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Magazine Chapter 15: Problem 99
For a simple pendulum, find the angular amplitude \(\theta_{m}\) at which the restoring torque required for simple harmonic motion deviates from the actual restoring torque by \(1.0 \%\). (See "Trigonometric Expansions" in Appendix E.)
Short Answer
Expert verified
The angular amplitude is approximately 0.346 radians.
Step by step solution
01
Understand Torque in a Pendulum
For a simple pendulum, the actual restoring torque is given by \( \tau = -mgL\sin(\theta) \), where \( m \) is the mass, \( g \) is the acceleration due to gravity, \( L \) is the length of the pendulum, and \( \theta \) is the angular displacement. For simple harmonic motion (SHM), this is approximated by \( \tau_{SHM} = -mgL\theta \).
02
Calculate First-Order Approximation
By using the Taylor series expansion for small angles, \( \sin(\theta) \approx \theta - \frac{\theta^3}{6} \). For small \( \theta \), the first order approximation is \( \sin(\theta) \approx \theta \). But, this approximation begins to deviate as \( \theta \) increases.
03
Determine Percentage Deviation
We want the deviation to be 1%. Thus, the condition is \( |\ \tau_{actual} - \tau_{SHM} \ | / \tau_{actual} = 0.01 \). Simplifying gives \( | mgL\theta - mgL\theta + mgL\frac{\theta^3}{6} | / mgL\sin(\theta) = 0.01 \).
04
Solve for \( \theta_m \)
Substitute the approximations for \( \sin(\theta) \) using the Taylor expansion: \[ \left| \frac{mgL\frac{\theta^3}{6}}{mgL\left(\theta - \frac{\theta^3}{6}\right)} \right| = 0.01 \]. Simplifying this equation, cancel terms, and solve for \( \theta \). This results in \( \theta^2 = 0.12 \times \theta \), resulting in \( \theta_m \approx 0.346 \) radians.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Angular Amplitude
Angular amplitude refers to the maximum angular displacement of a pendulum from the vertical rest position. Essentially, it is how far the pendulum swings away from its central, or equilibrium, position. Angular amplitude is crucial because it determines the energy stored in the pendulum's motion. The larger the amplitude, the more energy. In simple harmonic motion, which we will discuss shortly, energy conservation and small angle approximations are often used to simplify calculations.
- For a simple pendulum, the amplitude is usually small enough such that the sine of the angle can be approximated as the angle itself (&\( \sin(\theta) \approx \theta \) for small \( \theta \)).
- However, as angles increase, this approximation becomes less accurate, leading to differences between actual motion and the idealized simple harmonic motion.
Restoring Torque
In a simple pendulum, restoring torque is what brings the pendulum back to its equilibrium position. When a pendulum swings away from the vertical, gravity creates this torque, aiming to pull it back to the center. The restoring torque for a pendulum is calculated by (&\( \tau = -mgL\sin(\theta) \)), where \( m \) is the mass, \( g \) is the gravitational acceleration, \( L \) is the length of the pendulum, and \( \theta \) is the angle of displacement. The minus sign indicates that the torque acts in the direction opposite to the displacement.
- For small angles, it's approximated by (&\( \tau_{SHM} = -mgL\theta \)), assuming simple harmonic motion.
- This approximation works well when \( \theta \) is small because \( \sin(\theta) \approx \theta \).
- The deviation from simple harmonic motion starts becoming significant as \( \theta \) increases, which would require adjusting calculations for accuracy.
Simple Harmonic Motion
Simple harmonic motion, or SHM, is a type of periodic oscillatory motion where the force acting to restore an object to its equilibrium position is directly proportional to the displacement and acts in the opposite direction. It provides a simplified model of oscillations where, particularly in pendulums, can describe the behavior under assumptions of small angles.
- For a pendulum in SHM, the restoring torque equation simplifies from (&\( \tau = -mgL\sin(\theta) \)) to (&\( \tau_{SHM} = -mgL\theta \)) under small angle approximations.
- SHM assumes that the motion is a perfect sine wave, signifying constant energy distribution across its cycle.
- Any deviation in the actual restoring force from this approximation leads to the need to consider modifications, especially for angles where such simplifications break down.
Taylor Series Expansion
The Taylor Series Expansion is a mathematical technique used to approximate more complex functions through a series of simpler polynomial terms. In pendulum physics, we use it to simplify the equation for restoring torque when dealing with small angles. For example, the sine function is replaced by its Taylor expansion: (&\( \sin(\theta) \approx \theta - \frac{\theta^3}{6} \)) for small \( \theta \). This approximation significantly aids in the analysis by reducing the problem's complexity when using \( \theta \) alone instead of \( \sin(\theta) \).
- The first-order approximation (&\( \sin(\theta) \approx \theta \)) is used in conditions where high precision isn't necessary and \( \theta \) is minimal.
- In cases requiring more accuracy, such as the calculation of angular amplitude at which torque deviation reached 1%, the additional term \( \frac{\theta^3}{6} \) helps capture the deviation from true simple harmonic motion.
- This method is crucial in simplifying equations to make practical theoretical predictions about a pendulum's behavior.
