m il- R

A bead of mass m slides without friction along a curved wire with shape z = f(r) where r = Vr2 + y², i.e. the distance from the z-axis. The wire is rotated around the z-axis at a constant angular velocity w. Gravity acts downward along the z-axis with a constant acceleration g. a) Using Newton's second law in an inertial frame, derive an expression for radius ro of a fixed circular orbit (i.e. a solution with r = ro = const.). What is the normal force the wire applies to the bead to keep it in a circular orbit? b) Show that the equation of motion for r(t) (general equation not the circular motion) is F(1 + f'(r)²) + i² f'(r)f"(r) + gf'(r) – w?r = 0. Using this verify your answer to part (a). c) Consider small displacements from the circular orbit, r = ro+e(t). Derive a condition on the function f(r) such that a circular orbit at r = ro is stable. d) Find the force on the bead in the o direction, i.e. perpendicular to the plane of wire. The angular velocity is w = . Obtain the answer…

Assume a planet is a uniform sphere of radius R that (somehow) has a narrow radial tunnel through its center. Also assume we can position an apple anywhere along the tunnel or outside the sphere. Let Fp be the magnitude of the gravitational force on the apple when it is located at the planet's surface. How far from the surface (what multiple of R) is there a point where the magnitude of the gravitational force on the apple is 0.5 FR if we move the apple (a) away from the planet and (b) into the tunnel? (a) Number: Units: (b) Number: Units:

A geosynchronous Earth satellite is one that has an orbital period of precisely 1 day. Such orbits are useful for communication and weather observation because the satellite remains above the same point on Earth (provided it orbits in the equatorial plane in the same direction as Earth’s rotation). Calculate the radius of such an orbit based on the data for Earth in Appendix D.

Astronomers discover an exoplanet, a planet orbiting a star other than the Sun, that has an orbital period of 3.17 Earth years in a circular orbit around its star, which has a measured mass of 3.63×1030 kg. Find the radius r of the exoplanet's orbit.

An exotic planet Vogsphere is known to have a mass that is 1/81 that of the Earth and a radius 0.25 that of the Earth. Astrophysicist Trillian built a rocket and decided to leave the planet and never to return. Given that the escape speed from the Earth is 11.2 km/s, with what speed must Trillian achieve his goal?

Astronomers discover an exoplanet, a planet orbiting a star other than the Sun, that has an orbital period of 3.87 Earth years in a circular orbit around its star, which has a measured mass of 3.77×10^30 kg. Find the radius r of the exoplanet's orbit.

A 561-kg satellite is in a circular orbit about Earth at a height above Earth equal to Earth's mean radius. (a) Find the satellite's orbital speed.  ?m/s (b) Find the period of its revolution.  ?h (c) Find the gravitational force acting on it. ? N

The days on Mars and Earth are of nearly identical length. Earth's mass is 9.35 times Mars's mass, Earth's radius is 1.88 times Mars's radius, and Mars is on average 1.52 times farther away from the Sun than Earth is. The Martian year is 1.88 times longer than Earth's year. Assume that they are both uniform spheres and that their orbits about the Sun are circles. Estimate the ratio (Earth to Mars) of the following. (a) their spin angular momenta(b) their spin kinetic energies(c) their orbital angular momenta(d) their orbital kinetic energies

An asteroid is discovered to have a tiny moon that orbits it in a circular path at a distance of 147 km and with a period of 35.0 h. The asteroid is roughly spherical (unusual for such a small body) with a radius of 24.3 km. a) Find the acceleration of gravity at the surface of the asteroid. b) Find the escape velocity from the asteroid.