# Collar on a rim in a vertical frame [closed]

Suppose we have such construction as shown at the picture. The vertical frame starts from rest with constant acceleration $a.$ Find final angle $\theta,$ assuming no friction. Also plot $\dot\theta$ as function of $\theta.$

From Newton second law i have that $ma_{collar} = N\sin\theta = mg\cos\theta\sin\theta = \frac{1}{2}mg\sin{2\theta}.$ From this i can find $\theta_{maximum}.$ But i still need to plot the graph, i think i need to build some differential equation for it.

• I don't think you need to build a full differential equation: draw the forces on the mass and the relationship will follow because it's asking for $\dot\theta(\theta)$ and not $\dot\theta(t)$. I suppose you are allowed to neglect the moment of inertia of A - although it is clearly rotating about the axis perpendicular to the page and is of finite size as drawn... Jan 9, 2015 at 16:43
• clearly the mass A would like to follow the direction of the gravitational acceleration...combining this with the equivalence principle might lead you to solve this with just some basic trig... Jan 9, 2015 at 16:44
• @Floris thank you for noticing that i need to dance around $\theta$ not $t$. If you like you can issue your comment as an answer and i will accept.
– Yola
Jan 9, 2015 at 16:48

I don't think you need to build a full differential equation: draw the forces on the mass and the relationship will follow because it's asking for θ˙(θ) and not θ˙(t). In fact I wonder whether you can't simply rotate the frame of reference until you have a single "acceleration vector" (the combination of $a$ and $g$) pointing straight down, at which point you would be looking at a pendulum with non-small deflection...