To tackle this problem, we need to understand how the forces acting on the steel wire change when the brass block is immersed in a liquid. The extension of the wire is influenced by the weight of the block and the buoyant force acting on it when submerged. Let's break this down step by step.
Understanding the Forces at Play
Initially, when the brass block is hanging from the steel wire, the force exerted on the wire is simply the weight of the block. The weight (W) can be expressed as:
W = m * g
where m is the mass of the brass block and g is the acceleration due to gravity. The extension (e) of the wire is directly proportional to this weight according to Hooke's Law:
e = (W * L) / (A * Y)
Here, L is the original length of the wire, A is the cross-sectional area, and Y is the Young's modulus of the steel wire.
When the Block is Immersed
Now, when the brass block is fully immersed in a liquid of density ρl, the buoyant force (B) acting on the block comes into play. The buoyant force can be calculated using Archimedes' principle:
B = ρl * V * g
where V is the volume of the brass block. The effective weight of the block when submerged becomes:
W' = W - B
Substituting the expressions for weight and buoyant force, we get:
W' = (ρb * V * g) - (ρl * V * g)
This simplifies to:
W' = (ρb - ρl) * V * g
Calculating the Extensions
Now we can express the extensions in terms of the effective weights:
- The extension when the block is hanging freely is: e = (ρb * V * g * L) / (A * Y)
- The extension when the block is submerged is: e' = ((ρb - ρl) * V * g * L) / (A * Y)
Finding the Ratio
To find the ratio of the extensions, we can divide e by e':
e / e' = (ρb * V * g) / ((ρb - ρl) * V * g)
Notice that the volume V and acceleration due to gravity g cancel out:
e / e' = ρb / (ρb - ρl)
Final Answer
Thus, the ratio of the extensions is:
e / e' = ρb / (ρb - ρl)
From the options provided, this corresponds to option (a): ρb / (ρb - ρl).
This analysis illustrates how buoyancy affects the effective weight of an object and consequently the extension of a supporting wire. Understanding these principles is crucial in fields such as engineering and physics, where material properties and forces are constantly at play.
