Statics: find tension/wrap angle for pulley

[SoylentBlue]

Homework Statement

Given: T(sub-B) plus T(sub-B) prime = 36. Both T(sub-C) values are 0. T (sub-D) =36 and T (sub-D) prime = 12. Find values of T (sub-B) and T (sub-B) prime[/B]

Homework Equations

OK, so we need to find the wrap angle, which should be 180 + 30 -22, which comes to 188 degrees, which is 1.04 pi radians.
Then: T (tightside)/T (slackside) = e raised to wrap angle (in radians)
This yields a T (tightside)/T (slackside) of 2.84.
The book shows this ratio as 3.

Also, confused: the values of T (sub-D) are in the same ratio (3 to 1) yet their wrap value is clearly 180 degrees or 1 pi[/B]

Can someone point me in the right direction? Thank you.

The Attempt at a Solution

 

[Chestermiller]

This is a balance of moments problem, not a belt friction problem. But, in any event, in your equation for the ratio of the tensions, you left out the coefficient of friction.

Chet

 

[SoylentBlue]

Oh, so this is just a lever-arm problem? If we look at T (sub-D), it exerts a torque of (6 in)(36Lbs)=216 in-LB on the axle. Therefore, the tightside of the belt around pulley B, which is T (sub-B), must also exert 216 in-LB of force on the axle. We know its lever arm is 8 in, so the force must be 27LB. Is this the correct logic to apply here?

Thank you for helping...I am trying to wade through this statics course by myself, so I don't have access to professors or even other students to ask questions or debate logic.

 

[Chestermiller]

No. You need to include in the moment balance all four of these: T_B, T_B', T_D, and T_D'.

Chet