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[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.

Short Answer

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

Step by step solution

01

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 \( \phi_0 = 20^\circ \).
02

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: \( \theta(0) = \frac{\pi}{9} \).
03

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.
04

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 \( T \).
05

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.
06

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.
07

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.

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Key Concepts

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

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.

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Most popular questions from this chapter

Prove that if \(\mathbf{v}(t)\) is any vector that depends on time (for example the velocity of a moving particle) but which has constant magnitude, then \(\dot{\mathbf{v}}(t)\) is orthogonal to \(\mathbf{v}(t) .\) Prove the converse that if \(\dot{\mathbf{v}}(t)\) is orthogonal to \(\mathbf{v}(t),\) then \(|\mathbf{v}(t)|\) is constant. [Hint: Consider the derivative of \(\mathbf{v}^{2}\).] This is a very handy result. It explains why, in two- dimensional polars, \(d \hat{\mathbf{r}} / d t\) has to be in the direction of \(\hat{\boldsymbol{\phi}}\) and vice versa. It also shows that the speed of a charged particle in a magnetic field is constant, since the acceleration is perpendicular to the velocity.

Two vectors are given as \(\mathbf{b}=(1,2,3)\) and \(\mathbf{c}=(3,2,1)\). (Remember that these statements are just a compact way of giving you the components of the vectors.) Find \(\mathbf{b}+\mathbf{c}, 5 \mathbf{b}-2 \mathbf{c}, \mathbf{b} \cdot \mathbf{c},\) and \(\mathbf{b} \times \mathbf{c}\).

In case you haven't studied any differential equations before, I shall be introducing the necessary ideas as needed. Here is a simple excercise to get you started: Find the general solution of the firstorder equation \(d f / d t=f\) for an unknown function \(f(t) .\) [There are several ways to do this. One is to rewrite the equation as \(d f / f=d t\) and then integrate both sides.] How many arbitrary constants does the general solution contain? [Your answer should illustrate the important general theorem that the solution to any \(n\) th-order differential equation (in a very large class of "reasonable" equations) contains \(n\) arbitrary constants.]

Imagine two concentric cylinders, centered on the vertical \(z\) axis, with radii \(R \pm \epsilon,\) where \(\epsilon\) is very small. A small frictionless puck of thickness \(2 \epsilon\) is inserted between the two cylinders, so that it can be considered a point mass that can move freely at a fixed distance from the vertical axis. If we use cylindrical polar coordinates \((\rho, \phi, z)\) for its position (Problem 1.47 ), then \(\rho\) is fixed at \(\rho=R,\) while \(\phi\) and \(z\) can vary at will. Write down and solve Newton's second law for the general motion of the puck, including the effects of gravity. Describe the puck's motion.

Find the angle between a body diagonal of a cube and any one of its face diagonals. [Hint: Choose a cube with side 1 and with one corner at \(O\) and the opposite corner at the point (1,1,1) . Write down the vector that represents a body diagonal and another that represents a face diagonal, and then find the angle between them as in Problem 1.4.]

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