Problem 50 - Classical Mechanics

Question

[Computer] The differential equation (1.51) for the skateboard of Example 1.2 cannot be solved in terms of elementary functions, but is easily solved numerically. (a) If you have access to software, such as Mathematica, Maple, or Matlab, that can solve differential equations numerically, solve the differential equation for the case that the board is released from \(\phi_{\mathrm{o}}=20\) degrees, using the values \(R=5 \mathrm{m}\) and \(g=9.8 \mathrm{m} / \mathrm{s}^{2} .\) Make a plot of \(\phi\) against time for two or three periods. ( \(\mathbf{b}\) ) On the same picture, plot the approximate solution (1.57) with the same \(\phi_{\mathrm{o}}=20^{\circ}\) Comment on your two graphs. Note: If you haven't used the numerical solver before, you will need to learn the necessary syntax. For example, in Mathematica you will need to learn the syntax for "NDSolve”and how to plot the solution that it provides. This takes a bit of time, but is something that is very well worth learning.

Step-by-Step Solution

Answer

Use a numerical solver to plot the angle over time for a few periods and compare it with the approximate solution.

Step 1: Set Up the Differential Equation

First, identify the differential equation that you are dealing with from Example 1.2. The equation might typically look similar to \( \theta''(t) + f(R, g, \theta) = 0 \), where \( \theta(t) \) is the angle displacement over time, \( R \) is the radius, and \( g \) is the gravitational acceleration. For this problem, you have \( R = 5 \) meters and \( g = 9.8 \) m/s², and the initial condition is \( \phi_0 = 20^\circ \).

Step 2: Convert Initial Conditions

Convert the initial angle to radians as most computational software uses radians. So, \( \phi_0 = 20^\circ = \frac{\pi}{9} \) radians. Set this as the initial condition: \( \theta(0) = \frac{\pi}{9} \).

Step 3: Input Differential Equation in Software

Use software like Mathematica, MATLAB, or Maple to input the differential equation. For example, in Mathematica, you can use `NDSolve` to define the equation, initial conditions, and the time span over which you want to solve it. Ensure that you input the rotational equations of motion correctly with your given parameters.

Step 4: Solve Differential Equation Numerically

Execute the numerical solver to get the solution of \( \theta(t) \). In these kinds of numerical solutions, you specify the range of time you want to consider. Typically a few periods can be considered, where a period can be calculated using an approximate formula like \( T = 2\pi \sqrt{\frac{R}{g}} \), and solve for \( t \) ranging from \( 0 \) to a few multiples of \( T \).

Step 5: Plot the Solution

Use your chosen software to plot the solution \( \theta(t) \) over time. You should plot for at least two or three periods to visualize the oscillation of the angle.

Step 6: Plot Approximate Solution

On the same graph, plot the approximate solution given in equation (1.57) for comparison. If (1.57) is a linear approximation or a small-angle approximation, ensure to input it correctly in the plot.

Step 7: Compare and Analyze the Graphs

Visually compare the numerical solution to the approximate solution. Comment on the differences or similarities between the two. Note any discrepancies such as phase differences, amplitude differences, or damping effects that may not be present in approximate solutions.

Key Concepts

Mathematical Software

When tackling numerical differential equations, mastering mathematical software like Mathematica, Maple, or MATLAB can be incredibly beneficial. These tools are designed to handle complex calculations that are often intractable with manual methods. For example, in solving differential equations, features like `NDSolve` in Mathematica can numerically integrate equations to produce solutions over time. This function is particularly useful when elementary functions do not suffice.

Learning the syntax and capabilities of your chosen software can open up new avenues for solving complex problems. Spending time to understand how to input equations and initial conditions accurately will pay dividends in your ability to solve a wide range of numerical problems.

Using these tools, you can visualize solutions graphically. This visualization provides insights that pure numerical data could hide.

Initial Conditions

Initial conditions are crucial in solving differential equations because they define where the system starts. For instance, in the skateboard problem, the initial angle is given as 20 degrees. However, most mathematical software requires angles to be in radians. To convert degrees to radians, use the formula \[\phi_0 = 20^{\circ} = \frac{\pi}{9} \]Providing accurate initial conditions allows mathematical software to solve the equations correctly from the specified 'starting point.'

Without accurate initial conditions, your solution may be completely off, illustrating how sensitive some systems are to their starting values. Always double-check these inputs before running numerical solvers.

Graphical Analysis

Graphical analysis involves examining plots produced by numerical solutions. These plots represent the behavior of the system over time. For example, when you solve for \(\theta(t)\), the graph shows oscillations of the angle, offering a visual representation of the physical scenario modeled by the differential equation.

A plot can reveal critical information:

• Peak and trough points indicate maximum and minimum angle displacement.
• Comparison plots allow for easy identification of differences between numerical solutions and approximations.

Graphical analysis not only confirms the numerical results but also enhances understanding by making the trends and behaviors visually apparent.

Small-Angle Approximation

The small-angle approximation simplifies the analysis of oscillatory systems. It assumes that when angles are small, say less than 15 degrees, \(\sin\theta\approx\theta\) holds true. This approximation can convert complex trigonometric equations into simpler, more linear forms.

In the context of the skateboard example, an approximate solution might be used to compare against the numerical solution. By plotting both solutions together, you can see how closely the approximation matches the more accurate numerical model. Typically, differences become noticeable at larger angles, illustrating where the small-angle approximation breaks down.

This principle is beneficial for quick calculations, though its applicability is limited to scenarios with smaller angular displacements. Being aware of its limits is essential for accurate problem-solving.