Max Energy Stored in an Inductor

[Drakkith]

Homework Statement

Find the maximum energy stored in the inductor.

Homework Equations

$p(t) = 0.09375te^{-1000t}(1-500t)$

The Attempt at a Solution

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So, I've been working my way through a multi-apart problem and one part asks me to find the maximum energy stored in an inductor. Given the equation that I found for power above, I can find the maximum energy using my calculator by having it integrate from t=0 to the time where the graph crosses the x-axis. My calculator can also find the point that it crosses the x-axis, which turns out to be 0.02 0.002 seconds, however I'd like to know how to solve this by hand if possible.

 

[NascentOxygen]

It looks like it hinges on the term in brackets, that's where p(t) changes sign. Is that 500t or 50t?

 

[mfb]

If you want to integrate an exponential function multiplied with a polynomial (can you see how it has this shape?), partial integration is usually the best approach. Take the polynomial as part where you calculate the derivative and the exponential as part yo integrate, then you get something that is easier to integrate as next step.

 

[Drakkith]

It looks like it hinges on the term in brackets, that's where p(t) changes sign. Is that 500t or 50t?

It's 500t.

If you want to integrate an exponential function multiplied with a polynomial (can you see how it has this shape?), partial integration is usually the best approach.

Oh trust me I tried that. I have about half a page of work for it that I ended up putting a big X through when it turned out that I misunderstood what I was supposed to be doing (I thought it wanted me to integrate from 0 to infinity).

In any case, I think I solved the integral correctly, I just didn't have the right limits. What I'm more interested in is how to figure out the upper limit of the integral.

 

[mfb]

What I'm more interested in is how to figure out the upper limit of the integral.

The place where the energy in the inductor doesn't increase any more. p(t)=0. p(t) is a product of four factors, only two of them can become zero and do so at different points, these two points give you the integration limits (assuming the inductor has zero energy when the function first gets positive - that is not described in the part of the problem statement you showed).

 

[NascentOxygen]

It's 500t.

So (1 – 500t) crosses the x-axis at 0.002s (1/500) not your 0.02s?

 

[Drakkith]

So (1 – 500t) crosses the x-axis at 0.002s (1/500) not your 0.02s?

Crap, I forgot a zero when I typed it in. It's t=0.002.

The place where the energy in the inductor doesn't increase any more. p(t)=0. p(t) is a product of four factors, only two of them can become zero and do so at different points, these two points give you the integration limits (assuming the inductor has zero energy when the function first gets positive - that is not described in the part of the problem statement you showed).

Multiplying the terms in $p(t)$ gives me $p(t) = 0.009375te^{-1000t}-46.875t^2e^{-1000t}$.
Obviously the first limit is $t=0$.
Factoring out the exponential and then dividing it out gives me $0=0.09375t-46.875t^2$.
Solving for $t$ gives me $t=0$ and $t=0.002$

Wow. It's all so obvious now.

Thanks all!