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Chapter 9: Problem 49

Two masses \(m_{1}\) and \(m_{2}\) are connected by a spring of spring constant \(k\) and are placed on a frictionless horizontal surface. Initially the spring is stretched through a distance \(x_{0}\) when the system is released from rest. Find the distance moved by the two masses before they again come to rest.

Short Answer

Expert verified
Each mass moves a distance \(x_0\) before coming momentarily to rest.

Step by step solution

01

Understand the Problem

We have two masses, \(m_1\) and \(m_2\), connected by a spring with a spring constant \(k\). The spring is initially stretched by a distance \(x_0\) and then released. We need to find the distance each mass moves before they come to rest when the spring is neither compressed nor stretched.
02

Analyze Energy Conservation

Since the system is on a frictionless surface, mechanical energy is conserved. Initially, all energy is stored as potential energy in the spring: \(U_i = \frac{1}{2}kx_0^2\). When the masses come to rest, all this energy will have been converted into kinetic energy of the two masses, and soon back into potential energy at the other extreme, but we need the distances at first moment of rest where the system temporarily stops.
03

Calculate Final Potential Energy State

At the first rest moment after being released, the spring will be unstretched, and the masses will have moved to their positions. Thus, the new stretched or compressed state is zero, and spring potential energy is zero. But for the second temporal rest, we reach twice more stretched position as potential energy can be exerted back.
04

Determine Mass Relationship and Motion

Using the center of mass, the system's returns temporary rest means they trade kinetic energy for potential efficiently over a cycle twice stretched initially. The displacement focuses momentum interaction and energy transfer as they swing through equilibrium to contribute zero twice the distance via positions.
05

Utilize Center of Mass in Calculation

If no external forces act on the system, the center of mass remains unchanged. Assign center of mass symmetrically shared, calculated half energy shared as movement equatable \(x_f = x_0\). Write using the formulas as potential contributions are cycle animated.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conservation of Energy
In a spring-mass system located on a frictionless surface, energy conservation plays a crucial role. Initially, when the spring is stretched by a distance \(x_0\), all the mechanical energy in the system is in the form of spring potential energy: \(U_i = \frac{1}{2}kx_0^2\). As the system is released, this potential energy converts to kinetic energy and then oscillates between kinetic and potential energy states.
  • Mechanical energy is conserved due to the absence of friction.
  • Potential energy is at a maximum when the spring is fully stretched or compressed.
  • Kinetic energy is at a maximum when the spring passes through the equilibrium position.
When the masses move, the potential energy of the spring decreases while their kinetic energy increases until they finally come to a temporary stop towards another position cycle, forming a period of consistent rhythmic motion known as simple harmonic motion.
Center of Mass
The concept of the center of mass is essential when analyzing the movement of a spring-mass system. Since no external forces are acting on the system, the center of mass remains constant throughout the motion.
  • The center of mass defines the point where the entire mass of a system can be considered to concentrate.
  • This project influences how energy is exchanged between kinetic and potential forms.
In this system, you can think of the center of mass as a balance point, helping simplify the problem. By analyzing how the center of mass moves, we see that when the system comes to rest, the total displacement of the masses depends on balancing energy efficiently and symmetrically.
Harmonic Motion
Harmonic motion in a spring-mass system provides a rhythmic oscillation between positions of maximum stretch and compression. As the system passes through equilibrium, the energy stored as potential converts efficiently into kinetic energy.
  • It's characterized by periodic oscillations where forces accelerate masses towards equilibrium.
  • The motion is "harmonic" due to the sine wave pattern of displacement over time.
When released from a compressed or stretched position, the spring's force is proportional to the displacement, leading to a smooth back and forth motion. This is governed by Hooke's law, contributing to a motion that repeats in a cycle, called simple harmonic motion.

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Most popular questions from this chapter

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