If the 20 -kg cylinder is released from rest at h=0...

If the 20 -kg cylinder is released from rest at $h=0$ determine the required stiffness $k$ of each spring so that its motion is arrested or stops when $h=0.5 \mathrm{m} .$ Each spring has an unstretched length of $1 \mathrm{m}$

Key Concepts

Elastic Potential Energy

Elastic potential energy is the energy stored in a spring or any elastic material when it is stretched or compressed. According to Hooke’s law, the energy stored is proportional to the square of the displacement from its equilibrium position, scaled by the stiffness of the spring. This energy plays a key role when the deformation of springs is used to arrest motion.

Hooke's Law and Spring Constant

Hooke’s law relates the force exerted by a spring to its displacement from the equilibrium position, with the proportionality factor being the spring constant (stiffness). The spring constant measures how resistant a spring is to deformation, and it directly influences the elastic potential energy that can be stored for a given displacement.

Conservation of Energy Principle

This principle states that in an isolated system, energy can neither be created nor destroyed but only transformed from one form to another. In the context of mechanical systems involving springs and masses, gravitational potential energy lost by a falling object is converted into elastic potential energy stored in the spring as it deforms.

Gravitational Potential Energy

Gravitational potential energy is the energy an object possesses due to its position in a gravitational field. For an object at height h, this energy is calculated as the product of its mass, gravitational acceleration, and its height above a reference point. This concept is fundamental when analyzing the energy changes as an object moves vertically.

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