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A 10 kilogram object suspended from the end of a vertically hanging spring stretches the spring 9.8 centimeters. At time t = 0, the resulting mass-spring system is disturbed from its rest state by the force F(t) = 180 cos(8t). The force F(t) is expressed in Newtons and is positive in the downward direction, and time is measured in seconds. a. Determine the spring constant k. k = 1000 Newtons / meter b. Formulate the initial value problem for y(t), where y(t) is the displacement of the object from its equilibrium rest state, measured positive in the downward direction. (Give your answer in terms of y, y', y",t.) Differential equation: help (equations) Initial conditions: y(0) = 0 and y' (0) help (numbers) c. Solve the initial value problem for y(t). y(t) = help (formulas)

A 10 kilogram object suspended from the end of a vertically hanging spring stretches the spring 9.8 centimeters. At time t = 0, the resulting mass-spring system is disturbed from its rest state by the force F(t) = 180 cos(8t). The force F(t) is expressed in Newtons and is positive in the downward direction, and time is measured in seconds. a. Determine the spring constant k. k = 1000 Newtons / meter b. Formulate the initial value problem for y(t), where y(t) is the displacement of the object from its equilibrium rest state, measured positive in the downward direction. (Give your answer in terms of y, y', y",t.) Differential equation: help (equations) Initial conditions: y(0) = 0 and y' (0) help (numbers) c. Solve the initial value problem for y(t). y(t) = help (formulas)

Advanced Engineering Mathematics
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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**Spring-Mass System Problem** A 10 kilogram object suspended from the end of a vertically hanging spring stretches the spring 9.8 centimeters. At time \( t = 0 \), the resulting mass-spring system is disturbed from its rest state by the force \( F(t) = 180 \cos(8t) \). The force \( F(t) \) is expressed in Newtons and is positive in the downward direction, and time is measured in seconds. **a. Determine the Spring Constant \( k \):** \( k = \) 1000 Newtons/meter **b. Formulate the Initial Value Problem for \( y(t) \):** Where \( y(t) \) is the displacement of the object from its equilibrium rest state, measured positive in the downward direction. (Give your answer in terms of \( y, y', y'', t \).) - **Differential Equation:** - **Initial Conditions:** - \( y(0) = \) 0 - \( y'(0) = \) 0 **c. Solve the Initial Value Problem for \( y(t) \):** - \( y(t) = \) **d. Plot the Solution and Determine the Maximum Excursion:** Plot the solution and determine the maximum excursion from equilibrium made by the object on the time interval \( 0 \leq t < \infty \). If there is no such maximum, enter NONE. - **Maximum Excursion:** - \( \) meters (Note: Specific differential equations, solutions, and maximum excursion values are to be calculated based on the given initial and boundary conditions.)
Transcribed Image Text:**Spring-Mass System Problem** A 10 kilogram object suspended from the end of a vertically hanging spring stretches the spring 9.8 centimeters. At time \( t = 0 \), the resulting mass-spring system is disturbed from its rest state by the force \( F(t) = 180 \cos(8t) \). The force \( F(t) \) is expressed in Newtons and is positive in the downward direction, and time is measured in seconds. **a. Determine the Spring Constant \( k \):** \( k = \) 1000 Newtons/meter **b. Formulate the Initial Value Problem for \( y(t) \):** Where \( y(t) \) is the displacement of the object from its equilibrium rest state, measured positive in the downward direction. (Give your answer in terms of \( y, y', y'', t \).) - **Differential Equation:** - **Initial Conditions:** - \( y(0) = \) 0 - \( y'(0) = \) 0 **c. Solve the Initial Value Problem for \( y(t) \):** - \( y(t) = \) **d. Plot the Solution and Determine the Maximum Excursion:** Plot the solution and determine the maximum excursion from equilibrium made by the object on the time interval \( 0 \leq t < \infty \). If there is no such maximum, enter NONE. - **Maximum Excursion:** - \( \) meters (Note: Specific differential equations, solutions, and maximum excursion values are to be calculated based on the given initial and boundary conditions.)
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