Problem 14 - Master Resource Book in JEE Main Physics

Question

A block is submerged in vessel filled with water by a spring attached to the bottom of the vessel. In equilibrium, the spring is compressed. The vessel now moves downwards with an acceleration \(a(

Step-by-Step Solution

Answer

(c) will increase

Step 1: Understand the System at Equilibrium

Initially, the block is submerged in water and held by a spring. In this equilibrium state, the spring is compressed due to the balance between the buoyant force, the weight of the block, and the force exerted by the spring.

Step 2: Identify Forces Involved

The forces acting on the block are: the gravitational force (\(mg\)), the buoyant force (\(B\)), and the spring force (\(kx\), where \(x\) is the compression of the spring). On equilibrium, these forces are balanced such that \(B + kx = mg\).

Step 3: Analyze the Effect of Downward Acceleration

When the vessel accelerates downward with an acceleration \(a (< g)\), the apparent gravitational force changes to \((g-a)\). The new effective weight of the block is \(m(g-a)\). The buoyant force \(B\) remains the same initially since the volume of displaced water doesn't change immediately.

Step 4: Rebalance the Forces

Under this new condition, the forces need to rebalance. The equation now becomes \(B + kx' = m(g-a)\), where \(x'\) is the new compression of the spring. Since \(m(g-a) < mg\), this implies \(kx' < kx\). The spring compression must decrease to satisfy this inequality.

Step 5: Determine the Change in Spring Length

The decreased balance force means the spring compression reduces, implying the spring will decompress. Therefore, the length of the spring will increase.

Key Concepts

Spring Force

Spring force is an essential concept in physics, describing the force exerted by a spring when it is compressed or stretched. This force is governed by Hooke's Law, which states that the force exerted by a spring is directly proportional to the amount it is compressed or stretched. This can be mathematically described by the equation: \[ F_{spring} = kx \] where:

• \(F_{spring}\) is the force exerted by the spring.
• \(k\) is the spring constant, a property of the spring indicating its stiffness.
• \(x\) is the displacement of the spring from its equilibrium position.

In our exercise, the spring starts in a compressed state due to the balance of forces acting on the submerged block. Understanding how the spring force adjusts when the system's equilibrium is disturbed, such as by external acceleration, helps us analyze changes in the spring's compression or extension.

Equilibrium of Forces

In physics, equilibrium of forces refers to the situation where all the forces acting on an object are balanced, resulting in the object being in a state of rest or moving with a constant velocity. For our submerged block scenario, the equilibrium can be understood by the equation:\[ B + kx = mg \] Here:

• \(B\) is the buoyant force, which is the upward force exerted by the water, balancing the downward forces.
• \(kx\) is the spring force, representing the force supplied by the spring in its compressed state.
• \(mg\) is the weight of the block due to gravity.

At equilibrium, these forces balance out, maintaining the block's stationary position. When the vessel undergoes downward acceleration, the balance is disrupted, prompting a reevaluation of the forces to reach a new equilibrium.

Apparent Weight During Acceleration

The concept of apparent weight comes into play when an object is subject to additional forces beyond its inherent weight due to gravity. In our problem, as the vessel moves downward with an acceleration \(a\), the apparent weight of the block changes. The new force equation becomes:\[ m(g-a) \] where \(g\) is the acceleration due to gravity, and \(a\) is the vessel's downward acceleration.This change means that the block feels lighter since \((g-a)\) is lesser than \(g\). The buoyant force, however, remains unchanged in the initial moment following the downward movement. This leads to a decrease in spring compression, as the spring force adjusts to this new effective weight, allowing the spring to extend until a new balance of forces is achieved.