A mass m is held by two perpendicular identical springs in space in the x-y plane and is dropped from a height zo under the influence of gravity (let's call this the "dropped 2D harmonic oscillator"). The mass moves in a potential 1 U (x, y, z) = -k(x² + y²) + mgz 2 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 where the amplitudes A, B and the phase difference are determined by the initial conditions, and the frequency is w = √k/m. 9.81, write a program that Given A = 1, B = 2, p= π/3, w = 0.5, and g = 1, reads the initial drop height zo from user input;
A mass m is held by two perpendicular identical springs in space in the x-y plane and is dropped from a height zo under the influence of gravity (let's call this the "dropped 2D harmonic oscillator"). The mass moves in a potential 1 U (x, y, z) = -k(x² + y²) + mgz 2 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 where the amplitudes A, B and the phase difference are determined by the initial conditions, and the frequency is w = √k/m. 9.81, write a program that Given A = 1, B = 2, p= π/3, w = 0.5, and g = 1, reads the initial drop height zo from user input;
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![**Title: Dropped 2D Harmonic Oscillator**
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 the "dropped 2D harmonic oscillator"). The mass moves in a potential:
\[ U(x, y, z) = \frac{1}{2} k (x^2 + y^2) + mgz \]
where \( k \) is the spring constant and \( g \) is the acceleration due to gravity.
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, \( \mathbf{r}(t) = (x(t), y(t), z(t)) \), is given by:
\[ x(t) = A \cos(\omega t) \]
\[ y(t) = B \cos(\omega t + \phi) \]
\[ z(t) = -\frac{1}{2} g t^2 + z_0 \]
where the amplitudes \( A \), \( B \), and the phase difference \( \phi \) are determined by the initial conditions, and the frequency is \( \omega = \sqrt{\frac{k}{m}} \).
Given:
\[ A = 1 \]
\[ B = 2 \]
\[ \phi = \pi / 3 \]
\[ \omega = 0.5 \]
\[ g = 9.81 \]
Write a program that:
1. Reads the initial drop height \( z_0 \) from user input.
**Explanation of Equations and Variables:**
- \( x(t) \) and \( y(t) \) describe the oscillatory motion of the mass in the x and y directions due to the springs.
- \( z(t) \) describes the parabolic motion of the mass under gravity after being dropped from height \( z_0 \).
- \( A \) and \( B \) are the amplitudes of oscillation in the x and y directions, respectively.
- \( \phi \) is the phase difference between the x and y oscillations.
- \( \omega \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffe5146e3-5a58-4bea-b8ad-cd2d66a98667%2F02ffca6d-9ac6-4adf-8d87-8a672f8726db%2Fqxm5rxs_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Dropped 2D Harmonic Oscillator**
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 the "dropped 2D harmonic oscillator"). The mass moves in a potential:
\[ U(x, y, z) = \frac{1}{2} k (x^2 + y^2) + mgz \]
where \( k \) is the spring constant and \( g \) is the acceleration due to gravity.
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, \( \mathbf{r}(t) = (x(t), y(t), z(t)) \), is given by:
\[ x(t) = A \cos(\omega t) \]
\[ y(t) = B \cos(\omega t + \phi) \]
\[ z(t) = -\frac{1}{2} g t^2 + z_0 \]
where the amplitudes \( A \), \( B \), and the phase difference \( \phi \) are determined by the initial conditions, and the frequency is \( \omega = \sqrt{\frac{k}{m}} \).
Given:
\[ A = 1 \]
\[ B = 2 \]
\[ \phi = \pi / 3 \]
\[ \omega = 0.5 \]
\[ g = 9.81 \]
Write a program that:
1. Reads the initial drop height \( z_0 \) from user input.
**Explanation of Equations and Variables:**
- \( x(t) \) and \( y(t) \) describe the oscillatory motion of the mass in the x and y directions due to the springs.
- \( z(t) \) describes the parabolic motion of the mass under gravity after being dropped from height \( z_0 \).
- \( A \) and \( B \) are the amplitudes of oscillation in the x and y directions, respectively.
- \( \phi \) is the phase difference between the x and y oscillations.
- \( \omega \)
![### Calculating the Trajectory and Contact Points of an Oscillator
This tutorial will guide you through the process of calculating the trajectory of an oscillator, determining the contact point with the ground, and computing the average coordinates.
#### Step 2: Calculating Trajectory
2. **Trajectory Calculation:** The program calculates the trajectory \( \mathbf{r}(t) \) and stores the coordinates for discrete time steps \( \Delta t \) as a nested list `trajectory` that 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., \( z(t) \leq 0 \).
#### Step 3: Contact Points Computation
3. **Ground Contact Time:** Store the time for hitting the ground (i.e., the first time \( t \) when \( z(t) \leq 0 \)) in the variable `t_contact` and the corresponding positions in the variables `x_contact`, `y_contact`, and `z_contact`. Print the results as:
```plaintext
t_contact = 1.430
x_contact = 0.755
y_contact = -0.380
z_contact = ...
```
*(Output floating point numbers with 3 decimals using `format()`, e.g., `t_contact = {:.3f}.format(t_contact)`.)* The partial example output above is for \( z_0 = 10 \).
#### Step 4: Average Coordinates Calculation
4. **Calculate Average Coordinates:** The average \( x \)- and \( y \)-coordinates are calculated as follows:
\[
\bar{x} = \frac{1}{N} \sum_{i=1}^{N} x_i
\]
\[
\bar{y} = \frac{1}{N} \sum_{i=1}^{N} y_i
\]
where the \( x_i, y_i \) are the \( x(t), y(t) \) in the trajectory and \(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffe5146e3-5a58-4bea-b8ad-cd2d66a98667%2F02ffca6d-9ac6-4adf-8d87-8a672f8726db%2F4te0dpb_processed.png&w=3840&q=75)
Transcribed Image Text:### Calculating the Trajectory and Contact Points of an Oscillator
This tutorial will guide you through the process of calculating the trajectory of an oscillator, determining the contact point with the ground, and computing the average coordinates.
#### Step 2: Calculating Trajectory
2. **Trajectory Calculation:** The program calculates the trajectory \( \mathbf{r}(t) \) and stores the coordinates for discrete time steps \( \Delta t \) as a nested list `trajectory` that 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., \( z(t) \leq 0 \).
#### Step 3: Contact Points Computation
3. **Ground Contact Time:** Store the time for hitting the ground (i.e., the first time \( t \) when \( z(t) \leq 0 \)) in the variable `t_contact` and the corresponding positions in the variables `x_contact`, `y_contact`, and `z_contact`. Print the results as:
```plaintext
t_contact = 1.430
x_contact = 0.755
y_contact = -0.380
z_contact = ...
```
*(Output floating point numbers with 3 decimals using `format()`, e.g., `t_contact = {:.3f}.format(t_contact)`.)* The partial example output above is for \( z_0 = 10 \).
#### Step 4: Average Coordinates Calculation
4. **Calculate Average Coordinates:** The average \( x \)- and \( y \)-coordinates are calculated as follows:
\[
\bar{x} = \frac{1}{N} \sum_{i=1}^{N} x_i
\]
\[
\bar{y} = \frac{1}{N} \sum_{i=1}^{N} y_i
\]
where the \( x_i, y_i \) are the \( x(t), y(t) \) in the trajectory and \(
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