A horizontal tree branch is directly above another...

A horizontal tree branch is directly above another horizontal tree branch. The elevation of the higher branch is $9.65 \mathrm{~m}$ above the ground, and the elevation of the lower branch is $5.99 \mathrm{~m}$ above the ground. Some children decide to use the two branches to hold a tire swing. One end of the tire swing's rope is tied to the higher tree branch so that the bottom of the tire swing is $0.47 \mathrm{~m}$ above the ground. This swing is thus a restricted pendulum. Start. ing with the complete length of the rope at an initial angle of $14.2^{\circ}$ with respect to the vertical, how long does it take a child of mass $29.9 \mathrm{~kg}$ to complete one swing back and forth?

Key Concepts

Gravitational Acceleration

Gravitational acceleration is the acceleration imparted by the Earth’s gravitational field, typically approximated as 9.8 m/s² near the surface of the Earth. It plays a crucial role in determining the period of a pendulum, as the restoring force for the pendulum’s oscillation is directly proportional to gravity.

Independence from Mass

In the ideal model of a simple pendulum, the period of oscillation is independent of the mass of the bob. The mass cancels out in the derivation of the period formula, demonstrating that under ideal conditions, only the pendulum’s length and the acceleration due to gravity affect the period.

Small Angle Approximation

For small angular displacements from the vertical, the sine of the angle is approximately equal to the angle measured in radians. This simplification allows the pendulum’s motion to be treated as simple harmonic motion, leading to the standard formula for the period, which is valid for angles typically less than about 15°.

Simple Pendulum

A simple pendulum is an idealized physical system consisting of a mass suspended from a fixed point by a weightless, inextensible string or rod, which swings under the influence of gravity. Its period can be determined using the relationship between the length of the pendulum and gravitational acceleration, assuming no air resistance or friction.

Effective Length Determination

The effective length of a pendulum is the distance between the point of suspension and the center of mass of the swinging object. In practical applications such as a tire swing, this length is determined by subtracting the clearance from the full suspension height, taking into account any restrictions or attachments that adjust the free-swinging path.

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