Recent content by Haku

Right, but we don't know the direction of force for Fkx. We just know that they are static, so ma = 0. Hence the sum of the forces is 0. So should it just be Fax + Fkx = 0, then since we know Fax = -125N we can solve for Fkx right?

I have this question in my biomechanics class, and the way the teacher has solved it raised some questions with me. This is the snippet of work from the lecture slides: But, if we see the red variables acting as 'placeholders' for the value of respective forces, and the value of the force at...

I was wondering how you work out what values of s a Laplace transform exists? And what it actually means? The example given in class is an easy one and asks to calculate the Laplace transform of 3, = 3 * Laplace transform of 1 = 3 * 1/s. Showing this via the definition, where does the range of s...

My initial question was not really meant to ask about "If the limit exists, then necessarily the limit from either side exists" as this is straightforward. Rather it is this: We don't know the limit exists, yet we act as if it has (in some cases), but (from an analysis standpoint) we should...

If you are told something holds if the limit exists, and given a function f (specifically not piecewise defined), is it enough to show that the limit as x approaches c = the function evaluated at c? With a piecewise defined function, it is easy to check both sides of a potential discontinuity...

This is a link to the textbook https://www.google.co.nz/books/edition/Mathematical_Methods_for_Engineers_and_S/orOTiguKIR4C?hl=en&gbpv=0 We covered this in class also though. Fourier transforms, transform things in one domain to the frequency domain. So why would omega (angular frequency) be a...

But in this case they are asking for it to be transformed into the omega space, they ask for an expression for U(ω, t) where F{u} = U. In this case, do you think my expression for the resulting ODE (after taking Fourier transforms) is correct?

In my textbook I have the Fourier transform defined as follows (for f(t)): The question asks me to find an expression for U(ω, t), where U(ω, t) is the transformed equation (probably an ODE). Why would using omega be wrong? Isn't it just a dummy variable anyway? we have the transformed space as...

I know that e^-ix = cos(-x)-isin(x), but if we have e^-iwx does that equal cos(-wx) - isin(wx)? Thanks

My result seems to be right to me, when you solve via definition of Fourier transform you get F{uxx} + F{uxt} + F{utt} = 0 => Ʉtt + iωɄt - ω^2Ʉ = 0 (2nd order ODE) (Hence PDE has reduced to ODE) where F{u} = Ʉ. Ʉtt and Ʉt are the second and first order derivatives with respect to t (respectively).

I am wanting to transform the space coordinate. Where you said by definition u(x, t) = ..., what is that definition you are referring to? The Fourier transform? Because the way I have learned Fourier transforms is the following: The Fourier transform of f(t) is the integral over R of...

Ah I see. Thank you! Helped a lot!

Ah yes I see, I think I understand now, so by the second shifting property we have that the inverse Laplace transform of e^(-sc)F(s) = f(t-c)H(t-c), and in this example c = x^2/2 and F(s) = 1/s^2. So we get the inverse Laplace transform of e^(-sx^2/2)•1/s^2 = f(t-c)H(t-c), where c = x^2/2...

Ahhhh yes, it is because the lower limit of the integral is 0 right? So in this case you get; $$F(s) = \int_0^\infty t\,e^{-st}dt.$$ = 1/s^2 And we have the s-shifting variable e^-as, where as = sx^2/2 right? But in my definition of the s-shifting prop we used e^at as the shifting 'variable'...

Ah okay, I will try to integrate that now. As for F(s), why would it be tu(t)? the Laplace transform of t is 1/s^2. So why would the inverse of that change the original function? Also what was your easier way to calculate the inverse? Thank you!