To analyze the behavior of the two discs connected by a spring when the upper disc is displaced and released, we need to consider the forces acting on both discs and how they interact through the spring. Let's break down the problem step by step, focusing on the conditions given for the displacement of the upper disc.
Understanding the System
We have two discs: a heavier disc with mass 3m on the ground and a lighter disc with mass m on top, connected by a vertical spring with stiffness k. When the upper disc is pushed down by a distance δ and released, it will oscillate due to the restoring force of the spring.
Force Analysis
When the upper disc is displaced by δ, the spring is compressed, creating a force that acts to restore the system to equilibrium. The force exerted by the spring can be calculated using Hooke's Law:
- Spring Force, F_s = k * δ
This force acts upwards on the lighter disc and downwards on the heavier disc. The normal force from the ground on the lower disc will change depending on the displacement and the resulting motion of both discs.
Case Analysis
Now, let's analyze the different cases based on the value of δ:
Case A: If δ > 3mg/k
In this scenario, the force exerted by the spring when the upper disc is released is not sufficient to keep the lower disc in contact with the ground. The weight of the upper disc (mg) plus the spring force will exceed the weight of the lower disc (3mg), causing the lower disc to bounce up. Thus, this statement is true.
Case B: If δ = 2mg/k
Here, we need to calculate the maximum normal reaction force on the lower disc. When the upper disc is at its maximum displacement, the spring force will be:
The total force acting on the lower disc will be the weight of the lower disc plus the spring force:
- Normal Force, N = Weight of lower disc + Spring Force = 3mg + 2mg = 5mg
However, during the oscillation, the maximum normal force will be higher due to the dynamic effects. The maximum normal reaction from the ground on the lower disc can reach up to 6mg, making this statement true.
Case C: If δ = 2mg/k
This case is similar to Case B, but it claims that the maximum normal reaction is 4mg. Based on our previous calculation, we found that the maximum normal force can reach 6mg, not 4mg. Therefore, this statement is false.
Case D: If δ > 4mg/k
In this situation, similar to Case A, the displacement is large enough that the spring force will not be able to keep the lower disc in contact with the ground. The upper disc will push down with enough force to cause the lower disc to bounce up. Thus, this statement is also true.
Summary of Findings
To summarize:
- Statement A is true: The lower disc will bounce up if δ > 3mg/k.
- Statement B is true: The maximum normal reaction from the ground on the lower disc can reach 6mg when δ = 2mg/k.
- Statement C is false: The maximum normal reaction cannot be 4mg in this case.
- Statement D is true: The lower disc will bounce up if δ > 4mg/k.
Understanding these dynamics helps us appreciate how forces interact in a system with springs and masses, and how displacement affects the motion of connected objects.
