How to prove that the sum of two rotating vectors in an ellipse?

In summary, two vectors of lengths P and Q rotate in opposite directions around the z-axis with angular velocities ω and -ω. At t=0, these vectors have orientations with respect to the x-axis specified by θ1 and θ2. To find the orientation of the major axis of the resulting ellipse relative to the x-axis, expand the cosine and sine terms in the expressions for X and Y and collect the terms with cos(ωt) and sin(ωt). There is an easy method to find the angle of the principal axis by considering the time instant when the vectors are on the same line.
  • #1
ppoonamk
28
0

Homework Statement



Within the xy-plane, two vectors having lengths P and Q rotate around the z-axis with angular velocities ω and –ω. At t = 0,these vectors have orientations with respect to the x-axis specified by θ1 and θ2. How do I find the orientation of the major axis of the resulting ellipse relative to the x-axis.


The Attempt at a Solution



P=|p|cos(θ1+ωt) x^+ |p|sin(θ1+ωt) y^
Q=|q|cos(θ1+ωt) x^+ |q|sin(θ1+ωt) y^

x^- x hat
y^-y hat

How do I solve this after these 2 equations?
I tried to group the x and y vectors separately. But i could not figure out anything after that.

X= |p|cos(θ1+ωt)+|q|cos(θ1+ωt)
Y=|p|sin(θ1+ωt)+ |q|sin(θ1+ωt)

Without the θ terms I could have just squared both sides and added it. But now I am stuck. Thank you for the help :)
 
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  • #2
ppoonamk said:

Homework Statement



Within the xy-plane, two vectors having lengths P and Q rotate around the z-axis with angular velocities ω and –ω. At t = 0,these vectors have orientations with respect to the x-axis specified by θ1 and θ2. How do I find the orientation of the major axis of the resulting ellipse relative to the x-axis.

The Attempt at a Solution



P=|p|cos(θ1+ωt) x^+ |p|sin(θ1+ωt) y^
Q=|q|cos(θ1+ωt) x^+ |q|sin(θ1+ωt) y^

x^- x hat
y^-y hat

How do I solve this after these 2 equations?
I tried to group the x and y vectors separately. But i could not figure out anything after that.

X= |p|cos(θ1+ωt)+|q|cos(θ1+ωt)
Y=|p|sin(θ1+ωt)+ |q|sin(θ1+ωt)

Without the θ terms I could have just squared both sides and added it. But now I am stuck. Thank you for the help :)

The expressions are not correct.
The vectors have length P and Q, use them instead of |p| and |q|.
The vectors rotate in opposite directions (the angular velocities are ω and -ω).
One vector encloses θ1 angle with the x-axis at t=0, the other one encloses θ2.

Expand the cosine and sine terms in the expression for X and Y, collect the terms with cos(ωt) and sin(ωt).
There is an easy method to find the angle of the principal axis: Just think that the vectors rotate in opposite directions, and there is a time instant when they are on the same line, so the resultant has the longest length.
ehild
 
Last edited:

1. What is an ellipse?

An ellipse is a closed curved shape that is formed when a plane intersects a cone at an angle that is not perpendicular to the base. It is typically described as having two focal points, with the sum of the distances from any point on the curve to the two focal points being constant.

2. How do you represent rotating vectors in an ellipse?

Rotating vectors in an ellipse can be represented using polar coordinates, where the distance from the center of the ellipse to a point on its perimeter is the magnitude of the vector, and the angle between the vector and a fixed reference line is the direction of the vector.

3. What is the sum of two rotating vectors in an ellipse?

The sum of two rotating vectors in an ellipse is the vector that results from combining the magnitudes and directions of the two individual vectors. It represents the final position of an object that has moved along both vectors.

4. How can you prove that the sum of two rotating vectors in an ellipse is constant?

To prove that the sum of two rotating vectors in an ellipse is constant, we can use the Pythagorean theorem and trigonometric identities to show that the sum of the squares of the magnitudes of the two vectors is equal to the square of the distance between their starting and ending points. This distance is constant for all points on the ellipse, thus proving the constancy of the sum of the two vectors.

5. Are there any real-life applications of proving the sum of two rotating vectors in an ellipse?

Yes, there are many real-life applications of proving the sum of two rotating vectors in an ellipse, such as in celestial mechanics, where the motion of planets and satellites can be described using ellipses and their rotating vectors. It is also useful in navigation and engineering, where understanding the combined effects of multiple forces acting on an object is important.

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