Mechanics Help: Proving That θ = 45°?

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In summary, the conversation is about a mechanics question where the person is stuck on how to prove that θ = 45°. They have constructed a diagram and resolved forces horizontally and vertically, but are unsure of how to proceed. Another person mentions that the weight has been left out in one of the equations and asks the first person to consider the maximum horizontal force of friction for the beads. The first person is still confused and asks for further clarification.
  • #1
elle
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Mechanics help please!

Hi, can anyone help me with the following mechanics question:

http://tinypic.com/f1iogj.jpg"

I've attempted this question but I'm stuck on how i can prove that θ = 45° :confused:

I've constructed a diagram but I haven't scanned it but what I've got so far is from the diagram at point C, by resolving horizontally i find that the tensions, say T1 and T2 are equal to each other and I label this as T.

Resolving vertically I get 2T cos θ - w = 0

for the bead on the far left, by resolving horizontally I got:

F1 - T1 sin θ = 0 (F1 being the force acting towards the left)

vertically I get: N1 - T1 cos θ - w = 0


For the bead on the far right, by resolving horizontally I get:

F2 - t2 sin θ = 0

vertically:

N2 - T2 cos θ = 0

And this is where I'm stuck...I don't know where to go from here. Am I on the rght track? Can anyone help? Thanks! :confused: Sorry again that I haven't provided my diagram :frown:
 
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  • #2
Ok, a couple things here that you need to work on.

Sorry, I misread the question, ignore my first comment (which I have now deleted).

Second, you left out the weight in the vertical force equation for the point B on the far right.

Third, you need to figure out how the condition that the beads A and B be as far apart as possible plays into things. Here is a hint: what is the maximum horizontal force of friction that bead A can feel? What about bead B?
 
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  • #3
hmm I still don't understand...:confused:
 
  • #4
Where does the horizontal force on bead A that balances the tension come from? What is the maximum value this force can have? Answer the same thing for B.
 

Related to Mechanics Help: Proving That θ = 45°?

1. What is the equation for proving θ = 45° in mechanics?

In mechanics, the equation for proving that θ = 45° is usually derived from the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. This can be represented as c² = a² + b², where c is the hypotenuse and a and b are the other two sides.

2. How is the Pythagorean theorem used to prove θ = 45°?

In order to prove that θ = 45° using the Pythagorean theorem, we can set up a right triangle with sides of equal length, such as an isosceles right triangle. By substituting the lengths into the equation c² = a² + b² and simplifying, we can see that c = a√2. Since c is the hypotenuse and a is one of the other sides, this means that the ratio between the two sides is 1:√2. This ratio is only achieved when the angle opposite the hypotenuse is 45°.

3. Are there other ways to prove θ = 45° in mechanics?

Yes, there are other ways to prove θ = 45° in mechanics. One method is using trigonometric identities, specifically the identity tan(45°) = 1. By setting up a right triangle and using the tangent function to find the ratio between the sides, we can see that the angle θ opposite the hypotenuse is indeed 45°.

4. Why is proving θ = 45° important in mechanics?

In mechanics, proving that θ = 45° is important because it allows us to simplify calculations and make them more efficient. When an angle is equal to 45°, it means that the two sides adjacent to it are equal in length, making the calculations easier. Additionally, many real-life problems involve forces acting at 45° angles, so understanding this concept is crucial for solving these problems accurately.

5. Can θ ever be equal to a value other than 45° in mechanics?

Yes, θ can be equal to a value other than 45° in mechanics. This is because the Pythagorean theorem and trigonometric identities are not the only methods for proving angles in mechanics. There are also other geometric and algebraic techniques that can be used to find the value of θ in different scenarios. However, in many cases, θ is either assumed to be 45° or simplified to this value for the sake of convenience.

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