The Proof is 'not-so' Trivial - Physics Edition

Check now, I just edited the post, might be an error, I didn't do it from scratch.

Yeah that's what I got I'll probably post another question later

..

Would you like an A2 based problem here for you :P? I'm thinking something like Astro, perhaps around luminosity.

Edit: What level are you at? I'm not sure, I want to find a suitable question.

Technically I'm at As, but I know quite a bit of the A2 syllabus. It's just that I haven't had enough exposure to some of the maths. An astrophysics question would be interesting, thanks.

I'll post a question in a little bit. Just need to check that my solution is correct.

Are you content with the Stefan-Boltzmann equation for luminosity?

Problem 9- Difficulty *

Consider a regular prism with a cross section that is an n-sided polygon (for a six sided example, you could take an ordinary pencil). The prism is placed on an inclined plane such that it rests on one of it's faces. If the co-efficient of static friction between the face and the plane is $\mu$, determine what condition $\mu$ must satisfy so that the polygon will not slide down the plane.

I did this via inspection, so I'm hoping there is more. Is the answer:

$\mu > tan\left(\dfrac{90(n-2)}{n}\right)$

I think you've given the wrong angle in the triangle. I don't have the answers for the question but I can post the second part if you want (really not sure whether my solution to that part is correct though).

I've been silly, this:

$\mu \geq tan\left(\dfrac{180}{n}\right)$

yep. I'll edit in part 2, but I'm not 100% sure how to solve it.

Problem 11**

A wheel of diameter D is rolling along a muddy road at speed v. Particles of mud attached to the wheel are continuously thrown off from all points.

(a) If $v^2 > \frac{Dg}{2}$ show that the maximum height above the road attained by flying mud, $H_{max}$, is given by:

$H_{max} = \frac{D}{2} + \frac{v^2}{2g} + \frac{D^2 g}{8 v^2}$

(b) Show that the range of mud released at this angle is given by:

Problem 10 (Posted Earlier)

Consider a two state system with an orthonormal basis |1> and |2>. These two vectors are eigenvectors of the flavour operator, with |1> representing an electron neutrino and |2> representing a muon neutrino.

If neutrinos are massless, the Hamiltonian for a system like this would have been:
H = E|1><1| + E |2><2|

where E is a positive real constant. If the neutrino originally started in the state $| \psi(t=0)>$ = |1> what is the state of the system at later times t?

Difficulty: *** (But this is the first part to a longer question I will ask, this should be very simple to anyone with a little knowledge of quantum mechanics)

Problem 11**

A wheel of diameter D is rolling along a muddy road at speed v. Particles of mud attached to the wheel are continuously thrown off from all points.

(a) If $v^2 > \frac{Dg}{2}$ show that the maximum height above the road attained by flying mud, $H_{max}$, is given by:

$H_{max} = \frac{D}{2} + \frac{v^2}{2g} + \frac{D^2 g}{8 v^2}$

(b) Show that the range of mud released at this angle is given by:

Solution 10

I'll add here if I manage to do the others parts.

Using the statement that initially $|\Psi(t=0)\rangle = |1\rangle$, it's equal to one of the eigenvectors.

$|\Psi (t) \rangle = e^{\frac{-iEt}{\hbar}}|1\rangle$

Got the first bit, not been able to do b) yet...