planet orbits a star once every 2.84 10^7 s in a nearly circular orbit of radius 1.42 10^11 m.
(a) With respect to the star, determine the angular speed of the planet.
(b) With respect to the star, determine the tangential speed of the planet.
(c) With respect to the star, determine the magnitude and direction of the planet's centripetal acceleration.
To determine the answers to these questions, we can use the formulas for angular speed, tangential speed, and centripetal acceleration.
(a) The angular speed of an object moving in a circular path is given by the formula:
Angular Speed (ω) = (2π) / Time taken for one complete revolution (T)
We are given that the planet orbits the star once every 2.84 x 10^7 seconds. Therefore, the time taken for one complete revolution (T) is 2.84 x 10^7 seconds.
Substitute this value into the formula:
Angular Speed (ω) = (2π) / (2.84 x 10^7 s)
Calculate the angular speed to get the answer.
(b) The tangential speed of an object moving in a circular path is given by the formula:
Tangential Speed (v) = Radius (r) x Angular Speed (ω)
We are given that the radius of the orbit is 1.42 x 10^11 meters and we have already calculated the angular speed in part (a). Substitute these values into the formula to calculate the tangential speed.
(c) The centripetal acceleration of an object moving in a circular path is given by the formula:
Centripetal Acceleration (a) = (Tangential Speed (v))^2 / Radius (r)
We have already calculated the tangential speed in part (b) and we are given the radius of the orbit. Substitute these values into the formula to calculate the magnitude of the centripetal acceleration.
The direction of the centripetal acceleration is towards the center of the orbit, which in this case is towards the star.
Using these formulas and the given values, you can now calculate the angular speed, tangential speed, and centripetal acceleration of the planet with respect to the star.