A 0.150-kg frame, when suspended from a coil spring,...

A $0.150-\mathrm{kg}$ frame, when suspended from a coil spring, stretches the spring 0.050 $\mathrm{m}$ . A $0.200-\mathrm{kg}$ lump of putty is dropped from rest onto the frame from a height of 30.0 $\mathrm{cm}$ (Fig. 8.42$)$ . Find the maximum distance the frame moves downward from its initial position.

Key Concepts

Conservation of Energy in Oscillatory Systems

After the collision, the system’s motion can be analyzed using conservation of energy. The kinetic energy of the moving mass-spring system is gradually converted into spring (elastic) potential energy and gravitational potential energy, allowing one to calculate the maximum displacement where the system momentarily comes to rest before oscillating.

Conservation of Momentum in Inelastic Collisions

When two objects collide and stick together, as in an inelastic collision, the total momentum of the system is conserved, even though kinetic energy is not. This principle is used to determine the velocity of the combined masses immediately after the collision, which is a crucial step before applying energy conservation.

Gravitational and Elastic Potential Energy

In systems involving springs and gravity, energy transformations occur between gravitational potential energy (due to the height of an object) and elastic potential energy (stored in a deformed spring). Recognizing and quantifying these energy forms is key to understanding and solving for the amplitudes of motion in combined gravitational-spring systems.

Hooke's Law and Spring Constant Determination

Hooke’s Law relates the force exerted by a spring to its displacement from equilibrium through the formula F = kx. In problems where a mass is suspended from a spring, the equilibrium condition (mg = kx) allows one to determine the spring constant. Understanding this concept is essential for analyzing how the spring responds to additional forces or masses attached to it.

Equilibrium in Spring-Mass Systems

An object hanging from a spring reaches an equilibrium position where the downward gravitational force is exactly balanced by the upward spring force. This equilibrium point provides a reference for measuring additional displacements when extra weights are added or external forces act on the system.

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