2} \)).
- **Volume:** Remains constant.
- **Resistance:** Increases by a factor of 4.
- **Stress:** Increases due to the reduced cross-sectional area.
- **Strain:** Becomes 100% (since it’s stretched to double its length).
- **Young's Modulus:** Remains constant, as it is a material property.
- **Energy:** Elastic potential energy is stored due to the deformation.
In conclusion, stretching a wire to double its length has significant effects on its physical properties, especially on resistance and stress, while keeping properties like volume constant.
### 1. **Effect on Length:**
- The original length of the wire is \( L \).
- If the wire is stretched to double its length, the new length \( L' \) will be:
\[
L' = 2L
\]
### 2. **Effect on Cross-sectional Area:**
- As the wire is stretched, the material's volume remains constant (assuming the wire is made of a material that cannot be compressed or expanded significantly, like most metals).
- Volume of the wire, \( V \), before stretching is:
\[
V = A \times L
\]
where \( A \) is the cross-sectional area of the wire.
- After stretching, the volume of the wire remains the same, but the length increases. Since the volume remains constant, the cross-sectional area must decrease.
- Let the new cross-sectional area be \( A' \), and the new length be \( 2L \). The new volume is:
\[
V = A' \times 2L
\]
Since the volume remains constant, we have:
\[
A \times L = A' \times 2L
\]
Solving for \( A' \):
\[
A' = \frac{A}{2}
\]
So, the cross-sectional area of the wire is halved.
### 3. **Effect on Volume:**
- As mentioned earlier, the volume remains constant. This is because stretching a wire doesn’t add or remove any material, so the volume before and after stretching is the same:
\[
V_{\text{initial}} = V_{\text{final}}
\]
### 4. **Effect on Resistance:**
- The electrical resistance \( R \) of a wire is given by the formula:
\[
R = \rho \frac{L}{A}
\]
where:
- \( R \) is the resistance,
- \( \rho \) is the resistivity of the material (a constant),
- \( L \) is the length of the wire,
- \( A \) is the cross-sectional area.
- Before stretching, the resistance is:
\[
R_{\text{initial}} = \rho \frac{L}{A}
\]
- After stretching, the new resistance becomes:
\[
R_{\text{final}} = \rho \frac{2L}{\frac{A}{2}} = \rho \frac{2L \times 2}{A} = 4 \times R_{\text{initial}}
\]
- So, when the wire is stretched to double its original length, its resistance increases by a factor of 4.
### 5. **Effect on Mechanical Stress and Strain:**
- **Strain** is the measure of deformation or elongation in the wire and is defined as:
\[
\text{Strain} = \frac{\Delta L}{L}
\]
In this case, \( \Delta L = L \), so:
\[
\text{Strain} = \frac{L}{L} = 1
\]
This means the strain in the wire is 1 (or 100%), indicating the wire has been stretched by an amount equal to its original length.
- **Stress** is defined as the force applied per unit area and is calculated as:
\[
\text{Stress} = \frac{F}{A}
\]
Where \( F \) is the force applied, and \( A \) is the cross-sectional area.
Since the cross-sectional area \( A' \) has halved after stretching, the stress increases. If the same force is applied to the stretched wire, the stress doubles because the area is halved.
### 6. **Effect on Young's Modulus:**
- **Young's modulus (Y)** is a measure of the stiffness of a material, and it relates stress to strain by the formula:
\[
Y = \frac{\text{Stress}}{\text{Strain}}
\]
- Since the strain has increased and the area has decreased, the stress will increase. However, Young's modulus is a material constant and does not change because it only depends on the material itself, not on the dimensions.
### 7. **Energy Considerations:**
- When a wire is stretched, energy is stored in it as **elastic potential energy**. The work done to stretch the wire is stored as this potential energy.
- The amount of elastic potential energy stored in the wire is:
\[
U = \frac{1}{2} \times \text{Stress} \times \text{Strain} \times V
\]
As the wire is stretched, the strain and stress both increase, meaning more energy is stored in the wire.
### Summary of Changes When Wire is Stretched:
- **Length:** Doubles (\( L' = 2L \)).
- **Cross-sectional area:** Halves (\( A' = \frac{A}{2} \)).
- **Volume:** Remains constant.
- **Resistance:** Increases by a factor of 4.
- **Stress:** Increases due to the reduced cross-sectional area.
- **Strain:** Becomes 100% (since it’s stretched to double its length).
- **Young's Modulus:** Remains constant, as it is a material property.
- **Energy:** Elastic potential energy is stored due to the deformation.
In conclusion, stretching a wire to double its length has significant effects on its physical properties, especially on resistance and stress, while keeping properties like volume constant.
When a wire is stretched to double its original length, several important changes occur in its properties:
1. **Cross-Sectional Area**: If the wire is assumed to be stretched uniformly and the volume of the wire remains constant, the cross-sectional area will decrease. This is because the volume of the wire before and after stretching should be the same. For a wire of initial length \( L \) and initial cross-sectional area \( A \), the volume is \( L \times A \). When stretched to double its length (\( 2L \)), the new cross-sectional area \( A' \) can be found from:
\[
\text{Initial Volume} = \text{Final Volume}
\]
\[
L \times A = 2L \times A'
\]
\[
A' = \frac{A}{2}
\]
Therefore, the new cross-sectional area is half of the original area.
2. **Young’s Modulus**: Young's modulus (\( E \)) of the wire remains unchanged because it is a material property that depends on the material of the wire, not on its dimensions. Young's modulus is defined as:
\[
E = \frac{\text{Stress}}{\text{Strain}}
\]
where Stress = \(\frac{F}{A}\) and Strain = \(\frac{\Delta L}{L}\).
3. **Resistance of the Wire**: The electrical resistance \( R \) of a wire is given by:
\[
R = \rho \frac{L}{A}
\]
where \( \rho \) is the resistivity of the material, \( L \) is the length, and \( A \) is the cross-sectional area. When the length of the wire is doubled (\( 2L \)) and the cross-sectional area is halved (\( \frac{A}{2} \)), the new resistance \( R' \) can be calculated as:
\[
R' = \rho \frac{2L}{\frac{A}{2}} = 4 \rho \frac{L}{A} = 4R
\]
Hence, the resistance of the wire becomes four times its original resistance.
In summary, when a wire is stretched to double its length:
- The cross-sectional area is halved.
- The Young's modulus remains the same.
- The resistance of the wire becomes four times its original value.
1. **Cross-Sectional Area**: If the wire is assumed to be stretched uniformly and the volume of the wire remains constant, the cross-sectional area will decrease. This is because the volume of the wire before and after stretching should be the same. For a wire of initial length \( L \) and initial cross-sectional area \( A \), the volume is \( L \times A \). When stretched to double its length (\( 2L \)), the new cross-sectional area \( A' \) can be found from:
\[
\text{Initial Volume} = \text{Final Volume}
\]
\[
L \times A = 2L \times A'
\]
\[
A' = \frac{A}{2}
\]
Therefore, the new cross-sectional area is half of the original area.
2. **Young’s Modulus**: Young's modulus (\( E \)) of the wire remains unchanged because it is a material property that depends on the material of the wire, not on its dimensions. Young's modulus is defined as:
\[
E = \frac{\text{Stress}}{\text{Strain}}
\]
where Stress = \(\frac{F}{A}\) and Strain = \(\frac{\Delta L}{L}\).
3. **Resistance of the Wire**: The electrical resistance \( R \) of a wire is given by:
\[
R = \rho \frac{L}{A}
\]
where \( \rho \) is the resistivity of the material, \( L \) is the length, and \( A \) is the cross-sectional area. When the length of the wire is doubled (\( 2L \)) and the cross-sectional area is halved (\( \frac{A}{2} \)), the new resistance \( R' \) can be calculated as:
\[
R' = \rho \frac{2L}{\frac{A}{2}} = 4 \rho \frac{L}{A} = 4R
\]
Hence, the resistance of the wire becomes four times its original resistance.
In summary, when a wire is stretched to double its length:
- The cross-sectional area is halved.
- The Young's modulus remains the same.
- The resistance of the wire becomes four times its original value.