- #31
whitejac
- 169
- 0
I'm running out of variables here. The only other one not given is αm, which is something I currently do not know how to find specifically.haruspex said:
Kinematics Belt and Pulley Problem
- Thread starter whitejac
- Start date
Click For Summary
Discussion Overview
The discussion revolves around a kinematics problem involving two pulleys connected by a belt, focusing on the relationships between torque, angular acceleration, and the motion of the system. Participants explore the implications of no slip between the pulleys, the calculation of angular accelerations, and the challenges in determining time and velocity profiles for the motion.
Discussion Character
- Homework-related
- Technical explanation
- Debate/contested
Main Points Raised
- One participant begins by outlining their approach using torque equations and angular acceleration, leading to a calculation of time for a given angular displacement.
- Another participant questions the assumption that the angular accelerations of both pulleys are the same, prompting a discussion on the implications of this assumption.
- Some participants argue that each pulley must rotate through the same angle to avoid slip, while others clarify that this does not imply equal angular acceleration.
- There is a suggestion to equate the belt movements to relate displacement, velocity, and acceleration, but uncertainty remains about how to generalize this relationship.
- Participants express confusion about how to derive velocity from torque and angular acceleration, with some noting that the equations may not apply under non-constant acceleration conditions.
- A piecewise function is proposed to describe the angular acceleration during the motion, indicating a strategy for acceleration and deceleration phases.
- Concerns are raised about distinguishing between different radii, moments of inertia, and torques in the calculations, highlighting the complexity of the problem.
Areas of Agreement / Disagreement
Participants do not reach a consensus on the assumptions regarding angular acceleration and the application of equations under varying conditions. Multiple competing views remain regarding the relationships between the variables involved in the system.
Contextual Notes
Limitations include unresolved assumptions about the nature of acceleration, the dependence on specific definitions of torque and moment of inertia, and the lack of clarity on how to relate angular motion to time without additional relationships provided in the problem statement.
- #32
- 42,883
- 10,519
Concentrate on how the time depends on R. Your equation in post #23 does not include time. You need an equation that relates R to time.whitejac said:
- #33
whitejac
- 169
- 0
I know it doesn't, that's why taking derivatives isn't something I can currently do. I don't know where to put the t because it seems like I'd just be placing it arbitrarily. R isn't related to time as the pulley isn't growing or shrinking with relation to time. The Radius reduces the acceleration linearly, as shown in the relationship but I have not seen from the beginning how I can conjure a t and justify it as a logical relationship.haruspex said:
- #34
- 42,883
- 10,519
The time taken would be different with a different R, so the two are related.whitejac said:
If the load accelerates from rest at αL, how long will it take to turn through angle θ/2?
- #35
whitejac
- 169
- 0
I simply don't know.haruspex said:
- #36
- 42,883
- 10,519
It's completely analogous to linear motion. If an object accelerates at constant rate a, how far will it go from rest in time t? Use SUVAT or just integrate.whitejac said:
- #37
whitejac
- 169
- 0
haruspex said:
I've never heard of SUVAT, but this is what you mean? Integrating theta twice would yield the second equation regardless.
haruspex said:
Okay, a load accelerating from rest at αL would have the position equation:
θL = θ0L+ω0Lt+1/2αLt2
when θL =θL/2 and I.C.'s are applied:
θL = αLt2
And t = SQRT[θL/αL]
- #38
- 42,883
- 10,519
Right. So minimising time means maximising αL, which is not surprising but worth checking.whitejac said:
In your post #23, you eliminated αL and kept αM. But αL is what we need to maximise. So you need to revisit that and obtain the equation that relates αL to τM, r, R, JL and JM.
- #39
whitejac
- 169
- 0
haruspex said:
Okay, so I have done most of this but am out of town and away from the computer unail Sunday night.
I have rearranged Tm into an expression that has αL. Considering what you said about Radius R being the thing that changes the angular acceleration here, I took the derivative with respect to R and got an expression. To maximise alpha, I set the derivative equal to 0 and then solve for R. This became a heft fraction I cannot easily type on my phone.
Regardless, once I solve this, I will have the maximum apha and thus the Radius that minimizes the time required to go about a theta?
- #40
- 42,883
- 10,519
Yes, that should work.whitejac said:
- #41
whitejac
- 169
- 0
I've finally figured it out! I have to say, I am embarrassed how much I forgot of my multivariable calculator class. I thought I understood it better than this, but I suppose application is the real challenge.haruspex said:
Thank you so much for your patience. I will post back Sunday night with my attempt at part 2 and 3 but I believe it will not be too much of a challenge. It ought be similar in concept to this. The SUVAT has an expression with t, so I would relate it and the new found expression? Forgive me if this doesn't seem coherent. I'm not near paper to validate myself, but wanted to get the main idea straightened so I could mull it over during my trip.