To solve this problem, we need to analyze the forces acting on the two wires and how they relate to the Young's modulus of wire Y. Given that the wires are under tension due to the weight placed on the rod, we can use the relationship between stress, strain, and Young's modulus to find the unknown Young's modulus E2 for wire Y.
Understanding the Setup
We have two vertical wires, X and Y, connected by a horizontal rod at their lower ends. The weight of 30 N is applied at point O on the rod, which divides the rod into two segments in the ratio XO:OY = 1:2. This means that if we let the length of segment XO be x, then the length of segment OY will be 2x. The total length of the rod is 3x.
Calculating the Forces in Each Wire
Since the rod remains horizontal, the forces in the two wires must balance the weight applied. Let the tension in wire X be T1 and in wire Y be T2. The total weight is given as:
From the ratio of the lengths, we can express the tensions in terms of the Young's moduli and the lengths of the wires:
- For wire X: T1 = E1 * (A/l) * Δl1
- For wire Y: T2 = E2 * (A/l) * Δl2
Relating the Stretches
The stretches Δl1 and Δl2 are related to the lengths of the segments. Since XO:OY = 1:2, we can say:
- Δl1 = (1/3) * Δl (for wire X)
- Δl2 = (2/3) * Δl (for wire Y)
Setting Up the Equations
Now we can substitute these stretches into our tension equations:
- T1 = E1 * (A/l) * (1/3) * Δl
- T2 = E2 * (A/l) * (2/3) * Δl
Substituting T1 and T2 into the balance equation:
- E1 * (A/l) * (1/3) * Δl + E2 * (A/l) * (2/3) * Δl = 30 N
Simplifying the Equation
We can factor out common terms:
- (A/l) * Δl * (E1/3 + 2E2/3) = 30 N
Finding E2
Now, we can isolate E2. First, we need to express E2 in terms of known quantities:
- 30 N = (A/l) * Δl * (E1/3 + 2E2/3)
Rearranging gives:
- 30 N = (A/l) * Δl * (E1/3) + (2/3) * (A/l) * Δl * E2
Now, we can substitute E1 = 1.0 × 10^11 N/m²:
- 30 N = (A/l) * Δl * (1.0 × 10^11/3) + (2/3) * (A/l) * Δl * E2
To find E2, we can express E2 as:
- E2 = (30 N - (A/l) * Δl * (1.0 × 10^11/3)) * (3/(2 * (A/l) * Δl))
Final Calculation
Since we don't have the values for A, l, or Δl, we can express E2 in terms of these variables. However, if we assume that the cross-sectional area A and the length l are the same for both wires, we can simplify our calculations further. For practical purposes, if we assume that the elongation Δl is the same for both wires, we can find a numerical value for E2.
Ultimately, you will need to plug in the values for A, l, and Δl to find the specific Young's modulus E2 for wire Y. If you have those values, you can easily compute E2 using the derived formula.
