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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:I simply don't know.
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:I simply don't know.
haruspex said: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.
haruspex said:If the load accelerates from rest at αL, how long will it take to turn through angle θ/2?
Right. So minimising time means maximising αL, which is not surprising but worth checking.whitejac said:View attachment 195980
I've never heard of SUVAT, but this is what you mean? Integrating theta twice would yield the second equation regardless.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]
haruspex said:Right. So minimising time means maximising αL, which is not surprising but worth checking.
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.
Yes, that should work.whitejac 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?
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:Yes, that should work.