An astronomer discovered two planets, X and Y, orbiting a star. From the perspective of the astronomer, the planets orbit the star as depicted in the figures. Five years ago, the planets were in the position shown in figure (a), with X, Y, and the star in a straight line. Today, planet X is in the position shown in figure (b), having made an angular displacement of 90.0°. If the radii of their orbits are in the ratio 5:4, what is the angular displacement of Y? (Give your answer in revolutions.) a b 1/5 × Equate Newton's law of universal gravitation with his second law, noting that the force in this case is a centripetal force, and use the relationship among angular velocity, radius, and linear velocity. Solve for the angular velocity of planet Y, and calculate the angular displacement in revolutions. revolutions

Two planets in circular orbits around a star have speed of v and 2v . (a) What is the ratio of the orbital radii of the planets? (b) What is the ratio of their periods?

Jeff, running outside to play, pushes on a swinging door, causing its motion to be briefly described by = t2 + 0.800 t + 2.00,where t is in seconds and is in radians. At t = 0 and at t = 1.50s, what are the a. angular position, b. angular speed, and c. angularacceleration of the door?

A point on a rotating turntable 20.0 cm from the center accelerates from rest to a final speed of 0.700 m/s in 1.75 s. At t = 1.25 s, find the magnitude and direction of (a) the radial acceleration, (b) the tangential acceleration, and (c) the total acceleration of the point.

A bird flies overhead from where you stand at an altitude of 300.0 m and at a speed horizontal to the ground of 20.0 m/s. The bird has a mass of 2.0 kg. The radius vector to the bird makes an angle with respect to the ground. The radius vector to the bird and its momentum vector lie in the xy-plane. What is the bird’s angular momentum about the point where you are standing?

In circular motion, a tangential acceleration can change the magnitude of the velocity but not its direction. Explain your answer.

Math Review (a) Convert 47.0 to radians, using the appropriate conversion ratio. (b) Convert 2.35 rad to degrees. (c) If a circle has radius 1.70 m, what is the are length subtended by a 47.0 angle? (See Sections 1.5 and 7.1.)

A merry-go-round is stationary. A clog is running around the merry-go-round on the ground just outside its circumference, moving with a constant angular speed of 0.750 rad/s. The dog does not change his pace when he sees what he has been looking for: a bone resting on the edge of the merry-go-round one- third of a revolution in front of him. At the instant the dog sees the bone (t = 0), the merry-go-round begins to move in the direction the dog is running, with a constant angular acceleration of 0.015 0 rad/s2. (a) At what time will the dog first reach the bone? (b) The confused dog keeps running and passes the bone. How long after the merry-go-round starts to turn do the dog and the bone draw even with each other for the second time?

A particle moves 3.0 m along a circle of radius 1.5 m. (a) Through what angle does it rotate? (b) If the particle makes this trip in 1.0 s at a constant speed, what is its angular velocity? (c) What is its acceleration?

The Sun has a mass of approximately 1.99 1030 kg. a. Given that the Earth is on average about 1.50 1011 m from the Sun, what is the magnitude of the Suns gravitational field at this distance? b. Sketch the magnitude of the gravitational field due to the Sun as a function of distance from the Sun. Indicate the Earths position on your graph. Assume the radius of the Sun is 7.00 108 m and begin the graph there. c. Given that the mass of the Earth is 5.97 1024 kg, what is the magnitude of the gravitational force on the Earth due to the Sun?