Deriving the Fourth Mass in a Hanging Masses Problem

[Stochastic13]

Homework Statement

There are four masses hanging by a rope from the
ceiling in the simplest arrangement possible, mass 4 is attached by the rope to mass three right above it,
mass three is attached by a rope to mass 2 right above it, mass 2 is attached by the rope to mass 1 right above it
and mass one is attached by the rope to the ceiling. So the masses are hanging vertically from the ceiling attached by the rope.
Two of the tensions and three of the masses have been measured.
We know: T₁ T₂ m₁ m₂ m₃ Show that the fourth mass can be expressed as

m₄ = (m₁T₂/T₁ - T₂) - m₂ - m₃

Homework Equations

F = mg

The Attempt at a Solution

We know that m₄g + m₃g + m₂g = T₂
so m₄ = (T₂/g) - m₂ - m₃ since multiplying the first term by m₁/m₁ is the same as multiplying the term by one, we get m4 = (m₁T2/m₁g) - m2 - m3 using the fact that T₁ - T₂ = m₁g and substituting this equation in the denominator we get m₄ = (m₁T₂/T₁ - T₂) - m₂ - m₃ QED

Is this right? Did I answer the question properly? Just seems like I cheated. If you can point me in the direction of a better answer I'd greatly appreciate it.

 

[Doc Al]

There are four masses hanging by a rope from the
ceiling.

Please describe the arrangement in more detail. A diagram would help.

 

[Stochastic13]

It's the simplest arrangement possible, mass 4 is attached by the rope to mass three right above it, mass three is attached by a rope to mass 2 right above it, mass 2 is attached by the rope to mass 1 right above it and mass one is attached by the rope to the ceiling. So the masses are hanging vertically from the ceiling attached by the rope. I also forgot to mention that I used the fact that T₁ - T₂ = m₁g in the attempted solution portion of my post, then I used this equation in the last part of my answer to arrive at the final conclusion.

 

[Doc Al]

Show that the fourth mass can be expressed as

m₄ = (m₁T₂/T₁ - T₂) - m₂ - m₃

I assume you missed a parentheses and meant to write:
m₄ = m₁T₂/(T₁ - T₂) - m₂ - m₃

But your work looks fine to me.

Of course there are many ways to express m4; this is just one of them. For example, since you know T2, you should be able to express m4 in terms of T2, m2, and m3.

 

[Stochastic13]

Thanks, it just seemed way to simple, so I wasn't sure if I did it right.