An object is attached to a spring hanging from the ceiling. The object undergoes simple harmonic motion modeled by the differential equation my'' + ky = 0, where y(t) is the height of the object (relative to its equilibrium position) at time t, m is the mass of the object, and k is the spring constant. (a) Write the phase plane equivalent of the differential equation. The equation should be in dv terms of y,v, and where v = y'. dy dv mu + ky = 0 dy I (b) Integrate the phase plane equivalent equation with respect to y to find an equation relating y to v Write it in the form KE + PE = E k 1/1/2mv ² + 12/27 y ² where KE denotes kinetic energy, PE denotes potential energy, and E denotes total energy. = E (c) Suppose the mass is 5 kg and the spring constant is 3 kg/s2. If the spring is initially stretched 4 meters, held, and released, then determine the total energy and write the resulting equation describing the trajectory of the object in the phase plane. 37 v² + 3y² = 16
An object is attached to a spring hanging from the ceiling. The object undergoes simple harmonic motion modeled by the differential equation my'' + ky = 0, where y(t) is the height of the object (relative to its equilibrium position) at time t, m is the mass of the object, and k is the spring constant. (a) Write the phase plane equivalent of the differential equation. The equation should be in dv terms of y,v, and where v = y'. dy dv mu + ky = 0 dy I (b) Integrate the phase plane equivalent equation with respect to y to find an equation relating y to v Write it in the form KE + PE = E k 1/1/2mv ² + 12/27 y ² where KE denotes kinetic energy, PE denotes potential energy, and E denotes total energy. = E (c) Suppose the mass is 5 kg and the spring constant is 3 kg/s2. If the spring is initially stretched 4 meters, held, and released, then determine the total energy and write the resulting equation describing the trajectory of the object in the phase plane. 37 v² + 3y² = 16
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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![An object is attached to a spring hanging from the ceiling. The object undergoes simple harmonic motion modeled by the differential equation \( my'' + ky = 0 \), where \( y(t) \) is the height of the object (relative to its equilibrium position) at time \( t \), \( m \) is the mass of the object, and \( k \) is the spring constant.
**(a)** Write the phase plane equivalent of the differential equation. The equation should be in terms of \( y, v, \) and \( \frac{dv}{dy} \), where \( v = y' \).
\[ mv\frac{dv}{dy} + ky = 0 \]
**(b)** Integrate the phase plane equivalent equation with respect to \( y \) to find an equation relating \( y \) to \( v \). Write it in the form
\[ KE + PE = E \]
where \( KE \) denotes kinetic energy, \( PE \) denotes potential energy, and \( E \) denotes total energy.
\[ \frac{1}{2}mv^2 + \frac{k}{2}y^2 = E \]
**(c)** Suppose the mass is 5 kg and the spring constant is 3 kg/s². If the spring is initially stretched 4 meters, held, and released, then determine the total energy and write the resulting equation describing the trajectory of the object in the phase plane.
\[ \frac{5}{3}v^2 + \frac{3}{3}y^2 = 16 \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff57c5c9b-acb3-4a3d-a39b-55eeff7ee4ad%2F45cca40d-0efe-4069-9a40-edcb1a3a7ca4%2F5u8lzo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:An object is attached to a spring hanging from the ceiling. The object undergoes simple harmonic motion modeled by the differential equation \( my'' + ky = 0 \), where \( y(t) \) is the height of the object (relative to its equilibrium position) at time \( t \), \( m \) is the mass of the object, and \( k \) is the spring constant.
**(a)** Write the phase plane equivalent of the differential equation. The equation should be in terms of \( y, v, \) and \( \frac{dv}{dy} \), where \( v = y' \).
\[ mv\frac{dv}{dy} + ky = 0 \]
**(b)** Integrate the phase plane equivalent equation with respect to \( y \) to find an equation relating \( y \) to \( v \). Write it in the form
\[ KE + PE = E \]
where \( KE \) denotes kinetic energy, \( PE \) denotes potential energy, and \( E \) denotes total energy.
\[ \frac{1}{2}mv^2 + \frac{k}{2}y^2 = E \]
**(c)** Suppose the mass is 5 kg and the spring constant is 3 kg/s². If the spring is initially stretched 4 meters, held, and released, then determine the total energy and write the resulting equation describing the trajectory of the object in the phase plane.
\[ \frac{5}{3}v^2 + \frac{3}{3}y^2 = 16 \]
![### Graphing the Trajectory in the Phase Plane
**(d)** Graph the trajectory of the object in the phase plane:
The graph provided is a coordinate grid ranging from -5 to 5 on both the x-axis and y-axis. It is intended for plotting the trajectory of an object in a phase plane. Below the graph are tools to manipulate the graph, with options to "Clear All" and drawing options depicting initial conditions such as a point, ellipse, or trajectories intersecting at various points.
**(e)** Analyzing the Graph's Implications:
The graph in part (d) suggests that the solution can be expressed in the form:
\[ y = c_1 \cos(\omega t) + c_2 \sin(\omega t) \]
**Steps to Determine the Expression:**
1. **Determine Values**: Use assumptions from part (c) to find the values of constants \( c_1 \) and \( c_2 \).
2. **Calculate \( \omega \)**: Utilize the equation from part (c) to establish the value of \( \omega \).
3. **Expression for \( y \)**: Write the equation expressing \( y \), indicating the object's position over time.
**Important Note**: Initially, the spring is stretched, leading to a negative displacement.
**Hint**: To calculate \( \omega \), try using a \( t \) value other than 0.
\[ y = \, \text{(input box for solution)} \]
This section is designed to guide learners through the process of plotting phase plane trajectories and deriving the mathematical expression representing dynamic systems.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff57c5c9b-acb3-4a3d-a39b-55eeff7ee4ad%2F45cca40d-0efe-4069-9a40-edcb1a3a7ca4%2F6z5tguo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Graphing the Trajectory in the Phase Plane
**(d)** Graph the trajectory of the object in the phase plane:
The graph provided is a coordinate grid ranging from -5 to 5 on both the x-axis and y-axis. It is intended for plotting the trajectory of an object in a phase plane. Below the graph are tools to manipulate the graph, with options to "Clear All" and drawing options depicting initial conditions such as a point, ellipse, or trajectories intersecting at various points.
**(e)** Analyzing the Graph's Implications:
The graph in part (d) suggests that the solution can be expressed in the form:
\[ y = c_1 \cos(\omega t) + c_2 \sin(\omega t) \]
**Steps to Determine the Expression:**
1. **Determine Values**: Use assumptions from part (c) to find the values of constants \( c_1 \) and \( c_2 \).
2. **Calculate \( \omega \)**: Utilize the equation from part (c) to establish the value of \( \omega \).
3. **Expression for \( y \)**: Write the equation expressing \( y \), indicating the object's position over time.
**Important Note**: Initially, the spring is stretched, leading to a negative displacement.
**Hint**: To calculate \( \omega \), try using a \( t \) value other than 0.
\[ y = \, \text{(input box for solution)} \]
This section is designed to guide learners through the process of plotting phase plane trajectories and deriving the mathematical expression representing dynamic systems.
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