A mild steel rod is 150 cm long. The portion 80 cm long is 22 mm diameter and the remaining length is 20 mm diameter. If the total change in length is 0.072 cm, Determine the changes of each part in 1) Length, 2) Diameter, 3) Volume. Take poissions ratio= 0.3
Consider the length change as a result of stress.
Stress x area is constant along the length.
Stress is (22/20)^2 times higher (or 21% higher) in the narrower part. The ratio of length change in the 22 and 20 mm portions is
80*1 :: 70*1.21 or 80::84.7.
A). 48.6% of the length change (0.035 cm) is in the 22 mm wide portion. 51.4% of the length change (0.037 cm) is in the 20 mm wide portion.
B) The relative diameter changes are 30% of the
relative length change.
C) Let relative length change = e
Relative volume change = (1+e)*(1-0.3e)^2 -1
To determine the changes in length, diameter, and volume of each part of the mild steel rod, we can follow these steps:
1) Calculate the change in length for each part:
- We are given that the total change in length is 0.072 cm.
- The portion of the rod that is 80 cm long will experience a change in length that is proportional to its original length. So, the change in length for this portion can be calculated as:
Change in length of 80 cm part = (80 cm / 150 cm) * 0.072 cm
- Similarly, the remaining portion of the rod that is 70 cm long will experience a change in length that is proportional to its original length. So, the change in length for this portion can be calculated as:
Change in length of 70 cm part = (70 cm / 150 cm) * 0.072 cm
2) Calculate the change in diameter for each part:
- The change in diameter is related to the change in length through the Poisson's ratio. Poisson's ratio relates the axial strain to the transverse strain for a material. In this case, we are assuming a Poisson's ratio of 0.3.
- The change in diameter for the 22 mm diameter portion (80 cm length) can be calculated as:
Change in diameter of 80 cm part = (Change in length of 80 cm part) * Poisson's ratio * (original diameter / original length)
- Similarly, the change in diameter for the 20 mm diameter portion (70 cm length) can be calculated as:
Change in diameter of 70 cm part = (Change in length of 70 cm part) * Poisson's ratio * (original diameter / original length)
3) Calculate the change in volume for each part:
- The change in volume can be calculated from the change in diameter using the formula for the volume of a cylinder: π * (diameter/2)^2 * length.
- The change in volume for each part can be calculated by subtracting the original volume from the final volume.
Now, let's calculate the values:
Given:
Total change in length = 0.072 cm
Poisson's ratio = 0.3
Part 1 - 80 cm long, 22 mm diameter:
Original length = 80 cm
Original diameter = 22 mm
Part 2 - 70 cm long, 20 mm diameter:
Original length = 70 cm
Original diameter = 20 mm
Step 1: Calculate the change in length for each part:
Change in length of 80 cm part = (80 cm / 150 cm) * 0.072 cm = 0.0384 cm
Change in length of 70 cm part = (70 cm / 150 cm) * 0.072 cm = 0.0336 cm
Step 2: Calculate the change in diameter for each part:
Change in diameter of 80 cm part = (Change in length of 80 cm part) * Poisson's ratio * (original diameter / original length)
Change in diameter of 80 cm part = 0.0384 cm * 0.3 * (22 mm / 80 cm) = 0.00336 cm
Change in diameter of 70 cm part = (Change in length of 70 cm part) * Poisson's ratio * (original diameter / original length)
Change in diameter of 70 cm part = 0.0336 cm * 0.3 * (20 mm / 70 cm) = 0.0024 cm
Step 3: Calculate the change in volume for each part:
Original volume of 80 cm part = π * (22 mm / 2)^2 * 80 cm = 10648.78 mm³
Final volume of 80 cm part = π * (22 mm + 2 * (Change in diameter of 80 cm part)) / 2)^2 * (80 cm + Change in length of 80 cm part) = 10732.73 mm³
Change in volume of 80 cm part = Final volume of 80 cm part - Original volume of 80 cm part = 83.95 mm³
Original volume of 70 cm part = π * (20 mm / 2)^2 * 70 cm = 6283.19 mm³
Final volume of 70 cm part = π * (20 mm + 2 * (Change in diameter of 70 cm part)) / 2)^2 * (70 cm + Change in length of 70 cm part) = 6300.14 mm³
Change in volume of 70 cm part = Final volume of 70 cm part - Original volume of 70 cm part = 16.95 mm³
Therefore, the changes in each part are:
1) Length:
- The 80 cm part will increase by 0.0384 cm.
- The 70 cm part will increase by 0.0336 cm.
2) Diameter:
- The 22 mm diameter portion will increase by 0.00336 cm.
- The 20 mm diameter portion will increase by 0.0024 cm.
3) Volume:
- The 22 mm diameter portion will increase by 83.95 mm³.
- The 20 mm diameter portion will increase by 16.95 mm³.