Resultant of Forces Can Equal Zero: 1,2,3,4?

[gracy]

Homework Statement

Resultant of which of a following may be equal to zero?
1)10 N, 10 N , 10N
2) 10 N, 10 N , 25 N
3) 10 N, 10 N , 35N
4)None of these
More than one option can be correct.

Homework Equations

Addition of vectors

The Attempt at a Solution

1) Resultant of these three forces can equal to zero if two of the forces make 120 degrees angle with each other and their resultant is opposite in direction with the third force.
Answer at the back of textbook says option 1 is correct. I don't know how can we be so sure that other two options are wrong?

 

[TomHart]

Hint: If you have a single force, how much force in the opposite direction would be required to balance out (or counter) that single force?

 

[Isaac0427]

What if the question was this (really, the problem does boil down to this):
Which of the options could be side lengths of a triangle?

How would you figure this out?

 

[gracy]

How would you figure this out?

Triangle Inequality Theorem??

(really, the problem does boil down to this):

I fail to understand how is my problem reduced to side lengths of triangle. Care to explain?

 

[gneill]

I fail to understand how is my problem reduced to side lengths of triangle. Care to explain?

How might you go about summing a set of force vectors graphically? If the resultant of the sum is to be zero, where must the tip of the "last" vector end up?

 

[Isaac0427]

Triangle Inequality Theorem??

Ding ding ding

 

[gracy]

How might you go about summing a set of force vectors graphically?

I was adding the vectors using parallelogram.

 

[cnh1995]

I was adding the vectors using parallelogram.

Try the polygon method.

 

[gracy]

I don't understand how A + B + R will be zero ? shouldn't it be 2R?

 

[gracy]

View attachment 104715
I don't understand how A + B + R will be zero ? shouldn't it be 2R?

 

[cnh1995]

View attachment 104715
I don't understand how A + B + R will be zero ? shouldn't it be 2R?

A+B=R. R is the resultant of A and B.

 

[cnh1995]

You have three vectors and their resultant is given to be 0. How would you draw this situation using a polygon?

 

[Isaac0427]

View attachment 104715
I don't understand how A + B + R will be zero ? shouldn't it be 2R?

You want to do A+B-R, which will give you zero if A+B=R.

 

[gracy]

You want to do A+B-R,

But we want to add vectors why will there be negative sign??

 

[TomHart]

In the picture below, the top diagram shows the addition of three vectors (V1, V2, and V3) and the resultant vector (the sum of those three vectors). The lengths of the line segments represent the magnitude of the vector. The bottom diagram shows three vectors that, when added, produce a resultant vector of magnitude 0 (zero). (That is why there is no resultant vector R shown - because it is 0 magnitude.) So the question is, if you have three vectors of various magnitudes connected end-to-start (in other words, the end of one connected to the start of the next), and you can point each of those vectors in whatever direction you choose (so long as you keep them connected end-to-start), can there ever be a situation where you would NOT be able to connect the end of the third vector back to the start of the first vector - in other words a situation where it was impossible for 3 vectors to sum to 0?

 

[Isaac0427]

But we want to add vectors why will there be negative sign??

The answer from @TomHart is great, but in short, A+B-R is the same as A+B+(-R), where the last vector's magnitude is R (and the problem only gives you the magnitude), but it's direction is negative.

 

[David Lewis]

If you travel along the vectors connected head to tail, and end up back where you started, then the vector sum is zero.

 

[gracy]

Thank you so much for unambiguous explanations. If there are vectors a, b and c. Then a+b+c using the polygon law, we will get a quadrilateral with four sides as a, b, c and resultant r. If resultant r is 0, then and then only we will get a triangle after addition. Similarly if four vectors are given & their resultant is zero they will form quadrilateral with four sides otherwise quadrilateral with five sides, fifth side being the resultant. Right?

 

[cnh1995]

quadrilateral with five sides,

With five sides, it becomes a pentagon.

 

[gneill]

Only if all the vectors have the same magnitude can a regular figure (square, pentagon, hexagon,...) result. That's generally not the case. Further, when you go beyond three vectors there's no guarantee that the vector path won't cross itself or have concavities:

So irregular polygons can be formed (as on the right in the figure above) but so can objects that aren't polygons at all.

The case of three vectors that form a closed loop will always result in a triangle.

 

[gracy]

Magnitudes do not have to be equal for forming triangle, right?

 

[cnh1995]

Magnitudes do not have to be equal for forming triangle, right?

No. Only the sum of any two sides should be greater than the third.

 

[TomHart]

Magnitudes do not have to be equal for forming triangle, right?

Right.

 

[TomHart]

Maybe I missed it somewhere above, but Gracy, are you convinced why (1) is the correct answer? If not, @cnh1995 made a great point when he said:

No. Only the sum of any two sides should be greater than the third.

 

[gracy]

Yes , that's triangle inequality theorem.

 

[TomHart]

Yes, that's what I was going to say. Just kidding. Actually, I had to Google search to find out what "triangle inequality theorem" was.