A system of pulleys is loaded with an m1 = 106 pound weight and an m2 = 23 pound weight

Question

The tensions t₁ and t₂ in the ropes and the acceleration a of the 23-pound weight are modeled by the system below, where t₁ and t₂ are measured in pounds and a is in feet per second squared.

I need help setting this up, I keep reading it and my brain just can't comprehend it.

t₁ - 2t₂ + 0

t₁ - 2a = 106

t₂ + a = 23

Answer 1

There's a lot of info here in the description, so it's a bit confusing if you're trying to use all of it. Instead, I think, it's a system of simultaneous equations in 3 unknown variables, t₁ and t₂ and a.

(1) t₁ - 2t₂ = 0 (I think it's an = not a + which wouldn't give us any info and we can't solve from 2 equations only)

(2) t₁ - 2a = 106

(3) t₂ + a = 23

(2) and (3) both have an a term, so let's get rid of it by multiplying (3) by 2, and then adding (2) and (3) as follows:

t₁ - 2a = 106

+

2t₂ + 2a = 46

=

t₁ + 2t₂ + 0a = 152

(4) t₁ + 2t₂ = 152

Now we have 2 equations (1) and (4) in t₁ and t₂ both with one t₁ so we can subtract (4) from (1) to get rid of it:

t₁ - 2t₂ = 0

-

t₁ + 2t₂ = 152

=

0t₁ - 4t₂ = -152

=> t₂ = -152/-4 = 38

which you can substitute in (1) to get t₁ - 2(38) = 0 => t₁ = 76

and in (3) to get 38 + a = 23 => a = -15

Double-check by substituting found values in original equations to get (1) (76) - 2(38) = 0 as required, (2) (76) - 2(-15) = 106 as required, and (3) (38) + (-15) = 23 as required.