hanging masses problem

[CaptainBlood]

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

 

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.

[Hal.]

I'll suggest that you put a diagram here to allow responders to put a visual concept in their minds . I know it may be simple to imagine but help the responders and they will give their views .

[imatfaal]

There probably is a more stylistic/intuitive way - but I cannot see anything that markedly improves on your. I would also, in an answer, be more explicit about my rearrangements and bracketing, and mention that as system is in equilibrium there are no net forces (ie tension balances gravity). Also in vectors as net force equals zero then T and mg should be opposite signs.

[CaptainBlood]

OK, thanks imatfaal.

[DrRocket]

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

 

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.

 

 

That is just fine. I have no idea why you might feel that you cheated.

 

Imatfaal's stylistic comments are helpful and might make your explanation easier to follow, but that is just icing on the cake.

 

You might want to take a look at the idea of "free body diagrams" as a device to make the relations among forces clear. For some reason they are not always taught in physics courses (I recall showing the technique to a nuclear physicist who was teaching a basic non-majors class and getting a "gee that's neat" response), but they are routine in engineering mechanics. See an engineering text on mechanics or statics, or just talk to an engineering student. Engineering calculations quite often have to be shown to a large, critical audience for approval and therefore clarity of exposition is as important as is getting the right answer. Free body diagrams are a big help.

Edited April 7, 2011 by DrRocket

[CaptainBlood]

Thanks Doc.

 

P.S. I guess I felt that I cheated because the problem seemed too simple, also I don't know how to include the free body diagram so everybody can see it that's why I tried to explain the best I could.  I'd appreciate if someone could tell me how I could do that.

Edited April 16, 2011 by CaptainBlood