2.7: The Chain Rule: Problem Solving

The thermal expansion of a metal rod shows the application of the Chain Rule when one physical quantity depends on another that varies with time. As the rod is heated, its length changes according to linear thermal expansion, while the temperature of the system varies quadratically with time.

For linear thermal expansion, the length L of the rod depends on temperature T such that the rate of change of length with respect to temperature is constant:

where L₀ = 2 m is the initial length of the rod, and α is the coefficient of thermal expansion.

The temperature varies with time according to a quadratic relationship, so its rate of change with respect to time is linear:

Because length depends on temperature and temperature depends on time, the Chain Rule gives the rate of change of length with respect to time as

Substituting the expressions above results in

Evaluating this expression at t = 10 s yields the instantaneous rate of change of the rod’s length at that moment:

This expression gives the required rate at which the 2-meter metal rod is elongating at 10 seconds.

A metal rod, when heated, elongates due to linear thermal expansion. The length depends on the initial length, the material’s coefficient of thermal expansion, and the change in temperature of the system.

Here, the temperature of the rod changes quadratically with time.

The objective is to find the instantaneous rate of change of the length of a 2-meter metal rod with a known coefficient of thermal expansion at time exactly 10 seconds.

Because length depends on temperature, and temperature depends on time, the chain rule is applied.

To simplify the chain rule calculation, the differentiation is separated into two distinct components.

First, the length is differentiated with respect to temperature. This differentiation gives a constant value: the product of the initial length and the expansion coefficient.

Second, the temperature is differentiated with respect to time using the power rule, giving a linear rate of change.

According to the chain rule, the final rate is the product of these two components. Substituting the known values at 10 seconds gives the required instantaneous rate of change of the length.