Temperature and the period of a pendulum For oscillations of...

Temperature and the period of a pendulum For oscillations of small amplitude (short swings), we may safely model the relationship between the period $T$ and the length $L$ of a simple pendulum with the equation $$T=2 \pi \sqrt{\frac{L}{g}}$$ where $g$ is the constant acceleration of gravity at the pendulum's location. If we measure $g$ in centimeters per second squared, we measure $L$ in centimeters and $T$ in seconds. If the pendulum is made of metal, its length will vary with temperature, either increasing or decreasing at a rate that is roughly proportional to L. In symbols, with $u$ being temperature and $k$ the proportionality constant, $$\frac{d L}{d u}=k L$$ Assuming this to be the case, show that the rate at which the period changes with respect to temperature is $k T / 2 .$

Temperature and the period of a pendulum For oscillations
of small amplitude (short swings), we may safely model the relationship between the period $T$ and the length $L$ of a simple pendulum with the equation
$$T=2 \pi \sqrt{\frac{L}{g}}$$
where $g$ is the constant acceleration of gravity at the pendulum's
location. If we measure $g$ in centimeters per second squared,
we measure $L$ in centimeters and $T$ in seconds. If the pendulum
is made of metal, its length will vary with temperature, either
increasing or decreasing at a rate that is roughly proportional to
L. In symbols, with $u$ being temperature and $k$ the proportionality
constant,
$$\frac{d L}{d u}=k L$$
Assuming this to be the case, show that the rate at which the period changes with respect to temperature is $k T / 2 .$

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Key Concepts

Simple Pendulum Model

This concept refers to the mathematical model that describes the oscillatory motion of a pendulum for small amplitudes. The period of a simple pendulum is given by the formula T = 2??(L/g), which shows that the period T depends on the length L of the pendulum and the gravitational acceleration g. Understanding this model is essential for relating the pendulum’s properties to its oscillation period.

Thermal Expansion

Thermal expansion explains how the dimensions of materials change with temperature. In the context of the pendulum, the length L changes with temperature according to the differential equation dL/du = kL, where k is a constant representing the proportional rate of expansion or contraction. This concept is crucial for analyzing how temperature variations influence the physical dimensions of the pendulum.

Chain Rule in Differentiation

The chain rule is a fundamental concept in calculus used to differentiate composite functions. In this problem, the period T depends on the length L, which in turn depends on temperature u. By applying the chain rule, one differentiates T with respect to L and then multiplies by the derivative of L with respect to u (which is given by kL). This process enables the determination of how the period T changes as a function of temperature.

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