Problem 1 (Discontinuous Forcing). Suppose we have an undamped spring-mass system, where a 0.3 kg mass is attached to a spring with spring constant 30 N/m. At t = 0, the mass is disturbed from rest by an oscillating motor, which supplies a force of 3 cos 9.2t N to the system. After 10 seconds, the motor is switched off. We can model the forcing with the discontinuous function 3 cos 9.2t if 0 < t < 10 f(t) = %3D if t > 10. (a) Write down the initial value problem that describes this spring-mass system. (b) Solve the IVP from part (a) and express your answer as a piecewise function. Hint: First solve the IVP for 0 < t < 10, then for t > 10, and combine the two answers. Make sure the resulting function is differentiable at t = 10 (i.e. the functions and their derivatives must match up there). %3D
Problem 1 (Discontinuous Forcing). Suppose we have an undamped spring-mass system, where a 0.3 kg mass is attached to a spring with spring constant 30 N/m. At t = 0, the mass is disturbed from rest by an oscillating motor, which supplies a force of 3 cos 9.2t N to the system. After 10 seconds, the motor is switched off. We can model the forcing with the discontinuous function 3 cos 9.2t if 0 < t < 10 f(t) = %3D if t > 10. (a) Write down the initial value problem that describes this spring-mass system. (b) Solve the IVP from part (a) and express your answer as a piecewise function. Hint: First solve the IVP for 0 < t < 10, then for t > 10, and combine the two answers. Make sure the resulting function is differentiable at t = 10 (i.e. the functions and their derivatives must match up there). %3D
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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![**Problem 1 (Discontinuous Forcing)**
Suppose we have an undamped spring-mass system, where a 0.3 kg mass is attached to a spring with spring constant 30 N/m. At \( t = 0 \), the mass is disturbed from rest by an oscillating motor, which supplies a force of \( 3 \cos 9.2t \, \text{N} \) to the system. After 10 seconds, the motor is switched off. We can model the forcing with the discontinuous function:
\[
f(t) =
\begin{cases}
3 \cos 9.2t & \text{if } 0 \leq t < 10 \\
0 & \text{if } t \geq 10
\end{cases}
\]
(a) Write down the initial value problem that describes this spring-mass system.
(b) Solve the IVP from part (a) and express your answer as a piecewise function.
*Hint:* First solve the IVP for \( 0 \leq t < 10 \), then for \( t \geq 10 \), and combine the two answers. Make sure the resulting function is differentiable at \( t = 10 \) (i.e. the functions and their derivatives must match up there).
(c) Use graphing software (such as Desmos.com) to graph the solution from part (b), and use the graph to describe the motion of the spring-mass system.
(d) Repeat parts (a)–(c) with the forcing function:
\[
f(t) =
\begin{cases}
3 \cos 10t & \text{if } 0 \leq t < 10 \\
0 & \text{if } t \geq 10
\end{cases}
\]
What do you notice about the motion now?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3f0c8ec8-67d4-43a2-8c13-a70ac046cde5%2Fe985dd98-dae9-4eaa-939a-c8d8f6565dc4%2Fvangbh.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 1 (Discontinuous Forcing)**
Suppose we have an undamped spring-mass system, where a 0.3 kg mass is attached to a spring with spring constant 30 N/m. At \( t = 0 \), the mass is disturbed from rest by an oscillating motor, which supplies a force of \( 3 \cos 9.2t \, \text{N} \) to the system. After 10 seconds, the motor is switched off. We can model the forcing with the discontinuous function:
\[
f(t) =
\begin{cases}
3 \cos 9.2t & \text{if } 0 \leq t < 10 \\
0 & \text{if } t \geq 10
\end{cases}
\]
(a) Write down the initial value problem that describes this spring-mass system.
(b) Solve the IVP from part (a) and express your answer as a piecewise function.
*Hint:* First solve the IVP for \( 0 \leq t < 10 \), then for \( t \geq 10 \), and combine the two answers. Make sure the resulting function is differentiable at \( t = 10 \) (i.e. the functions and their derivatives must match up there).
(c) Use graphing software (such as Desmos.com) to graph the solution from part (b), and use the graph to describe the motion of the spring-mass system.
(d) Repeat parts (a)–(c) with the forcing function:
\[
f(t) =
\begin{cases}
3 \cos 10t & \text{if } 0 \leq t < 10 \\
0 & \text{if } t \geq 10
\end{cases}
\]
What do you notice about the motion now?
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