1.A brass wire 80 cm long and 0.3 cm in diameter is used to hang an object weighing 6.5 x 10 ^5 dynes. Find stress, strain and young's modules.
2. To one end of a wire 1.2 m long and 0.4 cm in diameter is hung on object with a mass of 10 kg. The wire is found to be elongate 1.34 x 10^4m. Calculate stress, strain and young's modules.
Stress
= force/area
1.
stres= 6.5*10^5 dynes / (π(0.3)²/4 cm²)
= 9195619 dynes/cm²
= 919.6 kPa
Young's modulus nor strain cannot be derived from the given information.
2.
Stress = force /area
10 kg * 9.8 m/s² ÷ (π0.004²/4 m²)
= 7.80 mPa
strain
=Elongation / original length
=1.34*10^(-4) m / 1.2 m (check typo)
=1.12 * 10^(-4)
Young's modulus
= stress/strain
= 7.8 *10^6 / (1.12*10^(-4))
= 6.96*10^10 Pa
= 69.6 GPa (corresponds to that of aluminium)
To find the stress, strain, and Young's modulus in both of these scenarios, we'll use the following formulas:
Stress (σ) = Force (F) / Cross-sectional area (A)
Strain (ε) = Change in length (ΔL) / Original length (L)
Young's modulus (E) = Stress (σ) / Strain (ε)
Now let's find the answers to the given questions step by step.
For question 1:
Given:
Length (L) = 80 cm = 0.8 m
Diameter (d) = 0.3 cm = 0.003 m
Weight of the object (W) = 6.5 × 10^5 dynes
1. First, calculate the cross-sectional area (A) of the wire.
The diameter (d) is given, so we can find the radius (r) by dividing it by 2:
Radius (r) = d/2 = 0.003 m/2 = 0.0015 m
Now, use the formula for the area of a circle:
A = πr^2 = π(0.0015 m)^2 ≈ 0.00000707 m^2
2. Calculate the stress (σ) by dividing the force (W) by the cross-sectional area (A):
σ = W / A = 6.5 × 10^5 dynes / 0.00000707 m^2 = 9.20 × 10^10 N/m^2 (or Pascal)
3. To find the strain (ε), we need to know the change in length (ΔL). If it is not given, we assume there is no change, i.e., ΔL = 0.
In that case, the strain (ε) will become 0, as well.
4. Calculate Young's modulus (E) by dividing stress (σ) by strain (ε):
E = σ / ε = 9.20 × 10^10 N/m^2 / 0 = undefined
Therefore, the stress is 9.20 × 10^10 N/m^2, the strain is 0, and Young's modulus is undefined for question 1.
For question 2:
Given:
Length (L) = 1.2 m
Diameter (d) = 0.4 cm = 0.004 m
Change in length (ΔL) = 1.34 × 10^4 m
Mass of the object (m) = 10 kg
1. Calculate the cross-sectional area (A) of the wire, similar to the previous calculation:
d/2 = 0.004 m/2 = 0.002 m
A = π(0.002 m)^2 ≈ 0.00001257 m^2
2. Calculate the force (F) acting on the wire using the weight formula:
Force (F) = mass (m) × acceleration due to gravity (g) = 10 kg × 9.8 m/s^2 = 98 N
3. Calculate the stress (σ) by dividing the force (F) by the cross-sectional area (A):
σ = F / A = 98 N / 0.00001257 m^2 ≈ 7.80 × 10^6 N/m^2 (or Pascal)
4. Calculate the strain (ε) by dividing the change in length (ΔL) by the original length (L):
ε = ΔL / L = 1.34 × 10^4 m / 1.2 m ≈ 1.12 × 10^4
5. Calculate Young's modulus (E) by dividing the stress (σ) by the strain (ε):
E = σ / ε = 7.80 × 10^6 N/m^2 / 1.12 × 10^4 = 6.96 × 10^2 N/m^2 (or Pascal)
Therefore, the stress is 7.80 × 10^6 N/m^2, the strain is 1.12 × 10^4, and Young's modulus is 6.96 × 10^2 N/m^2 (or Pascal) for question 2.