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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.

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
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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### 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 \(u(t)\).
Transcribed Image Text:### 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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