Recent content by JyJ

I thought the pressure would not have the ##\theta## component because of the symmetry and so there will be no ##\theta## component of the force as well. If ##\theta## is included I am stooped of what to do. I would assume in that case the flow velocity would have the tangential component, and...

Ahh, I see. I get that: $$v_r = \frac{r\frac{dh}{dt}}{2(\frac{r^2}{2}(\frac{1}{b}-\frac{1}{a} - h)}$$ Should I now use this in the Navier Stokes equation (without inertial terms) to find pressure?

Thanks for the corrections. I misunderstood what z_a is at first. I have calculated the volume again: $$ V = \pi r^2 (h+\frac{r^2}{4}(\frac{1}{a}-\frac{1}{b}))$$ Then, the rate of change of volume becomes: $$ \frac{dV}{dt} = \pi r^2 \frac{dh}{dt}$$ Could you please advise why we need it?

I am a bit confused as to why there is a minus sign in H(r) (the answer for the force has a plus). What I did is the following: if we position the spheres vertically, I thought of looking at the gap between the two spheres in a way that we move the bigger sphere upwards and the smaller one...

Thanks for your reply. Could you please hint me on how the first approximation for H can be achieved. I suppose since you mentioned for r small compared to the radii of the spheres, Taylor expansion should be used somewhere?

I solved a similar problem with a rigid plane and a circular disk of radius a. I followed the same procedure but for the limits of the integral it was easier to find them and they were just from 0 to a. However, here I am struggling to find the limits.

Homework Statement Show that the force resisting change of the minimum distance h between the surfaces of two rigid spheres of radii a and b which are nearly touching is: $$6\pi\frac{\mu}{h({a^{-1} + b^{-1}})^2}\frac{dh}{dt}$$ provided $$\frac{\rho h}{\mu}\frac{dh}{dt}$$ Homework Equations...

Thank you, I got it!

Yes, I did get an equation of a harmonic oscillator and, hence, found the frequency. However, my question is that is this sufficient for showing that ##\theta_0## is indeed stable equilibrium, i.e. that since we have equation of a harmonic oscillator, we conclude that it is stable equilibrium?

Thank you for your reply! Yes, you are correct there is a missing R in the denominator for w_0 and yes, I should not have treated θ_0 as a small quantity - my mistake. This all makes sense now though! The question also asks to show that θ_0 is a stable equilibrium, how can I can show this...

Homework Statement A circular hoop of radius R rotates with angular frequency ω about a vertical axis coincident with its diameter. A bead of mass m slides frictionlessly under gravity on the hoop. Let θ be the bead’s angular position relative to the vertical (so that θ = 0 corresponds to the...

Yes of course! Hope this will be sufficient:

Yes, I guess so since in what I have so far the units give something rather bizarre. When I was calculating v_0 I had: $$\vec{v_0} = r\dot{\theta}^2\mathbf{e}_\theta$$ in which I assumed $$\dot{r} = \dot{z} = 0$$ As for the angular momentum I don't quite see how factor m can appear in my...

Furthermore: an equation like $$ \dot{r} = \dot{r}\mathbf{e}_r + r\dot{\theta}\mathbf{e}_{\theta} + \dot{z}\mathbf{k}$$ is very confusing. How about$$ \dot{\vec r} = \dot{\rho}\mathbf{e}_\rho + \rho\dot{\theta}\mathbf{e}_{\theta} + \dot{z}\mathbf{k}\ \ ?$$ Yes I agree, it should be a vector -...

For the 3 component equations I obtained: $$ m(\ddot{r} - r(\dot{\theta})^2) = -\frac{Nr}{\sqrt(r^2+a^2)} \\ \frac{m} {r} d/dt(r^2\dot{\theta}) = 0 \\ m\ddot{z} = \frac{Na}{\sqrt(r^2+a^2)} - mg $$ where N is the magnitude of the normal force. After eliminating N and z, I indeed showed only one...