### Problem StatementA mass weighing \(32 \text{ lbs}\) stretches a spring \(3 \text{ inches}\). The mass is in a medium that exerts a viscous resistance of \(44 \text{ lbs}\) when the mass has a velocity of \(2 \text{ ft/sec}\).Suppose the object is displaced an additional \(7 \text{ inches}\) and released.Find an equation for the object's displacement, \(u(t)\), in feet after \(t\) seconds.\[u(t) = \] ### ExplanationThis problem involves finding the displacement of a mass attached to a spring over time, considering both the spring force and a resistance force due to the medium the mass is in. The given mass stretches the spring, and then additional displacement is applied and then released. The system is described by a differential equation that models the harmonic motion with damping. To solve for \(u(t)\), we need to set up and solve the corresponding differential equation considering:1. The initial displacement \(7 \text{ inches}\),2. The viscous resistance, and3. The spring constant.The problem provides all necessary constants:- Mass: \(32 \text{ lbs}\)- Initial stretch: \(3 \text{ inches}\)- Resistance: \(44 \text{ lbs}\) at \(2 \text{ ft/sec}\)We'll use these to derive and solve the equation for \(u(t)\).

Answered: Image /qna-images/answer/c17381de-7f75-499d-92c7-50815ece71ee.jpg

Answered: A mass weighting 32 lbs stretches a…

A mass weighting 32 lbs stretches a spring 3 inches. The mass is in a medium that exerts a viscous resistance of 44 lbs when the mass has a velocity of 2 ft/sec.

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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Concept explainers

Mechanical Properties of Solids

There are three types of matter- solid, liquid and gas. Out of them, solids have a rigid shaped and size. The intermolecular forces of attractions in solids are greatest. Mechanical properties are the physical properties of a material which it shows, when it is provided with certain amount of force or load. These properties are to be kept in mind for various purposes. When a force is applied to an object, the object may or may not get deformed depending upon its elastic property. But every time the object develops a restoring force against the force applied. This is commonly known as stress. If the object changes its dimensions due to the force applied it develops strain ( which is the ratio of change in dimension to the original dimension).

Question

### Problem Statement

A mass weighing \(32 \text{ lbs}\) stretches a spring \(3 \text{ inches}\). The mass is in a medium that exerts a viscous resistance of \(44 \text{ lbs}\) when the mass has a velocity of \(2 \text{ ft/sec}\).

Suppose the object is displaced an additional \(7 \text{ inches}\) and released.

Find an equation for the object's displacement, \(u(t)\), in feet after \(t\) seconds.

\[u(t) = \]

### Explanation

This problem involves finding the displacement of a mass attached to a spring over time, considering both the spring force and a resistance force due to the medium the mass is in.

The given mass stretches the spring, and then additional displacement is applied and then released. The system is described by a differential equation that models the harmonic motion with damping.

To solve for \(u(t)\), we need to set up and solve the corresponding differential equation considering:
1. The initial displacement \(7 \text{ inches}\),
2. The viscous resistance, and
3. The spring constant.

The problem provides all necessary constants:
- Mass: \(32 \text{ lbs}\)
- Initial stretch: \(3 \text{ inches}\)
- Resistance: \(44 \text{ lbs}\) at \(2 \text{ ft/sec}\)

We'll use these to derive and solve the equation for \(u(t)\).

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