The discussion revolves around a problem involving the time taken for an object to slide down a circular path with friction, framed within a mathematical context. The original poster expresses constraints against using conservation of energy, seeking a purely kinematic approach to the problem.
The discussion is ongoing, with various participants providing insights into the formulation of the differential equation and suggesting methods for manipulation. Some participants express uncertainty about the approach, while others offer guidance on substitutions and the use of the chain rule to relate angular velocity and time.
There are constraints noted regarding the use of conservation of energy, and participants are also grappling with the implications of friction on the problem's complexity. The original poster has indicated a lack of experience with ordinary differential equations, which may affect the depth of the discussion.
Now put it in standard form and solve with integration factor method.eddiezhang said:
I agree and energy considerations are not allowed as an approach to the solution.haruspex said:
Vice versa.kuruman said:
Well, that's hard to define. There are often simple algebraic ways to obtain an equation which is effectively the energy conservation equation without using it as a principle. Likewise momentum, angular momentum.kuruman said:
Thanks. I like the symmetry of the abbreviation which truly conveys the symmetry of the idea it's trying to express.haruspex said:
Was this problem given to you by a mathematician?eddiezhang said:
AI is technically allowed, but the point of the report is to show a) understanding of the math used and b) solving things from the ground up (which is why I was discouraged from using energy concepts).erobz said:
Thanks, I will investigate this.kuruman said:
I have to write this for a math class. This was the scenario I chose to investigate, but if it doesn't resolve itself nicely or is too much effort, I can always just change topics.kuruman said:
Energy is fundamental. Once you go beyond Newtonian mechanics, there are no forces and you need energy-based methods: Lagrangian, Hamiltomian etc.eddiezhang said:
I see. I would be curious to know whether the person in charge of your class has actually solved this problem analytically before presenting to you as a choice and, if so, how.eddiezhang said:
I used the "bottom" expression $$\omega(\theta)= \left \{\left[ \frac{v_0^2}{R^2}- \frac{2g}{R}\frac{\mu}{(4\mu^2+1)}\right]e^{-\mu \pi} + \frac{2g}{R} \frac{(2\mu^2\cos\theta -\mu\sin\theta+\cos\theta)e^{2\mu \theta}}{4\mu^2+1} \right\}^{1/2}e^{-\mu~\theta}\tag{1}$$ derived here. Equation (1) gives the angular speed of the sliding block as a function of ##\theta## starting from the horizontal position ##\left(\theta =-\frac{\pi}{2}\right)## with initial speed ##v_0.## Setting ##v_0=0## and ##\theta=0##, the angular speed at the bottom of the track for this case is $$\omega(0)= \left \{\left[- \frac{2g}{R}\frac{\mu}{(4\mu^2+1)}\right]e^{-\mu \pi}+ \frac{2g}{R} \frac{(2\mu^2 +1)}{4\mu^2+1} \right\}^{1/2}.\tag{2}$$ To find at what value of ##\mu## this is equal to zero, we set the right hand side of equation (2) equal to zero and see if we can solve the resulting equation for ##\mu.## After the obvious simplifications, we get $$\begin{align}haruspex said:
Which is clearly infeasible for sufficiently large ##\mu##. See my reply on that other thread.kuruman said:
Shown below is a revised plot of ##\omega(\theta)## using the corrected expression with ##\mu=0.62.## The plot is, of course, unrealistic because the mass should reach zero velocity asymptotically.haruspex said:
Asymptotically wrt time, yes, but that is consistent with your plot. Approaching ##\theta=0## it looks linear wrt ##\theta##, ##\omega\approx -c\theta##, so ##\theta \approx Ae^{-ct}##.kuruman said: