Two stars X and Y travel counterclockwise in circular orbits about a galaxy. The radii of their orbits are in the ratio 4:1 . At some time, they are aligned making a straight line with the center of the galaxy. Five years later, star X has rotated through 90°, By what angle has planet Y rotated through during this time
The period is proportional to Radius^(3/2)
so
Tx/Ty= (4R)^3/2 / R^3/2 = 2^3 = 8 times
8 * 90 = 360*2 = 720deg
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why Radius^(3/2)
mv^2/R = G m M/R^2
so if k^2 = GM
v = k R^-.5
T = 2 pi R/v = 2 pi R/kR^-.5 = constant*R^1.5
Tx/Ty= (4R)^4/2 / R^4/2 = 1^3 = 3 times
3 * 90 = 270 deg
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why Radius^(4/2)
mv^2/R = G m M/R^2
so if k^2 = GM
v = k R^-.5
T = 2 pi R/v = 2 pi R/kR^-.5 = constant*R^1.5
A circle has 360 degrees how can it be 720 degrees.
To determine the angle by which star Y has rotated, let's consider the ratio of their angular displacements.
The ratio of the radii of their orbits is given as 4:1. Let's assume that X and Y start at point A and align with the center of the galaxy at time t=0.
After 5 years, star X has rotated through 90°. This means X has moved from point A to point B, where AB is perpendicular to the line connecting the center of the galaxy and point A.
We can consider the angle subtended by star X at the center of the galaxy at time t=0 as the reference angle, which we'll call θ.
The angular displacement of X after 5 years is 90°, so the angle subtended by star X at the center of the galaxy after 5 years is θ + 90°.
Since the ratio of their radii is 4:1, star Y's orbit is four times smaller than X's orbit. This means that after 5 years, Y would have traveled a distance four times smaller than X.
Therefore, the angular displacement of Y can be calculated using the following ratio:
Angular displacement of X : Angular displacement of Y = Distance traveled by X: Distance traveled by Y
Since X has rotated 90°, the distance traveled by X is 90°.
Given that the distance traveled by Y is four times smaller, we have:
90° : Angular displacement of Y = 90° : 4(angular displacement of Y)
Now we can solve for the angular displacement of Y:
90° = 90° / 4(angular displacement of Y)
Dividing both sides by 90°:
1 = 1 / 4(angular displacement of Y)
Simplifying further:
angular displacement of Y = 4
Therefore, star Y has rotated through an angle of 4° during this time.
So, the answer is that star Y has rotated through an angle of 4°.