A 30 m tape actually measures 29.985m when it is supported at two ends only ,the temperature at 30C and tension of 70N.the tape weighs 12N and has a cross sectional area of 0.04cm^2 .if the field temperature is also 30C what tension should be applied so that tape measures exactly 30m when supported at two ends?

Question

Answer

Given Information:
- Length of the tape when supported at two ends: 29.985m
- Temperature: 30°C
- Tension: 70N
- Weight of the tape: 12N
- Cross-sectional area: 0.04cm²

Objective:
To determine the tension required for the tape to measure exactly 30m when supported at two ends, assuming a field temperature of 30°C.

Solution:

Step 1: Calculate the change in length due to temperature:
The change in length of the tape due to temperature can be calculated using the formula:

ΔL = αL₀ΔT

Where:
ΔL = Change in length
α = Coefficient of linear expansion (depends on the material)
L₀ = Initial length
ΔT = Change in temperature

Since the tape is made of a specific material, we can assume a value for the coefficient of linear expansion. Let's assume α = 1.2 x 10⁻⁵ (°C)⁻¹.

Using the given values:
L₀ = 29.985m
ΔT = 30°C - 30°C = 0°C

ΔL = (1.2 x 10⁻⁵ (°C)⁻¹)(29.985m)(0°C)
ΔL = 0m

Step 2: Calculate the change in length due to tension:
The change in length of the tape due to tension can be calculated using the formula:

ΔL = (F/A)(L₀/E)

Where:
ΔL = Change in length
F = Tension
A = Cross-sectional area
L₀ = Initial length
E = Young's modulus (depends on the material)

Since the tape is made of a specific material, we can assume a value for Young's modulus. Let's assume E = 2 x 10¹¹ N/m².

Using the given values:
F = 70N
A = 0.04cm² = 0.04 x 10⁻⁴ m²
L₀ = 29.985m
E = 2 x 10¹¹ N/m²

ΔL = (70N)/(0.04 x 10⁻⁴ m²)(29.985m)/(2 x 10¹¹ N/m²)
ΔL ≈ 0.074m

Step 3: Calculate the tension required:
To determine the tension required for the tape to measure exactly 30m when supported at two ends, we need to find the tension that compensates for the changes in length due to temperature and tension.

The total change in length is the sum of the changes due to temperature and tension:

Total ΔL = ΔL (due to temperature) + ΔL (due to tension)

Since we want the tape to measure exactly 30m, the total change in length should be zero:

Total ΔL = 0

Therefore, the tension required to balance the changes in length can be calculated as:

Tension required = -ΔL (due to temperature)/(ΔL (due to tension)) x Tension

Using the calculated values:
Tension required = -(0m)/(0.074m) x

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