A piece of art weighing 30 pounds is hung from two wires as shown below. Each wire is supporting half of the weight of the picture. However, since the wires extend out at an angle, they are also imparting force towards the left and right. Therefore, the force on each wire is greater than 15 pounds. Look at the diagram above with the vectors. Use trigonometry to find ||→ux||and ||→vx||.
||→ux||=_____ pounds ||→vx||= _______ pounds
What is the total force (in pounds) exerted on each wire?
||→u||= ______pounds ||→v||= ______ pounds
Well, this is a tricky one. Let me put on my thinking cap and try to answer it in a funny way!
Hmm, so we have a weighted piece of art that's hanging from two wires. It sounds like a delicate balancing act, just like trying to hold a pizza slice with way too many toppings. I hope those wires are strong!
Anyway, we're looking for the magnitudes of two vectors, ||→ux|| and ||→vx||. Now, magnitudes are all about size, like the size of a clown shoe or the size of the crowd when a famous clown enters the room.
To find these magnitudes, we'll have to use some trigonometry. Aww, trigonometry, the art of making circles and triangles get along. It's like the circus ringmaster trying to keep the acrobats and clowns from bumping into each other!
Now, the forces on the wires are greater than 15 pounds each because of the angle they're hanging at. It's like when a clown tries to ride a unicycle on a tightrope – they have to exert extra force to balance themselves!
But what is the total force exerted on each wire? Well, we need to find the magnitudes of the vectors ||→u|| and ||→v||. It's like measuring the weight of a clown car full of clowns – you gotta make sure it's sturdy enough!
So, let's put these math muscles to work and find those values:
||→ux|| = _______ pounds
||→vx|| = _______ pounds
||→u|| = _______ pounds
||→v|| = _______ pounds
Remember, math can be funnier than a clown slipping on a banana peel. I hope my humorous approach helps you crack a smile while solving this problem!
In order to find ||→ux|| and ||→vx|| using trigonometry, we need to use the given diagram with vectors. However, as an AI text-based bot, I cannot see or interpret visual diagrams.
To find the total force exerted on each wire, we can use the concept of vector addition. Since the weight of the picture is 30 pounds and each wire is supporting half of the weight, the force exerted on each wire is 15 pounds.
Therefore, ||→u|| = ||→v|| = 15 pounds.
To find the magnitudes ||→ux|| and ||→vx||, we can consider the given information about the weight of the art piece and the angles of the wires.
Let's denote the angle between the horizontal axis and the left wire as α, and the angle between the horizontal axis and the right wire as β.
Since each wire is supporting half of the weight of the art piece, we can say that the force acting on each wire is equal to 15 pounds.
Now, let's analyze the forces acting on each wire individually.
For the left wire:
The vertical component of the force is given by ||→u|| * sin(α), which is equal to 15 pounds. Therefore, we have:
||→u|| * sin(α) = 15
Similarly, the horizontal component of the force is given by ||→u|| * cos(α). However, in this case, the horizontal component is acting towards the left, so it is negative. Thus, we have:
||→u|| * cos(α) = - ||→ux||
For the right wire:
The vertical component of the force is given by ||→v|| * sin(β), which is equal to 15 pounds. Therefore, we have:
||→v|| * sin(β) = 15
Similarly, the horizontal component of the force is given by ||→v|| * cos(β), but in this case, the horizontal component is acting towards the right, so it is positive. Thus, we have:
||→v|| * cos(β) = ||→vx||
Now, we can solve these equations simultaneously to find the values of ||→ux|| and ||→vx||.
Solving the left wire equations:
From the second equation, we have:
||→u|| * cos(α) = - ||→ux||
Dividing this equation by the first equation, we get:
(||→u|| * cos(α))/(||→u|| * sin(α)) = (- ||→ux||)/15
Simplifying, we have:
tan(α) = - ||→ux||/15
Therefore, we can find ||→ux|| by taking the absolute value of 15 * tan(α).
Solving the right wire equations:
From the second equation, we have:
||→v|| * cos(β) = ||→vx||
Dividing this equation by the first equation, we get:
(||→v|| * cos(β))/(||→v|| * sin(β)) = ||→vx||/15
Simplifying, we have:
tan(β) = ||→vx||/15
Therefore, we can find ||→vx|| by taking the absolute value of 15 * tan(β).
To find ||→u|| and ||→v||, we can use the Pythagorean theorem. Since the vertical and horizontal components of each wire form a right triangle, we have:
For the left wire:
||→u|| = sqrt((||→u|| * sin(α))^2 + (||→u|| * cos(α))^2)
||→u|| = sqrt(15^2 + (- ||→ux||)^2)
For the right wire:
||→v|| = sqrt((||→v|| * sin(β))^2 + (||→v|| * cos(β))^2)
||→v|| = sqrt(15^2 + ||→vx||^2)
Now, you can substitute the values of α and β given in the question, and solve the equations to find the magnitudes ||→ux||, ||→vx||, ||→u||, and ||→v||.
this article should fill you in on the details
https://dewwool.com/the-formula-for-tension-in-a-rope-at-an-angle/