Pendulum

Overview

Engineering reference data for Pendulum in dynamics.

Key Formulas

Newton's Second Law

$F = ma$

Force = mass × acceleration.

Kinetic Energy

$E_k = \frac{1}{2}mv^2$

Energy of motion.

Momentum

$p = mv$

Mass × velocity.

Work

$W = Fd\cos\theta$

Force × displacement × cos(angle).

Variables

Symbol	Description	Unit
$F$	Force	N
$m$	Mass	kg
$a$	Acceleration	m/s²
$v$	Velocity	m/s
$E_k$	Kinetic energy	J

Example Calculations

Using the oscillation period formula, the required length for a pendulum to achieve a specific period can be calculated.

Derived Formula: $l = \left(\frac{T}{2\pi}\right)^2 a_g$

Sample Lengths:

• For T = 1 s (typical for a pendulum clock): l ≈ 0.249 m
• For T = 10 s: l ≈ 24.9 m
• For T = 100 s: l ≈ 2487 m

Note: Pendulum clocks served as the world standard for accurate timekeeping for 270 years until the invention of the quartz clock in 1927.

Types of Pendulums

Pendulums are commonly differentiated into four main types:

• Compound (Physical) Pendulum: A rigid body suspended from a fixed horizontal axis. The body may oscillate in a vertical plane due to the action of gravity.
• Simple (Mathematical) Pendulum: An idealized version where the mass is concentrated in a single point connected to a horizontal axis with a weightless chord, oscillating in a vertical plane.
• Conical Pendulum: Similar to the simple pendulum, except the suspended mass moves in uniform circular motion in a horizontal plane.
• Torsional Pendulum: Consists of a disk fixed to a slender rod, which is fastened to a fixed frame. When twisted, the disk oscillates back and forth about the rod's axis.

References

• Original Source