A ball is falling vertically through air over a vibrating surface. The position of the surface is \(x_{w}(t)=A \cos \omega t\), where \(A=1 \mathrm{~cm}\) and \(\omega\) is called the angular frequency of the vibrations. The ball starts from a position \(x=10 \mathrm{~cm}\) at \(t=0 \mathrm{~s}\). The acceleration of the ball is given as: $$ a(x, v, t)=\left\\{\begin{array}{cl} -g & x>x_{w} \\ -g-C\left(x-x_{w}\right) & x \leq x_{w} \end{array} .\right. $$ where \(g=9.81 \mathrm{~m} / \mathrm{s}^{2}\) and \(C=10000.0 \mathrm{~s}^{-2}\). (a) Write down the equation you need to solve to find the motion of the ball. Include initial conditions for the ball. (b) Write down the algorithm to find the position and velocity at \(t_{i+1}=t_{i}+\Delta t\) given the position and velocity at \(t_{i}\). Use Euler-Cromer's scheme. (c) Write a program to find the position and velocity of the ball as a function of time. (d) Check your program by comparing the initial motion of the ball with the exact solution when the acceleration is constant. Plot the results. (e) Check your program by first studing the behavior when the vibrating surface is stationary, that is, when \(A=0 \mathrm{~m}\) and \(x_{w}=0 \mathrm{~m}\). Plot the resulting behavior. Ensure that your timestep is small enough, \(\Delta t=10^{-5} \mathrm{~s}\). What happens if you increase the timestep to \(\Delta t=0.02 \mathrm{~s}\) ? (f) Finally, use your program to model the motion of the ball when the surface is vibrating. Use \(A=0.01 \mathrm{~m}, \omega=10 \mathrm{~s}^{-1}\), and simulate \(5 \mathrm{~s}\) of motion. Plot the results. What is happening? (g) What happens if you increase the vibrational frequency to \(\omega=30 \mathrm{~s}^{-1}\) ? Plot the results. Can you explain the difference from \(\omega=10 \mathrm{~s}^{-1}\) ?

Step by step solution

01

- Equation Set-Up (Part a)

To find the motion of the ball, start with the given acceleration equations. Define the position of the vibrating surface as: \[ x_w(t) = A \cos(\omega t) \] The ball starts at position: \[ x(0) = 10 \mathrm{ cm} = 0.1 \mathrm{ m} \] With the initial velocity: \[ v(0) = 0 \mathrm{ m/s} \]The acceleration equations are:\[ a(x, v, t) = \begin{cases} -g & x > x_w \ -g - C(x - x_w) & x \leq x_w \end{cases} \]Where: \[ g = 9.81 \mathrm{ m/s^2} \] \[ C = 10000.0 \mathrm{ s^{-2}} \]

02

- Euler-Cromer Scheme (Part b)

In the Euler-Cromer method, update the position and velocity iteratively. Use the time step \(\Delta t \). The update rules are: 1. Update velocity: \[ v(t_{i+1}) = v(t_i) + a(x(t_i), v(t_i), t_i) \Delta t \] 2. Update position: \[ x(t_{i+1}) = x(t_i) + v(t_{i+1}) \Delta t \]Where acceleration \(a(x, v, t)\) is calculated based on the current position and velocity.

03

- Implement the Program (Part c)

Write a program to compute the position and velocity over time using the Euler-Cromer scheme. Initialize necessary variables, include the equations from Step 1, and implement the iterative update process.

04

- Exact Solution Comparison (Part d)

For validation, compare the numerical solution to the exact solution for constant acceleration. Plot both results to ensure initial conditions match.

05

- Stationary Surface Test (Part e)

Test the program with a non-vibrating surface \(A = 0 \mathrm{ m}\) and verify timestep sensitivity. Set \( \Delta t = 10^{-5} \mathrm{ s}\) and plot, then increase to \( \Delta t = 0.02 \mathrm{ s}\) and observe changes in the motion.

06

- Vibrating Surface Test at \(\omega=10 \mathrm{ s^{-1}}\) (Part f)

Use the program to model the ball's motion with \( A = 0.01 \mathrm{ m} \) and \( \omega = 10 \mathrm{ s^{-1}} \) over a 5-second simulation. Plot the position and velocity against time.

07

- Higher Frequency Vibration (Part g)

Increase the vibrational frequency to \( \omega = 30 \mathrm{ s^{-1}} \) and plot the results over 5 seconds. Compare and explain differences in motion from the \( \omega = 10 \mathrm{ s^{-1}} \) case.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Numerical integration

Numerical integration involves approximating the values of integrals using computational methods. In the context of this exercise, it is used to solve the equations of motion for a ball falling on a vibrating surface. This is important where analytical solutions are not feasible. The Euler-Cromer method is one such technique that updates the velocity and position iteratively. This method assumes small intervals of time ∆t, making it quite suitable for time-dependent problems. We start by initializing variables with given initial conditions and apply iterative steps to compute the next state of the system. This way, numerical integration helps to approximate the complex dynamics of systems accurately.

Acceleration with position-dependence

In mechanics, acceleration that varies with position and time can greatly affect an object's motion. For the given exercise, the ball's acceleration depends on whether it is above or below the vibrating surface:
When the ball is above the surface \( x > x_w \), it experiences a constant acceleration \( -g \). Below the surface \( x \leq x_w \), the acceleration formula includes an extra term \( -C(x-x_w) \) that accounts for the spring-like interaction with the vibrating surface.
This introduces a scenario where forces change dynamically with the ball's position, complicating the motion significantly. Such position-dependent acceleration makes analytical solutions complex, thus reinforcing the need for numerical methods like Euler-Cromer in solving these problems.

Vibrational surface dynamics

The motion of a ball on a vibrating surface introduces unique dynamics. The surface's position is given by \( x_{w}(t) = A \cos(\omega t) \), where A is the amplitude, and \( \omega \) is the angular frequency. These vibrations cause the surface to move up and down over time, affecting the ball's position and velocity accordingly.
Understanding this interaction is crucial because the oscillation impacts the ball by altering forces at various points in its trajectory. By experimenting with different values of amplitude and frequency, one can see a spectrum of behaviors from regular oscillation to chaotic motion. This demonstrates the complex, sensitive dependence of motion on initial conditions and external driving frequencies.

Python programming for physics simulations

Programming is an invaluable tool for physics simulations, enabling analysts to model intricate dynamical systems that are otherwise challenging to solve analytically. In this exercise, Python is used due to its extensive libraries and simplicity.
Here, you'll write a Python script to implement the Euler-Cromer method, iterating over time steps to update position and velocity. By using libraries like NumPy for numerical operations and Matplotlib for plotting results, you can visualize the ball's motion over time.
Python's versatility allows you to test different initial conditions and parameters effortlessly, thereby getting a deeper understanding of the physical phenomena. Additionally, it reinforces coding skills alongside learning physics, which is instrumental in modern scientific research and educational settings.