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Two planets X and Y travel counterclockwise in circular orbits about a star, as seen in the figure.
The radii of their orbits are in the ratio 4:3. At some time, they are aligned, as seen in (a), making a straight line with the star. Five years later, planet X has rotated through 88.0°, as seen in (b). By what angle has planet Y rotated through during this time?

Two planets X and Y travel counterclockwise in circular orbits about a star as seen in the figure The radii of their orbits are in the ratio 43 At some time the class=

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Answer:  Two planets X and Y travel counterclockwise in circular orbits about a star, as seen in the figure. The radii of their orbits are in the ratio 4:3. At some time, they are aligned, as seen in (a), making a straight line with the star. Five years later, planet X has rotated through 88.0°, as seen in (b). Then, at an angle 135.48°, the planet Y rotated through during this time.

Explanation: To find the answer, we need to know about the Kepler's third law of planetary motion.

What is Kepler's third law of planetary motion?

  • Kepler's third law of planetary motion states that, the square of the period of revolution is proportional to the cube of the orbital radius of the elliptical path.
  • It can be expressed as,

                                      T² ∝ r³

How to solve the problem?

  • We have given with the ratio of the radii of their orbits as,4:3.
  • planet X rotated through an angle of 88°.
  • thus,

                 [tex]\frac{r_1}{r_2}=\frac{4}{5} \\\frac{T_1}{T_2} =(\frac{r_1}{r_2})^{3/2}\\[/tex]

  • As we know that,

                  [tex]T=\frac{2\pi }{w}[/tex] where, w is the angular velocity.

  • Angular displacement is the angle swept by the position vector of a particle in a given interval of time.

                           [tex]\alpha[/tex] =wt.

  • We can rewrite our equation as,

                   [tex]\frac{T_x}{T_y}=\frac{w_y}{w_x}\\thus,\\\frac{w_y}{w_x}=(\frac{r_1}{r_2})^{3/2}[/tex]

  • We have to find the angle rotated by planet Y during 5 yrs. So, we can rewrite the above equation in terms of angular displacement.

                     [tex]\frac{\alpha _y}{\alpha _x} = (\frac{r_1}{r_2})^{3/2}\\where,\\\alpha _x=\frac{88^0}{5 yrs} because,\\here, angle \beta_x =88^0.\\[/tex]

  • Thus, the angle rotated by planet Y during 5 yrs will be  [tex]\beta _y[/tex] =

                     [tex]\alpha _y=\frac{88}{5yrs} *(\frac{4}{3} )^{3/2}=\frac{135.48^0}{5yrs} .\\thus,\\\beta _y=135.48^0.[/tex]

Thus, we can conclude that the angle rotated by planet Y during 5 yrs will be 135.48 degrees.

Learn more about the Kepler's third law of planetary motion here:

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The planet Y then rotated through at this time at an angle of 135.48°.

In order to understand the solution, we must be familiar with Kepler's third law of planetary motion.

What does the third law of planetary motion by Kepler say?

  • According to Kepler's third law of planetary motion, the elliptical path's orbital radius is proportional to the cube of the square of the revolution's period.
  • It can be stated as follows:

                              T² ∝ r³

How can the issue be resolved?

  • The ratio of their orbital radii that we have provided is 4:3.
  • Planet X rotated at an 88° angle. thus,

                                  [tex]\frac{R_1}{R_2}=\frac{4}{5} \\\frac{T_1}{T_2}=(\frac{4}{5} ) ^{\frac{3}{2} }[/tex]

  • As we are aware,

                                [tex]T=\frac{2\pi }{w}[/tex]

where w is the angle of rotation per time.

  • The angle that a particle's position vector sweeps over in a specific amount of time is known as the angular displacement.

                                     [tex]\alpha[/tex]=wt.

  • Our equation can be rewritten as,

                                   [tex]\frac{w_y}{w_x} =(\frac{4}{5} ) ^{\frac{3}{2} }[/tex]

  • We have to find the angle that planet Y rotated at over the course of five years. Consequently, we can express the equation above in terms of angular displacement.

                                   [tex]\frac{\alpha _y}{\alpha _x}=(\frac{4}{3} ) ^{\frac{3}{2} } , where\\\alpha _x=\frac{88}{5yrs} \\[/tex]

  • So, during a period of five years, planet Y will rotate at an angle,

                           [tex]\alpha _y=\frac{88}{5yrs} *(\frac{4}{3} ) ^{\frac{3}{2} }=\frac{135.48}{5yrs}[/tex]

Thus, we may infer that planet Y will revolve at an angle of 135.48 degrees during the course of five years.

Learn more about the Kepler's law of planetary motion here:

https://brainly.com/question/28105769

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