Is tension in a string cumulative?

[MathewsMD]

Just to clarify, if a string has a threshold of 100N, can the string be bent so that the tension in different parts individually doesn't exceed 100N yet the total force of the tensions do?

My question arises because of this question:

http://imgur.com/90NGY5p
http://imgur.com/ObRq5QM

Essentially, in this question, the string is bent and has 2 different nodes and there are 3 tensions analyzed.
The maximum tension the string can withstand is 100N and if you look at the solution, the individual components do not exceed 100 N, yet the sum of them do.
Just wondering, is this allowed? I always thought the TOTAL tension the string could withstand is 100 N, and the solution for this questions says otherwise.

Any help with the problem would be great!

 

[arildno]

A string breaks LOCALLY, at the point where the (local) yield criterion is fulfilled first.

 

[MathewsMD]

A string breaks LOCALLY, at the point where the (local) yield criterion is fulfilled first.

In a massless string, wouldn't the tension be constant throughout?

Or do we analyze tension only in different sections when there are nodes in the string? I always thought tension was cumulative and that the sum of them, if over 100N in this case even if in they were in different segments (no single segment has a tension greater than or equal to 100N), would cause the string to break.

In this example:

In Case 1, if the top string has a tension of 75N and the bottom string had a tension of 50N, the tension due to the top mass alone would have been 25N, correct? The tension is cumulative, right?

I'm just struggling to understand if this same logic applies in Case 2. Wouldn't the tension in the left string be the addition of the tension of the lower (left) string's tension as well as the tension needed to pull the top (left) mass? Same idea with the right side? The circle is a pulley by the way, and I just wanted to confirm that tension is still in the string even after passing through the pulley. Any clarification on what I posted would be great.

 

[lasix]

My understanding of this question has 3 flower pots hanging from this string. probably a bad way to think of this question...

The diagram does not do the reality justice (IMO). All forces on the string do have different vectors so in theory the question/answer is accurate. However the total weight/gravitation pull on the flower pots, regardless of the hanging angle is still perpendicular to the ground.

In other words, the actual WEIGHT of the 3 flower pots is not effected by the tangential angle of the "force" applied. In conclusion the main thread holding the 3 in the air will exceed the 100N. Total weight does not get altered because of the angle the pots are hung on just because they are attacked to pullies - but not if they are individually tied off.

Hope that helped.

 

[lasix]

Just saw your picture.

case 1 string breaks

Case 2 string does not break because the force is shared.

 

[MathewsMD]

My understanding of this question has 3 flower pots hanging from this string. probably a bad way to think of this question...

The diagram does not do the reality justice (IMO). All forces on the string do have different vectors so in theory the question/answer is accurate. However the total weight/gravitation pull on the flower pots, regardless of the hanging angle is still perpendicular to the ground.

In other words, the actual WEIGHT of the 3 flower pots is not effected by the tangential angle of the "force" applied. In conclusion the main thread holding the 3 in the air will exceed the 100N. Total weight does not get altered because of the angle the pots are hung on just because they are attacked to pullies - but not if they are individually tied off.

Hope that helped.

Okay, thanks for the help. I think the one part I'm just struggling with is determining if the string would snap right now and also how to find out where it would snap? If there were two strings, does the top string's tension also depend on the bottom string's tension? Also, how exactly would tension be different if the pulley was frictionless and the string has mass? I don't necessarily need a full in-depth explanation here, but any reference to a good source would be appreciated!

 

[MathewsMD]

Just saw your picture.

case 1 string breaks

Case 2 string does not break because the force is shared.

Okay, and if in Case 2, the two strings were attached together at one point that then attached to the ceiling, the string would break then, right?

 

[lasix]

Okay, and if in Case 2, the two strings were attached together at one point that then attached to the ceiling, the string would break then, right?

Yes.

If there were two strings, does the top string's tension also depend on the bottom string's tension?

The force applied to the bottom string will be shared by the top string, however consider angles of how the force is being applied.

think of it this way. if you have 2 x 50N forces pulling perpendicular to the top string, then 100 N will be met.

If you have 2 x 50N pulling horizontal towards the center, then only 50N of force are applied to each end of the string and as the string stretches you would actually have slack in the middle of the string.

If you have a design like number 1, but with all the masses coming off on angles but still attached to a single vertical string, that is then attached to the main horizontal string the thread breaks.

Also, how exactly would tension be different if the pulley was frictionless and the string has mass?

friction of the pully decreases force transfer from the string. 50N down may = 48N with friction.

Weight of the string decreases the amount of total N that can be applied because the gravity on the string is adding to the force. In other words if you had a large enough string/ long enough string hanging from a sky scraper it would break under it's own weight.

 

[arildno]

No. Tension is NOT necessarily constant in a massless string.
That tension is constant along the string is only true when there are no other non-zero forces acting tangentially on the string.
If such other forces are present (friction, for example), tension will vary along the string even if it is massless.