Question 1195307: A line was found to be 1500ft was laid out using a 50-ft steel tape under a pull of 12lbs at a mean temperature of 20 degree Celsius. Determine the correct length of the line if the tape used is of standard length at 25°C under a pull of 10 lbs. Assume the cross-sectional area of tape to be 0.003 sq.in, 0.0000116 ft/ °C per as its coefficient of thermal expansion, and 30 x 10° psi as its elastic modulus.
Answer by ElectricPavlov(122)   (Show Source):
You can put this solution on YOUR website! ## Determining the Correct Length of the Line
We can solve this problem by considering the following factors that affect the length of the steel tape:
1. **Thermal Expansion:** As temperature increases, the steel tape expands.
2. **Tension:** Pulling the tape stretches it slightly.
We can use the following steps to find the correct length of the line:
**1. Calculate the Change in Length due to Thermal Expansion:**
* **Formula:** ΔL_temp = α * L₀ * ΔT
* ΔL_temp: Change in length due to temperature (°ft)
* α: Coefficient of thermal expansion (°ft/°C)
* L₀: Original length of the tape (ft)
* ΔT: Change in temperature (°C)
* **Values:**
* α = 0.0000116 ft/°C
* L₀ = 50 ft (assuming the tape is standard length)
* ΔT = 25°C - 20°C = 5°C
* **Calculation:**
* ΔL_temp = 0.0000116 ft/°C * 50 ft * 5°C
* ΔL_temp = 0.0029 ft ≈ 0.003 ft (round to 3 decimal places)
**2. Calculate the Change in Length due to Tension:**
* We need additional information about the relationship between tension and length change for this specific steel tape.
**Here, we can make two assumptions:**
**Assumption 1: Linear Relationship**
If the relationship between tension (T) and change in length (ΔL_tension) is linear, we can use the following formula:
* ΔL_tension = k * (T - T₀)
* k: proportionality constant (ft/lb) representing the change in length per unit change in tension
* T: Applied tension (lb)
* T₀: Reference tension (lb)
We don't have the value of k for this specific tape. However, we can acknowledge that a higher tension (12 lbs) will cause a greater change in length compared to the reference tension (assumed to be 10 lbs).
**Assumption 2: Information from Manufacturer**
The manufacturer might provide a table or graph showing the change in length for different tension values. This information can be used to determine ΔL_tension.
**3. Apply Corrections to Find the Correct Length:**
* Since the thermal expansion increases the length and the tension might decrease the length (depending on the tension-length relationship), we can assume a net **decrease** in length.
* **Correct Length (L_correct):**
* L_correct = L₀ - ΔL_temp - ΔL_tension
* **Note:** Due to the lack of information about ΔL_tension, we cannot calculate the exact correct length.
**Conclusion:**
* Based on the available information, we can determine that the steel tape likely experienced a slight **decrease** in length due to thermal expansion (0.003 ft).
* The tension might have caused a further decrease in length, but the magnitude is unknown without additional information.
Therefore, the actual length of the line (L_correct) is likely **shorter** than the measured length (1500 ft) by an unknown amount.
|
|
|
