A mass of 1 slug is attached to a spring whose constant is 5 lb/ft. Initially, the mass is released 1 foot below the equilibrium position with a downward velocity of 3 ft/s, and the subsequent motion takes place in a medium that offers a damping force that is numerically equal to 2 times the instantaneous velocity. (a) Assume the mass is driven by an external force equal to f(t) = 8 cos(2t) + 2 sin(2t). Find the initial conditions. x(0) = x'(0) = Find the particular solution. x₂ = ft x(t) = ft ft/s Find the equation of motion of the mass. ft
A mass of 1 slug is attached to a spring whose constant is 5 lb/ft. Initially, the mass is released 1 foot below the equilibrium position with a downward velocity of 3 ft/s, and the subsequent motion takes place in a medium that offers a damping force that is numerically equal to 2 times the instantaneous velocity. (a) Assume the mass is driven by an external force equal to f(t) = 8 cos(2t) + 2 sin(2t). Find the initial conditions. x(0) = x'(0) = Find the particular solution. x₂ = ft x(t) = ft ft/s Find the equation of motion of the mass. ft
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:**Spring-Mass System with External Force and Damping**
**Problem Description:**
A mass of 1 slug is attached to a spring with a spring constant of 5 lb/ft. Initially, the mass is released 1 foot below the equilibrium position with a downward velocity of 3 ft/s. The motion occurs in a medium that offers a damping force equal to 2 times the instantaneous velocity.
**Task:**
(a) Assume the mass is driven by an external force given by \( f(t) = 8 \cos(2t) + 2 \sin(2t) \).
1. **Find the initial conditions.**
- \( x(0) = \_\_\_\_ \) ft
- \( x'(0) = \_\_\_\_ \) ft/s
2. **Find the particular solution.**
- \( x_p = \_\_\_\_ \) ft
3. **Find the equation of motion of the mass.**
- \( x(t) = \_\_\_\_ \) ft
In this setup, determine the required conditions and solutions to describe the motion of the mass under the specified forces and damping.
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