- #1
eddiezhang
- 39
- 8
- Homework Statement
- I am currently reading this paper (https://www.researchgate.net/publication/373607009_Exact_Solution_of_the_Problem_of_Brachistochrone_with_Allowance_for_the_Coulomb_Friction_Forces), where the author models the brachistochrone (least time of descent) curve for a circular ball while considering coulomb friction.
I'm confused by the penultimate step (the working behind equation 18).
- Relevant Equations
- .5mv^2 + mgy = mgH
Others contained in file / attached in attempt
Conservation of energy is invoked, which I understand, but the author omits most of the steps. He arrives at (18):
I've tried to substitute (13) and (17) into the LHS of the energy conservation equation, but the result is quite messy and my gut feeling is that it doesn't simplify in a nice way to achieve equation (18). For example:
My Maths teacher has looked at this and is also confused as to how the author arrives at (18).
I realise that having to slog your way through this paper is not a trivial task, so any help or insight into how the author arrives at (18) is deeply appreciated. My gut feeling is that there is a simple physics principle applicable that I've missed from which (18) follows.
Relevant symbols used in the paper (for your convenience):
- μ is the coefficient of friction
- α is the angle formed between the unit tangent vector of the point on the curve being investigated; φ I believe is the same, but specifically as a function of time. The notation used is a little confusing.
- C1 is introduced as a constant of integration in (13)
- H is the starting height of of the ball above (0,0), i.e. the maximum y-value the ball possesses
Many thanks
I've tried to substitute (13) and (17) into the LHS of the energy conservation equation, but the result is quite messy and my gut feeling is that it doesn't simplify in a nice way to achieve equation (18). For example:
My Maths teacher has looked at this and is also confused as to how the author arrives at (18).
I realise that having to slog your way through this paper is not a trivial task, so any help or insight into how the author arrives at (18) is deeply appreciated. My gut feeling is that there is a simple physics principle applicable that I've missed from which (18) follows.
Relevant symbols used in the paper (for your convenience):
- μ is the coefficient of friction
- α is the angle formed between the unit tangent vector of the point on the curve being investigated; φ I believe is the same, but specifically as a function of time. The notation used is a little confusing.
- C1 is introduced as a constant of integration in (13)
- H is the starting height of of the ball above (0,0), i.e. the maximum y-value the ball possesses
Many thanks