Final answer:
The diameter of the hex bar frame needs to be approximately 28 mm to prevent it from stretching more than 1.0 cm when a 2.5 × 10^4-kg truck passes by it.
Step-by-step explanation:
To determine the diameter of the hex bar frame, we need to consider the weight of the truck and the maximum allowed stretching distance. Since the rod supports all of the weight of the truck, we can use Hooke's Law to find the diameter.
Hooke's Law states that the force applied to an object is directly proportional to its extension if the material obeys Hook's law within its elastic limit, which is the case for steel.
The formula to calculate the extension is given by:
ΔL = (F * L) / (A * E)
Where ΔL is the extension, F is the force applied, L is the length of the rod, A is the cross-sectional area of the rod, and E is the elastic modulus of the material. We can rearrange this formula to find the diameter:
d = ((F * L) / (ΔL * π * E))^(1/2)
Substituting the given values:
d = ((2.5 × 10^4 * 9.8 * 1.5) / (1 * 10^-2 * π * 2.06 × 10^11))^(1/2)
d ≈ 0.028 meters = 28 mm