3 stars in an equilateral triangle

In summary, the conversation discusses the calculation of the speed of three identical stars forming an equilateral triangle and rotating around its center while moving in a common circle. The method involves using Kepler's law and the equation for radius, and results in the equation v=sqrt (GM/R), where R is the distance between the stars. However, the answer given in the book is v=sqrt (GM/L), leading to confusion about whether it is a typo or not.
  • #1
burianek
3
0
Three identical stars of mass M form an equilateral triangle that rotates around the triangle's center as the stars move in a common circle about that center. The triangle has edge length L. What is the speed of the stars?


All I've been able to come up with is they rotate around the center. I took Kepler's law, T^2 = (4pi^2*r^3)/(GM), and replaced T with 2pi/angvelocity. Then, I replaced angvelocity = velocity/radius, and put the equation for radius (radius = (L*sqrt(3))/3) back in...

v=sqrt (GM/R) = sqrt (3GM/(L*sqr(3))) - - - the answer given is sqrt (GM/L) ... not sure where I went wrong, because I know that R isn't equal to L. Does anyone have any direction, or do you think this is just a typo in the book's answer key?
 
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  • #2
Welcome to PF!

burianek said:
Three identical stars of mass M form an equilateral triangle that rotates around the triangle's center as the stars move in a common circle about that center. The triangle has edge length L. What is the speed of the stars?

Hi burianek! Welcome to PF! :smile:

Calculate the forces from each of the two other two stars separately, add them, and then equate to the centripetal force.
 
  • #3


I would first like to commend you for attempting to use Kepler's laws to solve this problem. However, I believe there may be a few errors in your calculations. First, when using Kepler's third law (T^2 = (4pi^2*r^3)/(GM)) to find the orbital period, T represents the time it takes for one complete orbit, not the angular velocity. So, the equation should be T^2 = (4pi^2*R^3)/(GM). Second, when substituting for the angular velocity, you should use the equation v = ω*R, where ω is the angular velocity and R is the radius. In this case, R is the distance from the center of the triangle to one of the stars, which can be found using trigonometry as R = (L/2)/sin(60°) = L/√3. Therefore, the correct equation for the speed of the stars would be v = ω*R = (2pi/T)*R = (2pi/√(GM))*√(L/√3) = √(GM/L). I hope this helps clarify any confusion and leads to the correct answer.
 

FAQ: 3 stars in an equilateral triangle

What is the significance of 3 stars in an equilateral triangle?

The number 3 has long been seen as a symbol of balance and harmony, and the equilateral triangle represents stability and strength. Combining these two elements, 3 stars in an equilateral triangle can symbolize a powerful and harmonious unity.

Is there a specific meaning behind the placement of the stars in the equilateral triangle?

The placement of the stars in an equilateral triangle can vary depending on the context. In astronomy, the stars may represent the positioning of celestial bodies. In symbolism, the placement could represent the balance or hierarchy of the elements or concepts represented by the stars.

Are there any cultural or religious associations with 3 stars in an equilateral triangle?

Yes, there are many cultures and religions that use 3 stars in an equilateral triangle as a symbol. For example, in Christianity, the three stars can represent the Holy Trinity. In some Native American cultures, the stars may represent the three aspects of life: past, present, and future.

What does the color of the stars in the equilateral triangle symbolize?

The color of the stars can add another layer of meaning to the symbol. In general, gold or yellow stars can represent divinity, enlightenment, or success. Silver or white stars can symbolize purity, innocence, or spirituality. Red stars may represent passion, courage, or strength.

Do the stars in an equilateral triangle have a specific orientation?

No, the orientation of the stars in an equilateral triangle does not have a fixed meaning. Some may interpret the top point as the most important or dominant element, while others may see the bottom point as representing a strong foundation. Ultimately, the orientation can be interpreted in various ways depending on the context and personal beliefs.

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