Tension in two wires at unknown angles

[MickeyBlue]

Homework Statement

An object at rest is suspended from two strings (A and B) as shown in the diagram, with A shorter than B. The object pulls on the point O with a force of 20 N. Each of the strings also exerts a force on the Point O. The angle between the strings at O is 90^o.

Which of the statements below are true? (If A and E are true, and the others are false, enter TFFFT).

A) The force exerted by B on point O points downward and to the left.
B) The magnitude of the force exerted by B on point O must be smaller than that of A.
C) The magnitude of the vector sum of the forces exerted by A and B is greater than 20 N.
D) The vector sum of the forces exerted by A and B points upward and to the right.
E) The magnitude of the force exerted by B must be 20 N or less.

Homework Equations

F = ma

The Attempt at a Solution

I know that the net forces in the horizontal and vertical planes will add up to 0 respectively, and that double angles and trig can be used to narrow down the given data into two unknowns (one value for tension and the angle). I think that the final answer is False, False, False, True, True. I'm at a bit of a loss as to how to think about the problem further.

 

[Merlin3189]

Perhaps you could give a reason for each answer.
(BTW I disagree with only one of your answers. One other answer I might say the opposite, but I can see how one could say T and F for it!)

 

[MickeyBlue]

A) F - In component form, the forces will act up and to the right
B) F - I think considering cos of this force to the block will make it bigger than A
C) F - The vector sum should be equal to 20N to keep the system in equilibrium
D) F? - The forces in the horizontal should be opposite and equal, which means the vector sum should be up in the vertical
E) T - If B were greater than 20N, there would most likely be movement in this direction

 

[Merlin3189]

A) F - In component form, the forces will act up and to the right - force in B is up and right. I'm not sure where components come in?
B) F - I think considering cos of this force to the block will make it bigger than A - not clear to me. cos of which angle? how big is that angle?
C) F - The vector sum should be equal to 20N to keep the system in equilibrium - Agree
D) F? - The forces in the horizontal should be opposite and equal, which means the vector sum should be up in the vertical - Agree
E) T - If B were greater than 20N, there would most likely be movement in this direction - I don't follow your logic here. If both A and B were greater than 20N pulling mainly sideways, they'd cancel each other and only their smaller vertical components would add to 20N. But I do think B is less than 15N.

 

[MickeyBlue]

A) F - In component form, the forces will act up and to the right - force in B is up and right. I'm not sure where components come in?
B) F - I think considering cos of this force to the block will make it bigger than A - not clear to me. cos of which angle? how big is that angle?
C) F - The vector sum should be equal to 20N to keep the system in equilibrium - Agree
D) F? - The forces in the horizontal should be opposite and equal, which means the vector sum should be up in the vertical - Agree
E) T - If B were greater than 20N, there would most likely be movement in this direction - I don't follow your logic here. If both A and B were greater than 20N pulling mainly sideways, they'd cancel each other and only their smaller vertical components would add to 20N. But I do think B is less than 15N.

• A: where the resultant B can be resolved into vectors on the x- and y-axes
• B: I'm sorry, I confused this with determining magnitude of work.
• E: As far as this goes I know it can be said that Ay + By = 20N to counter the weight, regardless of which one is greater. If this refers to force in the vertical, then B would have to be less than 20N with A making up the difference. It's just the mathematical relationship between B and the acute angle that made me unsure. I understand it now.

Thank you for sacrificing some of your Saturday night!

 

[haruspex]

B: I'm sorry, I confused this with determining magnitude of work.

You need to be careful with B. I draw your attention to two facts:

• You are not told angle alpha
• The word "must" in the question

Write out the balance of forces equations for the horizontal and vertical directions.

E: As far as this goes I know it can be said that Ay + By = 20N to counter the weight, regardless of which one is greater. If this refers to force in the vertical, then B would have to be less than 20N with A making up the difference. It's just the mathematical relationship between B and the acute angle that made me unsure. I understand it now

I don't think you do.
To answer this question you need to use the given 90 degree angle.

 

[CWatters]

I note it says A is shorter than B. So the length of A is in the range 0 to slightly less than B.

 

[CWatters]

I think that makes statement B true.

 

[CWatters]

For statement E... I think force B would be a maximum when length A is a maximum. The max length of A is just less than length B. Eg the angle is 45 degrees.

 

[CWatters]

At the risk of giving too much help.. if A is less than B what does that mean for Tan(alpha) and hence alpha.

 

[haruspex]

it says A is shorter than B

Yes, I missed that.

 

[MickeyBlue]

I completely forgot about Pythagoras. Thanks for all the help.