Kinematics Belt and Pulley Problem

[whitejac]

But it also changes the acceleration, so maybe you don't want the minimum length.

I'm running out of variables here. The only other one not given is α_m, which is something I currently do not know how to find specifically.

 

[haruspex]

I'm running out of variables here. The only other one not given is α_m, which is something I currently do not know how to find specifically.

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]

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.

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]

R isn't related to time as the pulley isn't growing or shrinking with relation to time.

The time taken would be different with a different R, so the two are related.
If the load accelerates from rest at α_L, how long will it take to turn through angle θ/2?

 

[whitejac]

The time taken would be different with a different R, so the two are related.
If the load accelerates from rest at α_L, how long will it take to turn through angle θ/2?

I simply don't know.

 

[haruspex]

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]

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.

I've never heard of SUVAT, but this is what you mean? Integrating theta twice would yield the second equation regardless.

If the load accelerates from rest at αL, how long will it take to turn through angle θ/2?

Okay, a load accelerating from rest at α_L would have the position equation:
θ_L = θ_0L+ω_0Lt+1/2α_Lt²
when θ_L =θ_L/2 and I.C.'s are applied:
θ_L = α_Lt²
And t = SQRT[θ_L/α_L]

 

[haruspex]

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 α_L would have the position equation:
θ_L = θ_0L+ω_0Lt+1/2α_Lt²
when θ_L =θ_L/2 and I.C.'s are applied:
θ_L = α_Lt²
And t = SQRT[θ_L/α_L]

Right. So minimising time means maximising α_L, which is not surprising but worth checking.

In your post #23, you eliminated α_L and kept α_M. But α_L is what we need to maximise. So you need to revisit that and obtain the equation that relates α_L to τ_M, r, R, J_L and J_M.

 

[whitejac]

Right. So minimising time means maximising α_L, which is not surprising but worth checking.

In your post #23, you eliminated α_L and kept α_M. But α_L is what we need to maximise. So you need to revisit that and obtain the equation that relates α_L to τ_M, r, R, J_L and J_M.

Okay, so I have done most of this but am out of town and away from the computer unail Sunday night.

I have rearranged T_m into an expression that has α_L. 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?

 

[haruspex]

Okay, so I have done most of this but am out of town and away from the computer unail Sunday night.

I have rearranged T_m into an expression that has α_L. 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?

Yes, that should work.

 

[whitejac]

Yes, that should work.

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.
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.