Multivariable Calc Help Needed

[lost zenith]

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[indigo plinth]

@queen shale sorry to ping but if u get a chance, u mind taking a look at this?

[queen shale]

you can sorta assume omega is constant for now as you're told you can assume friction is negligible

10sin(theta)=0 theta=0,pi which makes sense as for theta=0 the GPE is as low as it'll get and at pi it's gone from increasing to decreasing in GPE (going up to coming down)

[indigo plinth]

The critical points have nothing to do with the gradient right?

[queen shale]

critical points are when the gradient is 0

(or undefined, but cos is a nice friendly function)

[indigo plinth]

I'm confused here because usually when we find gradient we take partial derivatives or x and y

but there is no x or y here

so instead do I take w.r.t theta and omega?

[queen shale]

refer to this

but yes, your variables are theta and omega this time

E'(theta, omega)=(10sin(theta), omega)

[indigo plinth]

Ok so that will be the gradient and I set that = 0 to find the critial points?

[queen shale]

yes

so you have either when omega=0 or when sin(theta)=0 which is at theta=0,pi

[indigo plinth]

hmm those r the only critical points?

[queen shale]

yeah

visualise it, just like i did with the theta derivative

when the rotational velocity is 0 it's gonna either stop forever or turn around, which is a critical point

[indigo plinth]

ahh ok ok

[queen shale]

stopping forever is stable equilibrium, instantaneous 0 velocity is an unstable equilibrium

https://www.khanacademy.org/math/multivariable-calculus/applications-of-multivariable-derivatives/optimizing-multivariable-functions/a/second-partial-derivative-test

use this

and you need to set up pairs of theta and omega

(0,0), (π,0) are your critical points

[indigo plinth]

ok

[indigo plinth]

Do u know how to do part c) above?

[queen shale]

if it's at an unstable equilibrium then it's got just GPE but no KE, E=10, then at the bottom we have theta=0 so we have 10=1/2 omega^2 -10 so omega is 2sqrt(10)

[indigo plinth]

What is GPE and KE?

[queen shale]

kinetic energy and gravitational potential energy

you calculate the total energy at the top and then have that equal to the total energy at the bottom because negligible energy was lost to friction

[indigo plinth]

But doesn't it say we r ignoring any eneregy added via perturbation or removed via friction?

[queen shale]

so you do E(pi, 0)=E(0, omega)

[queen shale]

yes

that's what i did

just have potential at the top, just have kinetic at the bottom

[indigo plinth]

ok

Also, when finding the critical points, we r setting the partial deriviatves = 0 right?

[queen shale]

yes

refer to the khan academy link

[indigo plinth]

I don't need to use Hessian Matrix to determine whether it's a saddle point or local max/min right?

[queen shale]

you can just use the equation on the khan academy page using f_xx • f_yy - (f_xy)^2

[indigo plinth]

Ok so I got that:

a) grad is (10sin(thetha), w) like u said
critical points r (0,0 and (pi, 0) like u said

b) I got that (0,0) is a local minimum point and (pi, 0) is a saddle point

I'm still not too sure what is going on for c)...

[queen shale]

local minimum means that no matter how you nudge it, it'll fall back down the 'slope' of your original function, saddle point or maximum means you can nudge it into a lower energy state if you want @indigo plinth

it wants to reach the lowest energy

[indigo plinth]

hmm ok

[indigo plinth]

The point at which the equilbrium is unstable is (pi, 0) right since that is the point which isn't the local minimum?

[queen shale]

take a look at this

you can see your critical points

if you were to nudge theta a little larger or a little smaller it would slip down into those pits

[queen shale]

yes

but at (0,0) you can see if you nudge it it's already at the lowest point it can be

[indigo plinth]

oh

Is our velocity equation simply E(thetha, omega)?

[queen shale]

no, that's our energy equation

hence that graph is a graph of the energy

and since things are at their most stable when they have the least potential energy

[indigo plinth]

yeah that's right

[queen shale]

it tracks, makes sense?

[indigo plinth]

yeah but then

how is it asking us for velocity without giving us a velocity equation 😭?

[queen shale]

velocity is omega!

you use conservation of energy when you know the starting point

[indigo plinth]

ohh, sorry sorry, i'm dumb

[queen shale]

you know the start and end points, so you plug those in and use conservation of energy

start: E(π,0)=10
end: E(0,w)=½w²-10
start=end energy
10=½w²-10
40=w²
w=2√10

[indigo plinth]

ohhh, given that the pendelum reaches the bottom, theta = 0?

[queen shale]

yes

[indigo plinth]

So we start the point where the equilibrium is unstable?

[queen shale]

yes

[queen shale]

it mentions you want to find the point on the same level curve, why do you think it needs to be on the same level curve?

@indigo plinth answer soon, please, I need to sleep imminently

[indigo plinth]

because the pendelum comes back?

[queen shale]

not quite

what actually is the function representing

[indigo plinth]

total energy

[queen shale]

so staying on the same level curve means you're keeping what constant?

[indigo plinth]

we can't lose energy, it has to remain the same

[queen shale]

correct

I need to sleep now, good luck with any additional parts

post them in a separate thread if you need any additional help, someone else will probably help out

[indigo plinth]

ok thanks!

gn!

[queen shale]

np, cya

[indigo plinth]

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