Tension forces of two wires in comparison to the gravitational force

[volcore]

Homework Statement

A sign is suspended by two wires as shown in the (Figure 1).
Is the tension in each wire larger than, equal to, or smaller than the gravitational force exerted on the sign by Earth?

The tension in the left wire is smaller than the gravitational force that Earth exerts on the sign. The tension in the right wire is larger than the gravitational force that Earth exerts on the sign.

The tension in the left wire is equal to the gravitational force that Earth exerts on the sign. The tension in the right wire is smaller than the gravitational force that Earth exerts on the sign.

The tension in both wires is larger than the gravitational force exerted on the sign by Earth.
The tension in the left wire is larger than the gravitational force that Earth exerts on the sign.

The tension in the right wire is smaller than the gravitational force that Earth exerts on the sign.
The tension in both wires is equal to the gravitational force exerted on the sign by Earth.

The tension in the left wire is smaller than the gravitational force that Earth exerts on the sign.
The tension in the right wire is equal to the gravitational force that Earth exerts on the sign.

Relevant Equations

None

The correct answer is the second one. I honestly have no idea why this is so. I understand that the right rope has less tension that the left one since it's at a shallower angle from real world experience, but I don't really know why this is so, let alone how the forces compare to gravitational force.

 

[Orodruin]

Did you try drawing a free body diagram and make an equilibrium analysis?

 

[Doc Al]

I understand that the right rope has less tension that the left one since it's at a shallower angle

I would question your reasoning here.

Do what @Orodruin suggests. Label the two tensions and do a force analysis.

 

[volcore]

I would question your reasoning here.

Do what @Orodruin suggests. Label the two tensions and do a force analysis.

So create a free body diagram of all the forces acting on the block?

 

[volcore]

Did you try drawing a free body diagram and make an equilibrium analysis?

Sorry, what exactly do you mean by an equilibrium analysis?

 

[Doc Al]

So create a free body diagram of all the forces acting on the block?

Exactly.

 

[volcore]

Exactly.

would there be three forces acting on the block, 2 forces pointing down, one to the right, and one to the left representing the left and right strings respectively, with the third force being gravity?

 

[Doc Al]

would there be three forces acting on the block, 2 forces pointing down, one to the right, and one to the left representing the left and right strings respectively, with the third force being gravity?

Yes. Three forces act on the block.

 

[volcore]

Yes. Three forces act on the block.

How would I tell their relation to gravity though?

 

[Doc Al]

How would I tell their relation to gravity though?

Once you have your free body diagram, set up equations describing the forces acting on the box. Since the box in equilibrium, what must be the net force on it?

 

[volcore]

Once you have your free body diagram, set up equations describing the forces acting on the box. Since the box in equilibrium, what must be the net force on it?

Net force would be zero right?

 

[Doc Al]

Net force would be zero right?

Yes.

 

[Lnewqban]

The correct answer is the second one. I honestly have no idea why this is so. I understand that the right rope has less tension that the left one since it's at a shallower angle from real world experience, but I don't really know why this is so, let alone how the forces compare to gravitational force.

The fourth answer does not make sense.
Answers 1, 3 and 5 show that the right wire or rope has more or equal tension, which contradicts your real world experience.
By elimination, answer 2 should be correct.

Using only your real world experience, how the tensions on each rope (TL and TR) compare to the weight of the sign (W) if:
1) Both ropes are vertical.
2) Both ropes diverge in a symmetrical way, each at 45 degrees from vertical.
3) Left rope is perfectly vertical while right rope is perfectly horizontal.

 

[Merlin3189]

The fourth answer does not make sense.
Answers 1, 3 and 5 show that the right wire or rope has more or equal tension, which contradicts your real world experience.
By elimination, answer 2 should be correct.

Disagree about 3, " ... 3 ... show that the right wire or rope has more or equal tension"
It says, "The tension in the left wire is larger than the gravitational force " and "The tension in both wires is larger than the gravitational force." Therefore it makes no comparison between the two forces.

Obviously, by horizontal consideration, T_left >T_right, so:

1 - T_ left wire is smaller than gravitational force. T_ right wire is larger than gravitational force.
=> T_left <T_right => wrong

2 - T_left wire is equal to the gravitational force. T_right wire is smaller than the gravitational force .
=> T_left >T_right => possible

3 - The tension in both wires is larger than the gravitational force.
The tension in the left wire is larger than the gravitational force - redundant.
=> no comparison yet

4 - T_right is smaller than the gravitational force.
T_both wires is equal to the gravitational force.
=> self contradictory

5 - T_left wire is smaller than the gravitational force.
T_right wire is equal to the gravitational force.
=> T_left <T_right => wrong

So 2 and 3 are the only possible answers. Try to eliminate one:

BUT drawing a "Freebody" diagram is the best way. When you have that, draw a triangle of forces and the answer is obvious, without any calculation (other than trivial arithmetic to calculate the angles in the corners.)

 

[PeroK]

Why can't they just ask you to calculate the tension in each wire?

 

[Merlin3189]

1 - Multiple choice question.
2 - Maybe they're hoping you'll use triangle of forces, especially as no weight is given.

We rarely get the true context of such questions. If it came up during work on triangle of forces, perhaps the tutor thought this might be an interesting Q to investigate student's intuitive understanding. Sometimes they want to get away from the, "use the same formula as all the other questions and just put the numbers in."
===============
Edit: Oops! Sorry. Just realized it probably wasn't an exasperated criticism of the question setter, but rather a hint to the OP.

 

[PeroK]

1 - Multiple choice question.
2 - Maybe they're hoping you'll use triangle of forces, especially as no weight is given.

We rarely get the true context of such questions. If it came up during work on triangle of forces, perhaps the tutor thought this might be an interesting Q to investigate student's intuitive understanding. Sometimes they want to get away from the, "use the same formula as all the other questions and just put the numbers in."
===============
Edit: Oops! Sorry. Just realized it probably wasn't an exasperated criticism of the question setter, but rather a hint to the OP.

It was exasperation with the question setter.

 

[volcore]

Disagree about 3, " ... 3 ... show that the right wire or rope has more or equal tension"
It says, "The tension in the left wire is larger than the gravitational force " and "The tension in both wires is larger than the gravitational force." Therefore it makes no comparison between the two forces.

Obviously, by horizontal consideration, T_left >T_right, so:

1 - T_ left wire is smaller than gravitational force. T_ right wire is larger than gravitational force.
=> T_left <T_right => wrong

2 - T_left wire is equal to the gravitational force. T_right wire is smaller than the gravitational force .
=> T_left >T_right => possible

3 - The tension in both wires is larger than the gravitational force.
The tension in the left wire is larger than the gravitational force - redundant.
=> no comparison yet

4 - T_right is smaller than the gravitational force.
T_both wires is equal to the gravitational force.
=> self contradictory

5 - T_left wire is smaller than the gravitational force.
T_right wire is equal to the gravitational force.
=> T_left <T_right => wrong

So 2 and 3 are the only possible answers. Try to eliminate one:

BUT drawing a "Freebody" diagram is the best way. When you have that, draw a triangle of forces and the answer is obvious, without any calculation (other than trivial arithmetic to calculate the angles in the corners.)

Sorry, what exactly do you mean by a triangle of forces? Should I separate the two tension forces into triangles, like a 50 40 90 triangle for the left wire, and a 20 90 and 70 triangle for the right?

 

[Merlin3189]

Left is something like a Freebody diagram. Right is like a triangle of forces: a set of forces in equilibrium can be added (nose to tail) in a closed polygon. (Triangle really refers to slightly different process of adding two vectors to find a resultant. )

 

[volcore]

Left is something like a Freebody diagram. Right is like a triangle of forces: a set of forces in equilibrium can be added (nose to tail) in a closed polygon. (Triangle really refers to slightly different process of adding two vectors to find a resultant. )
View attachment 257392

Ah, so from the right hand diagram I can see that L and W are equal in length and therefore equal in magnitude, and R is less than W, giving me the second answer. Thanks a lot!

 

[Merlin3189]

Even if the result hadn't been so easy, the largest force is opposite the largest angle, the smallest force opposite the smallest angle.
If you need a numerical relation, then the lengths of the sides are proportional to the sine of the opposite angle.

 

[PeroK]

Even if the result hadn't been so easy, the largest force is opposite the largest angle, the smallest force opposite the smallest angle.
If you need a numerical relation, then the lengths of the sides are proportional to the sine of the opposite angle.

That would be the "law of signs" in this case!

 

[volcore]

Even if the result hadn't been so easy, the largest force is opposite the largest angle, the smallest force opposite the smallest angle.
If you need a numerical relation, then the lengths of the sides are proportional to the sine of the opposite angle.

Right, thanks for the help

 

[haruspex]

Answers 1, 3 and 5 show that the right wire or rope has more or equal tension

No, 3 permits left > right > weight, which is indeed possible if the left hand wire were a little more towards horizontal.

@Merlin3189's geometric method is neat. Algebraically, it is not obvious what the relationship between the angles has to be to make left = weight. If the angles the wires make to the vertical are A (left) and B then the balance of forces gives $\tan(B)=\frac{\sin(A)}{1-\cos(A)}$.
The trick is to express the RHS in terms of A/2, which produces $\tan(B)=\cot(A/2)$, so $2B+A=\pi$.

Note that where the mass centre of the sign must be for system to hang so. Maybe that contributes to the deceptiveness of the problem.

Option 4 looks like a typo. Maybe copied incorrectly by the OP.
The second part of option 3 is redundant.