A chandelier of mass 250 kg is hung vertically from a ceiling using a 1.25 m long brass rod of square cross-section of sides 0.4 cm each. How much will the rod stretch when the chandelier is hung? By what factor is the minimum breaking load greater than the weight of the chandelier?
The cross sectional area is
A = 0.16 cm^2 = 1.6*10^-5 m^2
The stress in the road is
sigma = M*g/A = 1.53*10^8 N/m^2
The amount of stretch is
delta L = L*sigma/E
where E is Young's modular for brass.
The breaking load is
P = A* sigmau
where sigmay is the yield stress for brass.
You will have to look up the modulus and yield stress values. They want you to compute the factor P/(M*g)
To find how much the rod will stretch when the chandelier is hung, we need to consider the stress and strain in the rod.
1) The stress (σ) in the rod can be calculated using the formula:
σ = F / A
where F is the force applied (weight of the chandelier) and A is the cross-sectional area of the rod.
2) The strain (ε) in the rod can be calculated using the formula:
ε = ΔL / L
where ΔL is the change in length of the rod and L is the original length of the rod.
Let's calculate these values step by step:
Step 1: Calculate the force applied (weight of the chandelier):
F = m * g
where m is the mass of the chandelier and g is the acceleration due to gravity (approximately 9.8 m/s^2).
F = 250 kg * 9.8 m/s^2
Step 2: Calculate the cross-sectional area of the rod:
A = w * h
where w is the width of the square cross-section of the rod and h is the height of the square cross-section of the rod.
A = (0.4 cm)^2
Step 3: Calculate the stress in the rod:
σ = F / A
Step 4: Calculate the strain in the rod:
ε = ΔL / L
Step 5: Calculate the change in length (ΔL) of the rod:
ΔL = ε * L
Finally, calculate the minimum breaking load (F_b) by multiplying the weight of the chandelier and the factor (k):
F_b = k * F
To find the factor by which the minimum breaking load (F_b) is greater than the weight of the chandelier, divide F_b by F:
Factor = F_b / F
Let's calculate the values using the given information:
Step 1: Calculate the force applied (weight of the chandelier):
F = 250 kg * 9.8 m/s^2 = 2450 N
Step 2: Calculate the cross-sectional area of the rod:
A = (0.4 cm)^2 = (0.004 m)^2 = 1.6 x 10^-5 m^2
Step 3: Calculate the stress in the rod:
σ = F / A
Step 4: Calculate the strain in the rod:
ε = ΔL / L
Step 5: Calculate the change in length (ΔL) of the rod:
ΔL = ε * L
Finally, calculate the minimum breaking load (F_b) and the factor:
F_b = k * F
Factor = F_b / F
By following these steps, you can calculate the stretch of the rod when the chandelier is hung and find the factor by which the minimum breaking load is greater than the weight of the chandelier.