Any physics geeks able to help out a lowly math geek?
It’s getting close to the end of the school year and have covered all that I need to in my senior essentials of calculus class. So I decided to have them research a topic that related to calculus and teach it to the class. Many of these topics relate to physics - something I haven’t had in years.
One particular student has decided to focus on average and instantaneous current. We’ve found relevant formulas here but are having difficulty linking this to your typical high school physics class where the only current formula they’ve dealt with is V = IR.
I also need them to make real-world connections, something he’s struggling with as well.
The electric devices that would be the most understandable to an intro calculus student are capacitors and inductors. The voltage across a capacitor, for example, is V = Q / C, where Q is the charge on the capacitor; and the current flowing onto (or off of) the capacitor is I = dQ/dt. Similarly, the voltage across an inductor is proportional to the derivative of the current through it. You might also want to look into RC circuits or RLC circuits, if they’ve seen any basic differential equations.
Root-mean-square (RMS) values of sinusoidal currents and voltages illustrate another use of the calculus in electrical work.
The RMS value sought is the square root of the average square of a sinusoidal wave.
The average of the squares is the integral over 1 cycle, i.e. from 0 to 2π, of A[sup]2[/sup]sin[sup]2[/sup]xdx with this integral divided by 2π. This is equal to ½A[sup]2[/sup] and its square root is 0.707A.
For extra credit, there is an alternative way to solve the problem using trigonometric identities and knowing that the average of a sine wave over one cycle, or as the angle approaches infinity, is zero.
RLC circuits are especially nice, if the goal is to make connections, and if they’ve seen mass-spring systems. An RLC circuit is exactly analogous to a forced, damped harmonic oscillator. Swap out mass for inductance, spring constant for capacitance, and drag for resistance, and the equations (and their solutions) are all exactly the same.
To clarify, though, is this senior high school or senior college? If the former, they probably haven’t seen enough diffeq to do much with this, but if the latter, they should have.
A time delay relay (a relay with a capacitor across the coil to slow it down) is effectively an RLC circuit. Going through some basic calculus to figure out if the RLC circuit is overdamped, underdamped, or critically damped has the real world application of telling you whether or not the relay will chatter when opening and closing.
Another good example of real world calculus is PID feedback loops.
I don’t think they are going to be covering DiffEQs in essentials of calculus. Besides the RMS value I can’t think of any electrical concepts that can be solved without differential equations.
An AC circuit with a resistor R and voltage V(t) has current I(t) = V(t)/R. The power lost in the resistor (say, a lightbulb filament) is P=IV.
If the voltage is sinusoidal, say
V(t) = V[sub]0[/sub]sin(wt + d)
then the power loss is
P(t) = V[sub]0[/sub][sup]2[/sup]/R*sin[sup]2[/sup](wt + d)
For temperature concerns, the time-averaged power loss is what is interesting, and one can calculate it via an integral of P(t). One can turn it into a practical problem by asking what it the largest voltage amplitude V[sub]0[/sub] allowed if the resistor can get rid of no more than 1 W = 1 J/s of generated heat (which is where all that power is going.)
Well, calculus could be used to compute power and to show that for out-of-phase voltage and current the power is modified by the cosine of the phase difference.
Given V = Asin(x) and I = Bsin(x) the average power is the product of VI integrated over one cycle and divided by 2π. That’s the integral from 0 to 2π of ABsin[sup]2[/sup]xdx/2π. From the computation of RMS value above we know this = AB/2. That can be written as (A/√2)*(B/√2) which is the product of the RMS values of voltage and current.
For the case of a phase difference between voltage and current. V = Asin(x) and I = Bsin(x + a). Using the trigonometric formula, I can be written as Bsinxcosa + Bcosxsina. When this is multiplied by V we get VI = ABsin[sup]2[/sup]xcosa + ABsinxcosxsina. Notice that the first term is just the power for the in phase case above multiplied by the cosine of the phase difference or V[sub]RMS[/sub]*I[sub]RMS[/sub]*cosa. The second term when integrated over one cycle = 0.
Probably not helpful, but if you take a capacitor, an inductor, and a battery you’ve got the same thing as a damped, driven pendulum. Then use calculus to solve the pendulum.
Approximately. The pendulum is only a harmonic oscillator in the limit of small deflection angle. Mass on a spring (touched on by Chronos in post 5) is perhaps a cleaner analogy.