gravity (let's call this the "dropped 2D harmonic oscillator"). The mass moves in a potential U (x, y, z) = k(x² + +mgz and the motion stops when the mass hits the ground at z = 0. (You don't need U(x, y, z) for solving the problem and it is only provided for the exact physics context.) The trajectory of the mass, i.e, its curve in space, r(t) = (x(t), y(t), z(t)) is x(t) = A cos(wt) y(t) = B cos(wt + p) 1 z(t) = - =gt² + 20 are determined by the initial conditions, and the frequency is w = √k/m. where the amplitudes A, B and the phase difference Given A = 1, B = 2,0 = π/3, w = 0.5, and g = 9.81, write a program that
gravity (let's call this the "dropped 2D harmonic oscillator"). The mass moves in a potential U (x, y, z) = k(x² + +mgz and the motion stops when the mass hits the ground at z = 0. (You don't need U(x, y, z) for solving the problem and it is only provided for the exact physics context.) The trajectory of the mass, i.e, its curve in space, r(t) = (x(t), y(t), z(t)) is x(t) = A cos(wt) y(t) = B cos(wt + p) 1 z(t) = - =gt² + 20 are determined by the initial conditions, and the frequency is w = √k/m. where the amplitudes A, B and the phase difference Given A = 1, B = 2,0 = π/3, w = 0.5, and g = 9.81, write a program that
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN:9780133594140
Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
Section: Chapter Questions
Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
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![### Trajectory Calculation and Averaging Coordinates
#### Step 2: Calculating the Trajectory
The code calculates the trajectory \( \mathbf{r}(t) \) and stores the coordinates for time steps \( \Delta t \) as a nested list `trajectory`, which contains:
\[ [ [x_0, y_0, z_0], [x_1, y_1, z_1], [x_2, y_2, z_2], \ldots ] \]
- Start from time \( t = 0 \) and use a time step \( \Delta t = 0.01 \).
- The last data point in the trajectory should be the time when the oscillator "hits the ground", i.e., when \( z(t) \leq 0 \).
#### Step 3: Time and Position at Ground Contact
The code calculates the time for hitting the ground (i.e., the first time \( t \) when \( z(t) \leq 0 \)) and stores it in the variable `t_contact`. The corresponding positions are stored in the variables `x_contact`, `y_contact`, and `z_contact`.
**Example Output:**
```plaintext
t_contact = 1.430
x_contact = 0.755
y_contact = -0.380
z_contact = ...
```
*(The partial example output is for \( z_0 = 10 \). Floating point numbers are outputted with 3 decimals using `format()`, e.g., `t_contact = {:.3f}.format(t_contact)`.)*
#### Step 4: Calculating the Average Coordinates
The code calculates the average x- and y-coordinates using the formulas:
\[ \overline{x} = \frac{1}{N} \sum_{i=1}^{N} x_i \]
\[ \overline{y} = \frac{1}{N} \sum_{i=1}^{N} y_i \]
where \( x_i, y_i \) are the coordinates \( x(t), y(t) \) in the trajectory, and \( N \) is the number of data points that you calculated.
Store the result as a list in the variable `center`:
\[ \text{center} = [\overline{x}, \overline{y}] \]
**Example Output:**
```plaintext
center = [0.917, ....](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffe5146e3-5a58-4bea-b8ad-cd2d66a98667%2Ff393124c-d710-4726-8ae0-bfbc0003ee9b%2F3ez7d1r_processed.png&w=3840&q=75)
Transcribed Image Text:### Trajectory Calculation and Averaging Coordinates
#### Step 2: Calculating the Trajectory
The code calculates the trajectory \( \mathbf{r}(t) \) and stores the coordinates for time steps \( \Delta t \) as a nested list `trajectory`, which contains:
\[ [ [x_0, y_0, z_0], [x_1, y_1, z_1], [x_2, y_2, z_2], \ldots ] \]
- Start from time \( t = 0 \) and use a time step \( \Delta t = 0.01 \).
- The last data point in the trajectory should be the time when the oscillator "hits the ground", i.e., when \( z(t) \leq 0 \).
#### Step 3: Time and Position at Ground Contact
The code calculates the time for hitting the ground (i.e., the first time \( t \) when \( z(t) \leq 0 \)) and stores it in the variable `t_contact`. The corresponding positions are stored in the variables `x_contact`, `y_contact`, and `z_contact`.
**Example Output:**
```plaintext
t_contact = 1.430
x_contact = 0.755
y_contact = -0.380
z_contact = ...
```
*(The partial example output is for \( z_0 = 10 \). Floating point numbers are outputted with 3 decimals using `format()`, e.g., `t_contact = {:.3f}.format(t_contact)`.)*
#### Step 4: Calculating the Average Coordinates
The code calculates the average x- and y-coordinates using the formulas:
\[ \overline{x} = \frac{1}{N} \sum_{i=1}^{N} x_i \]
\[ \overline{y} = \frac{1}{N} \sum_{i=1}^{N} y_i \]
where \( x_i, y_i \) are the coordinates \( x(t), y(t) \) in the trajectory, and \( N \) is the number of data points that you calculated.
Store the result as a list in the variable `center`:
\[ \text{center} = [\overline{x}, \overline{y}] \]
**Example Output:**
```plaintext
center = [0.917, ....
![**Study of Dropped 2D Harmonic Oscillator**
**Introduction:**
A mass \( m \) is held by two perpendicular identical springs in space in the \( x\)-\(y \) plane and is dropped from a height \( z_0 \) under the influence of gravity. Let's call this setup the "dropped 2D harmonic oscillator." The mass moves in a potential described by:
\[ U(x, y, z) = \frac{1}{2}k(x^2 + y^2) + mgz \]
The motion ends when the mass hits the ground at \( z = 0 \). The exact expression for \( U(x, y, z) \) is given for context but is not necessary for solving the problem directly.
**Trajectory of the Mass:**
The mass follows a trajectory, which is its curve in space, \( \mathbf{r}(t) = (x(t), y(t), z(t)) \), given by:
\[
\begin{align*}
x(t) &= A \cos(\omega t) \\
y(t) &= B \cos(\omega t + \phi) \\
z(t) &= -\frac{1}{2}gt^2 + z_0
\end{align*}
\]
where the amplitudes \( A \), \( B \), and the phase difference \( \phi \) are determined by the initial conditions, and the frequency \( \omega \) is:
\[ \omega = \sqrt{\frac{k}{m}} \]
**Given Conditions:**
- \( A = 1 \)
- \( B = 2 \)
- \( \phi = \pi/3 \)
- \( \omega = 0.5 \)
- \( g = 9.81 \)
**Task:**
Develop a program that:
1. Reads the initial drop height \( z_0 \) from user input.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffe5146e3-5a58-4bea-b8ad-cd2d66a98667%2Ff393124c-d710-4726-8ae0-bfbc0003ee9b%2Fyulwjw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Study of Dropped 2D Harmonic Oscillator**
**Introduction:**
A mass \( m \) is held by two perpendicular identical springs in space in the \( x\)-\(y \) plane and is dropped from a height \( z_0 \) under the influence of gravity. Let's call this setup the "dropped 2D harmonic oscillator." The mass moves in a potential described by:
\[ U(x, y, z) = \frac{1}{2}k(x^2 + y^2) + mgz \]
The motion ends when the mass hits the ground at \( z = 0 \). The exact expression for \( U(x, y, z) \) is given for context but is not necessary for solving the problem directly.
**Trajectory of the Mass:**
The mass follows a trajectory, which is its curve in space, \( \mathbf{r}(t) = (x(t), y(t), z(t)) \), given by:
\[
\begin{align*}
x(t) &= A \cos(\omega t) \\
y(t) &= B \cos(\omega t + \phi) \\
z(t) &= -\frac{1}{2}gt^2 + z_0
\end{align*}
\]
where the amplitudes \( A \), \( B \), and the phase difference \( \phi \) are determined by the initial conditions, and the frequency \( \omega \) is:
\[ \omega = \sqrt{\frac{k}{m}} \]
**Given Conditions:**
- \( A = 1 \)
- \( B = 2 \)
- \( \phi = \pi/3 \)
- \( \omega = 0.5 \)
- \( g = 9.81 \)
**Task:**
Develop a program that:
1. Reads the initial drop height \( z_0 \) from user input.
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