Taylor Series and Oscillating Buildings 2

Summary

In this video, the speaker derives the differential equation for the motion of a simple pendulum using the Taylor series approximation for the sine function. The system is modeled with a mass hanging from a frictionless, massless string, with gravity as the only force. The differential equation is simplified by approximating sine(theta) as theta for small displacements, leading to the equation of a simple harmonic oscillator. This approximation holds for angles up to about 15 degrees, at which point the pendulum behaves like an oscillator, transferring between potential and kinetic energy as it swings.

Takeaways

• 😀 A simple pendulum consists of a mass hanging from an ideal, massless string with no friction or air resistance, and only gravity affects its motion.
• 😀 The goal is to derive the differential equation describing the motion of the pendulum and connect it to the behavior of tall buildings.
• 😀 The motion of the pendulum can be modeled by the angle of displacement (theta), and the primary force at play is gravity.
• 😀 The gravity vector has two components: one parallel to the string (perpendicular to the ball's motion) and one tangential to the motion of the ball.
• 😀 The tangential force acting on the ball is given by -mg * sin(theta), where the negative sign represents gravity acting as a restorative force.
• 😀 Displacement (s) is defined as l * theta, where l is the length of the string and theta is the angle of displacement.
• 😀 By differentiating displacement with respect to time, a second-order differential equation for the system is derived: l * theta'' = -mg * sin(theta).
• 😀 After simplifying, the equation becomes theta'' = - (g/l) * sin(theta), which is a second-order differential equation involving a transcendental function.
• 😀 To simplify the equation, the Taylor series approximation for sine is used, which approximates sin(theta) as theta when theta is small (small angle approximation).
• 😀 The resulting simpler equation, theta'' = -k * theta (where k = g/l), describes simple harmonic motion, valid for small angles (around 15 degrees or less).

Q & A

What is the goal of the video?

The goal of the video is to derive the differential equation that describes the motion of a simple pendulum and connect it to the motion of tall buildings.

What assumptions are made in the pendulum system?

The assumptions are that the string is massless and does not stretch, and that there is no friction or air resistance. The only force considered is gravity.

How is the gravity vector broken down in the pendulum system?

The gravity vector is broken into two components: one parallel to the string (perpendicular to the ball's motion) and one tangential to the ball's motion. The tangential component is responsible for the ball’s acceleration.

Why is the tangential force considered negative?

The tangential force is negative because gravity acts as a restorative force, always pointing opposite to the direction of displacement.

How is displacement related to the angle of the pendulum?

The displacement (s) is related to the angle (theta) by the formula s = l * theta, where l is the length of the string.

What differential equation is derived for the pendulum's motion?

The derived differential equation is θ'' = - (g/l) * sin(θ), where g is the acceleration due to gravity, l is the length of the string, and θ is the angle of displacement.

What is the issue with solving the derived differential equation?

The differential equation is a second-order equation involving the transcendental function sin(θ), which makes it difficult to find a closed-form solution.

How is the problem with the equation resolved?

The issue is resolved by approximating sin(θ) with θ using the small angle approximation, where higher-order terms like θ³ are neglected.

What does the simplified differential equation describe?

The simplified differential equation, θ'' = -k * θ (where k = g/l), describes a simple harmonic oscillator, where the acceleration is proportional to the displacement, similar to a mass-spring system.

For what range of θ does the small angle approximation hold?

The small angle approximation holds for angles less than approximately 15 degrees or 1/4 radian, beyond which the approximation becomes less accurate.