**Problem 1 (Discontinuous Forcing):** Suppose we have an undamped spring-mass system, where a 0.3 kg mass is attached to a spring with a spring constant of 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: \[ f(t) = \begin{cases} 3 \cos 9.2t & \text{if } 0 \le t < 10 \\ 0 & \text{if } t \ge 10 \end{cases} \] Tasks: (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 \( 0 \le t < 10 \), then for \( t \ge 10 \), 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: \[ f(t) = \begin{cases} 3 \cos 10t & \text{if } 0 \le t < 10 \\ 0 & \text{if } t \ge 10 \end{cases} \] **What do you notice about the motion now?**
**Problem 1 (Discontinuous Forcing):** Suppose we have an undamped spring-mass system, where a 0.3 kg mass is attached to a spring with a spring constant of 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: \[ f(t) = \begin{cases} 3 \cos 9.2t & \text{if } 0 \le t < 10 \\ 0 & \text{if } t \ge 10 \end{cases} \] Tasks: (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 \( 0 \le t < 10 \), then for \( t \ge 10 \), 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: \[ f(t) = \begin{cases} 3 \cos 10t & \text{if } 0 \le t < 10 \\ 0 & \text{if } t \ge 10 \end{cases} \] **What do you notice about the motion now?**
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 a spring constant of 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:
\[
f(t) =
\begin{cases}
3 \cos 9.2t & \text{if } 0 \le t < 10 \\
0 & \text{if } t \ge 10
\end{cases}
\]
Tasks:
(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 \( 0 \le t < 10 \), then for \( t \ge 10 \), 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:
\[
f(t) =
\begin{cases}
3 \cos 10t & \text{if } 0 \le t < 10 \\
0 & \text{if } t \ge 10
\end{cases}
\]
**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%2F75028c1c-a227-48b0-b78f-1f7c52b8e4d2%2Feegbfp.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 a spring constant of 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:
\[
f(t) =
\begin{cases}
3 \cos 9.2t & \text{if } 0 \le t < 10 \\
0 & \text{if } t \ge 10
\end{cases}
\]
Tasks:
(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 \( 0 \le t < 10 \), then for \( t \ge 10 \), 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:
\[
f(t) =
\begin{cases}
3 \cos 10t & \text{if } 0 \le t < 10 \\
0 & \text{if } t \ge 10
\end{cases}
\]
**What do you notice about the motion now?**
Expert Solution
Solution: a
The undamped spring mass system is given by .
Here,
The undamped spring mass system becomes .
b. For , the non-homogeneous equation is .
It can be written as .
Solve the homogeneous equation:
The characteristic equation is
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