For oscillations of small amplitude (short swings), we may...

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 $k T / 2$.

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 $k T / 2$.

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.

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.

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.

Transcript

00:01 Section 3 .6, problem 105, we're dealing with a problem that deals with the temperature and the period of a pendulum.

00:09 So if we're given the period of a pendulum, we're given its length, and we have a consonant with the force of gravity, then we're told that the period of this pendulum is 2 pi square root of l over g.

00:29 Now, if this is a metal pendulum, it changes with temperature.

00:33 And so we're also given the differential equation d l d u is equal to some constant times l so the where u is the temperature so it tells me the length of the pendulum is the change in the length is proportional to the length okay so as it changes with temperature what we're asked here to find is d l excuse me the we're asked to find is the change of the period with respect to the temperature.

01:13 So what we're asked to find here is they ask us to find dt, d .u.

01:23 How does the period change with the temperature? so i know according to the chain rule, dt, dt, dt, d .u is dtdl times dldu.

01:39 And i write it though because i already know dldu.

01:42 So this will give me dt d .u so let's go to a new page and let's figure out what we need to know then so what we have is t is equal to 2 pi square root of l over in g and we're given that dl d u so dl d u is equal to k l so we're asked to find is dt d u which we recognize is dt dl times d l d u so first step find d t d l okay so that means we need to differentiate this expression with respect to l so it's two pi and then you've got l over g to the one -half so you're differentiating that with respect to l so what that will give me is 2 pi times one -half and then you've got l over g to the my minus one half the derivative of l over g with respect to l which is one over g so that is d t d l and that's multiplied by d l well let's just go ahead and simplify this this is all of this right here that is d t d l and then it's multiplied by d l d u so if we simplify this what happens is you've got a two and a two that cancels and then what you end up with here is what pi over g...

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