Related Rates: Chain Rule, Motion, and Magnetic Flux

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Related rates

Problems that connect rates of changing quantities through an equation relating those quantities.

Chain rule for time rates

If F=f(x) and x varies with time, then dF/dt=(df/dx)(dx/dt).

Related-rates procedure

Identify quantities, write their relationship, differentiate, substitute known values, then solve and interpret.

Why differentiate before substituting?

Substituting too early can treat changing quantities as constants and erase their rates.

Negative rate

A negative rate means the measured quantity is decreasing under the chosen sign convention.

Quiz questions

1. Two vehicles are 4 mi east and 3 mi north of an intersection. Their signed coordinate rates are dx/dt = -80 mi/hr and dy/dt = 60 mi/hr. At that instant, how fast is their separation changing?Answer: -28 mi/hr
2. With east chosen as the positive x-direction, why is dx/dt negative for a vehicle traveling west?Answer: Its x-coordinate is decreasing.
3. A loop with area A lies in a magnetic field of strength B. If θ is the angle between the field and the loop's normal, what is the magnetic flux?Answer: Φ = BA cos(θ)
4. What equation expresses Faraday's law for the induced electromotive force E and magnetic flux Φ?Answer: E = -dΦ/dt
5. A loop of area A rotates in a constant magnetic field B at constant angular velocity ω, with θ = ωt. What is its induced electromotive force?Answer: E = ωBA sin(ωt)