Recent content by wrobel

yes My solution is as follows. Introduce a frame ##OXYZ## such that the axis ##OZ## is directed upwards. The sides of the groove a the planes defined by equations $$X=Z\tan\alpha ,\quad X=-Z\tan\alpha .$$ The center of the coin has the coordinates ##(0,y,z).## Let ##\varphi## stand for the...

See Chapter 3 Section 6: F. Riesz' Representation Theorem and Corollary 1 therein. In the case of ##\mathbb{C}## the isomorphism between ##X## and ##X'## is not linear but conjugate linear. Sometimes (in Sobolev spaces for example) the dual space can be realized in nontrivial way. In this...

On the other hand consider a real Hilbert space ##F## of sequences ##x=(x_1,x_2,\ldots)## with inner product $$(x,y)_F=\sum_{k=1}^\infty k^2x_ky_k,\quad \|x\|_F^2=\sum_{k=1}^\infty k^2x_k^2<\infty.$$ It is easy to see that the dual space ##F'## consists of sequences ##x'=(x_1',\ldots)## such...

Hope you are speaking about Hilbert spaces. If it is so you should separate two cases: a Hilbert space over ##\mathbb{R}## and over ##\mathbb{C}##. In the second case be careful with duality. For details see for example Yosida: Functional Analysis, Riesz Representation Theorem and near it

it is not simple harmonic motion

it is hard to answer questions that contain incorrect prerequisites in their formulations :)

the equation of motion has already been written what else do you want ?

Here I would like to draw colleagues attention to a class of problems that are simple from physics viewpoint but have pure geometric obstacles. I believe such problems can be discussed in basic courses of analytic geometry as well. A coin of radius ##r## and of mass ##m## is put in a chute such...

Actually it is a pendulum. The equations of motion are the same as ones for the pendulum.

You ask irrelevant question. Analyze formulas and make sure that ##as(t)\in [0,\pi]## for all ##t\ge 0##; ##s(t)## is a periodic function

the 2nd Newton is ##m\ddot s=F\cos(as)## and ##s(0)=0,\quad \dot s(0)=0##. multiply both sides of the equation by ##\dot s##: ##m\ddot s\dot s=F\dot s\cos(as)## or equivalently $$\frac{d}{dt}\Big(\frac{1}{2}m\dot s^2-\frac{F}{a}\sin(as)\Big)=0$$ finish it by yourself

picture is needed

A mechanical system consists of rigid bodies and particles. Sometimes we a priori know something about motion of some components of the system. For example a pendulum can only rotate or a particle can only slide along the inclined plane. These restrictions are called constraints. In more...

This is a well-known and very old fact: https://en.wikipedia.org/wiki/Tusi_couple This problem is kinematic. Let's solve it by kinematic means. At the picture ##O## is a center of the big circle; ##C## is a contact point of the circles. We must prove that a point ##A## on the rim of the small...