hc verma problem 30 (i didn't understand the force on clamp by pulley.because shouldn't it be the resultant of 2T horizontally and t vertically?)

:a block is kept on the floor of an elevator at rest.the elevator starts descending with an acceleration of 12m/s^2.find when the block will strike the top of the elevator if its length is "h" metres

my approach

let it strike the top after t seconds

h=(g-2)_/2t² and this would give the answer.how to solve it without the concept of relative acceleration?

To tackle this problem effectively, let's break it down step by step. The scenario involves a block resting on the floor of an elevator that begins to descend with a specific acceleration. The goal is to determine when the block will strike the top of the elevator without relying on the concept of relative acceleration. Instead, we will analyze the forces acting on the block and the motion of both the block and the elevator.

Understanding the System

Initially, the elevator is at rest, and the block is stationary on the floor. When the elevator starts descending with an acceleration of 12 m/s², we need to consider how this affects the block.

Forces Acting on the Block

While the elevator accelerates downwards, the block experiences two main forces:

• The gravitational force acting downwards, which is equal to mg, where m is the mass of the block and g is the acceleration due to gravity (approximately 9.81 m/s²).
• The normal force exerted by the elevator floor on the block, which we will denote as N.

Since the elevator is accelerating downwards, the effective acceleration acting on the block can be considered as:

g - a, where a is the acceleration of the elevator (12 m/s²). Thus, the effective acceleration acting on the block becomes:

g - 12 m/s².

Motion of the Block

As the elevator descends, the block will also start to fall due to gravity. However, since the elevator is accelerating downwards faster than the gravitational pull on the block, the block will eventually strike the ceiling of the elevator.

Setting Up the Equation

Let’s denote the height of the elevator as h. The time taken for the block to hit the top of the elevator can be calculated using the following kinematic equation:

h = (1/2) * (g - a) * t².

Substituting the values, we have:

h = (1/2) * (9.81 - 12) * t².

This simplifies to:

h = (1/2) * (-2.19) * t².

Rearranging gives:

t² = -h / (1.095).

Finding the Time

To find the time t when the block strikes the top of the elevator, we can solve for t:

t = √(-h / (1.095)).

However, since time cannot be negative, we need to ensure that the height h is such that the block will indeed hit the top before the elevator descends too far. This means that the block will strike the top of the elevator after a certain time, which can be calculated based on the height of the elevator and the effective acceleration.

Final Thoughts

In summary, by analyzing the forces acting on the block and using kinematic equations, we can determine the time it takes for the block to strike the top of the elevator without needing to rely on the concept of relative acceleration. The key is to understand how the downward acceleration of the elevator affects the motion of the block within it.