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Chapter 3: Problem 39

An idealized pendulum consists of a pointlike mass \(m\) on the end of a massless, rigid rod of length \(L\). Its amplitude, \(\theta\), is the angle the rod makes with the vertical when the pendulum is at the end of its swing. Write a numerical simulation to determine the period of the pendulum for any combination of \(m, L\), and \(\theta\). Examine the effect of changing each variable while manipulating the others.

Short Answer

Expert verified
Set up parameters, simulate motion using an integration method, and calculate the period for various \( m, L, \theta \) values to observe the effects on the period.

Step by step solution

01

Set Up Parameters

First, define the parameters of the pendulum: the mass \( m \), rod length \( L \), gravitational constant \( g \), and initial angle \( \theta \). Here, take \( g = 9.81 \text{ m/s}^2 \).
02

Establish the Equations of Motion

Use the small-angle approximation for simplicity, where the angular motion is described by \( \frac{d^2\theta}{dt^2} = -\frac{g}{L} \sin(\theta) \). For small angles, \( \sin(\theta) \approx \theta \).
03

Discretize the Motion

Apply numerical methods to simulate the pendulum's motion. Use the Euler method or the Runge-Kutta method to discretize the equations and iterate over small time steps, \( \Delta t \).
04

Implement Numerical Integration

In your simulation, update the angle and angular velocity using: \( \omega_{i+1} = \omega_i + \frac{d^2\theta}{dt^2} \Delta t \) and \( \theta_{i+1} = \theta_i + \omega_i \Delta t \). Iterate until the pendulum completes a full swing.
05

Measure the Period

Determine the period by tracking the time taken for one complete oscillation from \( \theta \) to \(-\theta \) and back. Sum the time steps \( \Delta t \) for one complete cycle.
06

Analyze the Effects of Changing Variables

Adjust \( m, L, \) and \( \theta \) one at a time while running the simulation to observe changes in the period. Note that for small amplitudes, the period is mostly affected by \( L \) and not \( m \).

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Key Concepts

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

Numerical Simulation
A numerical simulation involves using computational algorithms to predict the outcome of a physical system. In the context of this pendulum exercise, the goal is to determine the pendulum's period by using these techniques. Instead of analytically solving complex equations, which can be difficult or even impossible for non-linear systems, numerical simulations break the problem into many small, manageable steps.

Why use simulations? They offer a way to understand how changes in various parameters affect a system without solving equations by hand.
  • Support experimentation by allowing you to change mass, length, or angles to see different outcomes
  • Provide insights where analytical solutions are complicated
  • Enable the exploration of systems under conditions that are hard to replicate physically
Through these simulations, you can effectively model the pendulum's motion and gather data like its period under various circumstances.
Harmonic Motion
Harmonic motion is a type of periodic motion where the force acting on an object is proportional to the displacement of the object from its equilibrium position. In the case of a pendulum, it exhibits harmonic motion under certain conditions.

For the pendulum:
  • The restoring force is due to gravity and is proportional to the sine of the displacement angle.
  • For small angles, the motion is approximately simple harmonic, following the equation: \[\frac{d^2\theta}{dt^2} = -\frac{g}{L} \sin(\theta)\]
  • As the amplitude increases beyond certain limits, the motion becomes anharmonic, and more complicated calculations are required.
The pendulum's motion, when treated as harmonic, simplifies analysis and computation, allowing predictions of its behavior over time.
Small-Angle Approximation
The small-angle approximation is a useful mathematical simplification that assumes \(\theta\) is sufficiently small such that \(\sin(\theta) \approx \theta\). This approximation is pivotal in simplifying the equations of motion for pendulums, especially when dealing with harmonic motion.

Why use the small-angle approximation?
  • Simplifies calculations by reducing non-linear terms to linear ones
  • Valid for angles typically less than about 15 degrees, beyond which accuracy diminishes
  • Makes solving differential equations by hand or with simple numerical methods feasible
This leads to formulating the pendulum's motion as a simple harmonic oscillator. Although it limits accuracy for larger angles, it's often used to provide quick and practical insight into the system's dynamics.
Runge-Kutta Method
The Runge-Kutta method is a popular technique for solving ordinary differential equations numerically. It's more accurate than simpler methods, such as Euler's method, especially for systems with non-linear dynamics like pendulums.

How does the Runge-Kutta method work?
  • Calculates intermediate steps within a single time step to achieve higher accuracy
  • More stable and precise than basic methods
  • Widely used in simulations due to its ability to handle a wide range of differential equations effectively
For the pendulum exercise, the Runge-Kutta method allows us to accurately compute how the angle and velocity evolve over discrete time intervals, producing reliable estimates of the pendulum's period without significant numerical error.

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Most popular questions from this chapter

When you buy a helium-filled balloon, the seller has to inflate it from a large metal cylinder of the compressed gas. The helium inside the cylinder has energy, as can be demonstrated for example by releasing a little of it into the air: you hear a hissing sound, and that sound energy must have come from somewhere. The total amount of energy in the cylinder is very large, and if the valve is inadvertently damaged or broken off, the cylinder can behave like bomb or a rocket. Suppose the company that puts the gas in the cylinders prepares cylinder A with half the normal amount of pure helium, and cylinder B with the normal amount. Cylinder B has twice as much energy, and yet the temperatures of both cylinders are the same. Explain, at the atomic level, what form of energy is involved, and why cylinder B has twice as much.

(a) A mass \(m\) is hung from a spring whose spring constant is \(k\). Write down an expression for the total interaction energy of the system, \(U\), and find its equilibrium position. Wwhint \\{hwhint:hangfromspring\\} (b) Explain how you could use your result from part a to determine an unknown spring constant.

Explain in terms of conservation of energy why sweating cools your body, even though the sweat is at the same temperature as your body. Describe the forms of energy involved in this energy transformation. Why don't you get the same cooling effect if you wipe the sweat off with a towel? Hint: The sweat is evaporating.

(a) You release a magnet on a tabletop near a big piece of iron, and the magnet leaps across the table to the iron. Does the magnetic energy increase, or decrease? Explain. (b) Suppose instead that you have two repelling magnets. You give them an initial push towards each other, so they decelerate while approaching each other. Does the magnetic energy increase, or decrease? Explain.

The following table gives the amount of energy required in order to heat, melt, or boil a gram of water. $$\begin{array}{ll}\hline \text { heat } 1 \mathrm{~g} \text { of ice by } 1^{\circ} \mathrm{C} & 2.05 \mathrm{~J} \\ \text { melt } 1 \mathrm{~g} \text { of ice } & 333 \mathrm{~J} \\ \text { heat } 1 \mathrm{~g} \text { of liquid by } 1^{\circ} \mathrm{C} & 4.19 \mathrm{~J} \\ \text { boil } 1 \mathrm{~g} \text { of water } & 2500 \mathrm{~J} \\ \text { heat } 1 \mathrm{~g} \text { of steam by } 1^{\circ} \mathrm{C} & 2.01 \mathrm{~J} \\ \hline\end{array}$$ (a) How much energy is required in order to convert \(1.00 \mathrm{~g}\) of ice at \(-20^{\circ} \mathrm{C}\) into steam at \(137^{\circ} \mathrm{C}\) ? (answer check available at lightandmatter.com) (b) What is the minimum amount of hot water that could melt \(1.00 \mathrm{~g}\) of ice? (answer check available at lightandmatter.com)
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