To solve the problem involving the block B and the two springs, we need to analyze the forces acting on the block when it is displaced and how these forces relate to the displacements x and y. Let's break this down step by step.
Understanding the System
We have a block B connected to two springs: S1 with spring constant k and S2 with spring constant 4k. When the block is displaced towards wall 1 by a distance x, both springs will exert forces on the block. The key here is to understand how these forces change as the block moves.
Forces Acting on the Block
When the block is displaced by a distance x, the force exerted by each spring can be calculated using Hooke's Law, which states that the force exerted by a spring is proportional to its displacement:
- Force from spring S1: F1 = -kx
- Force from spring S2: F2 = -4k(x + y)
Here, y is the additional displacement towards wall 2 when the block moves back after being released. The total force acting on the block when it is displaced by x is the sum of the forces from both springs:
Total Force, F = F1 + F2 = -kx - 4k(x + y)
Equilibrium and Maximum Displacement
When the block reaches its maximum displacement y towards wall 2, the forces will again be acting on the block, but now the spring S1 will be compressed by a distance of (x + y) and spring S2 will be stretched by a distance of y. The forces at this point are:
- Force from spring S1: F1 = -k(x + y)
- Force from spring S2: F2 = -4ky
Setting the total force to zero at maximum displacement gives us:
-k(x + y) - 4ky = 0
Solving for the Ratio x/y
Rearranging the equation, we get:
k(x + y) = -4ky
Dividing through by k (assuming k is not zero), we have:
x + y = -4y
This simplifies to:
x = -4y - y = -5y
Now, to find the ratio x/y, we can express it as:
x/y = -5
However, since we are looking for the absolute ratio, we take the positive value:
|x/y| = 5
Final Thoughts on the Ratio
Given the options provided (A) 4, (B) 2, (C) 1/2, (D) 1/4, it seems there might be a misunderstanding in the interpretation of the problem or the options given. The derived ratio does not match any of the options directly. However, if we consider the nature of the springs and their constants, we can conclude that the behavior of the system indicates that the ratio of displacements is influenced by the spring constants. The correct interpretation of the problem may lead to a different understanding of the maximum displacements.
In summary, the analysis shows that the relationship between the displacements x and y is crucial in understanding the dynamics of the system. The derived ratio indicates a significant relationship between the two springs and their respective constants, which ultimately affects the motion of the block.
