A steel rod of length 60 cm and radius 2.5 mm hangs vertically. If a mass of 2.5 kg
is attached to the end of the rod, calculate the increase in the length of the rod
No, I am not going to Google Young's modulus of steel for you, E
L = 0.6 m
r = 0.0025 m
A = pi (.0025)^2
delta L/L = 2.5(9.81)/(E A)
delta L = 0.6*2.5*9.81/(E*pi*.0025^2)
To calculate the increase in the length of the rod due to the weight attached at the end, we can use the concept of strain and the Young's modulus of steel.
Step 1: Calculate the original cross-sectional area of the rod.
The radius of the rod is given as 2.5 mm. The cross-sectional area of a rod is given by the formula:
A = πr^2
where r is the radius and π is a constant approximately equal to 3.14.
Therefore, the cross-sectional area can be calculated as:
A = π(0.0025)^2
A = 0.000019625 m^2
Step 2: Calculate the original volume of the rod.
The volume of a cylinder (rod) is given by the formula:
V = Ah
where A is the cross-sectional area and h is the height or length of the rod.
Therefore, the original volume can be calculated as:
V = 0.000019625 m^2 * 0.6 m
V = 0.000011775 m^3
Step 3: Calculate the original stress on the rod.
Stress (σ) is defined as the force (F) applied per unit area (A). The stress formula is given by:
σ = F / A
where F is the force applied (mass * gravitational acceleration) and A is the cross-sectional area.
The gravitational acceleration (g) is approximately 9.8 m/s^2.
Therefore, the force applied (F) can be calculated as:
F = mass * g
F = 2.5 kg * 9.8 m/s^2
F = 24.5 N
Therefore, the original stress can be calculated as:
σ = 24.5 N / 0.000019625 m^2
σ = 1,248,101.27 Pa
Step 4: Calculate the Young's modulus of steel.
The Young's modulus (E) is a measure of the stiffness or elasticity of a material.
For steel, the Young's modulus is approximately 200 GPa (Gigapascals), which is equivalent to 200 × 10^9 Pa.
Step 5: Calculate the increase in length of the rod.
The increase in length (ΔL) can be calculated using Hooke's Law, which states that the strain (ε) is proportional to the stress (σ) and the Young's modulus (E). The formula for strain is given as:
ε = σ / E
Therefore, the increase in length (ΔL) can be calculated as:
ΔL = ε * L
where ε is the strain, L is the original length of the rod.
The strain can be calculated as:
ε = σ / E
ε = 1,248,101.27 Pa / (200 × 10^9 Pa)
ε = 0.000006240507 m/m
Therefore, the increase in length can be calculated as:
ΔL = 0.000006240507 m/m * 0.6 m
ΔL = 0.000003744304 m or 0.37 mm
Therefore, the increase in length of the rod due to the weight attached at the end is approximately 0.37 mm.