6. A uniform circular disc of mass m and radius 2a . centre O, is smoothly pivoted at a point A, where OA=a. (i) Find the moment of inertia of the disc about an avis through A perpendicular to the plane of the disc. The disc is free to rotate in a vertical plane about the axis through A. Given that the disc is held with O directly above A and then slightly displaced so that it swings in a vertical plane, (ii) show that in the ensuing motion, de 3a dt = 29(1 – cos0), %3D where o is the angle AO makes with the upward vertical.

14–83. A rocket of mass m is fired vertically from the surface of the earth, i.e., atr = r. Assuming that no mass is lost as it travels upward, determine the work it must do against gravity to reach a distance n. The force of gravity is F = GM,m/² (Eq. 13–1), where M is the mass of the earth and r the distance between the rocket and the center of the earth.

(a) Imagine that a space probe could be fired as a projectile from the Earth's surface with an initial speed of 5.48 x 10 m/s relative to the Sun. What would its speed be when it is very far from the Earth (in m/s)? Ignore atmospheric friction, the effects of other planets, and the rotation of the Earth. (Consider the mass of the Sun in your calculations.) Your response differs from the correct answer by more than 10%. Double check your calculations. m/s (b) What If? The speed provided in part (a) is very difficult to achieve technologically. Often, Jupiter is used as a "gravitational slingshot" to increase the speed of a probe to the escape speed from the solar system, which is 1.85 x 10 m/s from a point on Jupiter's orbit around the Sun (if Jupiter is not nearby). If the probe is launched from the Earth's surface at a speed of 4.10 x 10 m/s relative to the Sun, what is the increase in speed needed from the gravitational slingshot at Jupiter for the space probe to escape the solar…