Why Small Angle Swing is Preferred for Pendulums in Simple Harmonic Motion
The recommendation to keep the angle of swing of a pendulum small is primarily based on the principles of simple harmonic motion (SHM). This article explores the reasons why keeping the angle of swing small is preferred and how it affects the behavior of the pendulum.
Linear Approximation and Simple Harmonic Motion
For small angles, typically less than about 15 degrees, the sine of the angle in radians is approximately equal to the angle itself. This means that the restoring force, which is proportional to the sine of the angle, behaves linearly. Consequently, the pendulum's motion can be accurately modeled as SHM, leading to predictable and consistent behavior.
The formula for the period of a pendulum in SHM is given by:
Period Independence
When the angle is small, the period of the pendulum, the time it takes to complete one full swing, is independent of the amplitude (angle of swing). This means that the pendulum will have a consistent period, making it easier to measure time accurately.
With a small angle, the period is given by the formula:
T 2π√[(L/g)]
Reduced Energy Loss
At larger angles, the pendulum experiences greater air resistance and potential energy loss due to friction at the pivot point. Keeping the angle small minimizes these energy losses, allowing the pendulum to maintain its motion more effectively. This is crucial for maintaining accuracy in timekeeping and other applications.
Simplified Calculations
When analyzing the motion of a pendulum using small angles, the mathematical calculations become simpler. This makes it easier to predict the behavior of the pendulum over time. Simplified calculations are essential for both theoretical understanding and practical applications.
Implications of Large Swing Angles
At large swing angles, a large amount of swing non-linearity occurs due to the changing interaction with gravity and leverage. This can make the pendulum's behavior less predictable and more difficult to analyze. However, a pendulum can be used as a means of tapping into gravity energy as a power source by inputting a force when the pendulum has a high angle and minimum velocity and then extracting force again at the bottom of the swing when the pendulum has maximum velocity. Sometimes science refuses to observe the simple but important stuff!
Pendulum Motion and Simple Harmonic Motion
Pendulum motion is not true Simple Harmonic Motion (SHM). In SHM, the force that tries to bring the mass back to the zero position is always proportional to the amount of displacement. However, for a pendulum, the movement is angular, and the restoring force, which pulls the bob back to the zero position, is not proportional to the angular displacement (θ). Instead, it is proportional to sinθ.
The time period of a pendulum in SHM is given by:
Small Angle Approximation
As a result, the time period changes depending on the angle of swing and it cannot be used as a perfect time-keeper. However, if we keep θ small, the pendulum can be made to approximate SHM. The relationship between the arc length and the perpendicular line can be seen in the diagram below:
Consider the following sketch showing three arcs each of radius 1. The angle θ in radians is the same as the length of the arc. The sinθ is the same as the line drawn perpendicular to the base. The arc is invariably longer than the perpendicular line. However, as the angle θ becomes smaller and smaller, the arc length θ comes closer to and becomes more or less coincident with the perpendicular line representing sinθ. Eventually, as θ approaches 0, sinθ becomes equal to θ.
Using the small angle approximation, sinθ ≈ θ, the pendulum's motion can be more closely modeled as SHM, making predictions and analyses easier and more accurate.
In conclusion, keeping the angle of swing small ensures that the pendulum behaves in a predictable manner, allowing for accurate measurements and analyses. This principle is crucial in various scientific and practical applications of pendulums.