An ideal spring with a spring constant of 15 N / m is suspended vertically. A body of mass 0.60

step 1 In this scenario we are given a mass of 0 .6 kilograms being attached to a spring vertically and

An ideal spring with a spring constant of 15 N / m is...

An ideal spring with a spring constant of $15 \mathrm{N} / \mathrm{m}$ is suspended vertically. A body of mass $0.60 \mathrm{kg}$ is attached to the unstretched spring and released. (a) What is the extension of the spring when the speed is a maximum? (b) What is the maximum speed?

Key Concepts

Conservation of Mechanical Energy

This principle states that in the absence of non-conservative forces, the total mechanical energy of a system (the sum of kinetic energy and potential energy) remains constant. In a vertically suspended spring-mass system, energy is continually exchanged between gravitational potential energy, elastic potential energy stored in the spring, and kinetic energy as the mass oscillates.

Equilibrium Position in Spring-Mass Systems

The equilibrium position is reached when the net force acting on the mass is zero, meaning that the upward spring force exactly balances the downward gravitational force. In oscillatory motion, the mass reaches its maximum speed as it passes through or near this equilibrium position, where the stored potential energies are minimal and most energy is in the form of kinetic energy.

Spring Force and Hooke’s Law

Hooke’s Law quantifies the force exerted by an ideal spring as directly proportional to its displacement from the unstretched length, with the spring constant as the proportionality factor. This relation is crucial for determining both the elastic potential energy stored in the spring and the dynamics of the mass as it oscillates.

Simple Harmonic Motion

Simple harmonic motion describes periodic motion where the restoring force is proportional to the displacement from equilibrium. In the context of a vertically suspended spring with an attached mass, the system oscillates about the equilibrium position in a manner that can be approximated as simple harmonic motion, which explains characteristics such as maximum speed at the equilibrium point.

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