\text { The moment of inertia for an extended body is } I=\int r^{2} d m \text {. Prove that for a circular disc of } \text { mass } M \text { and radius } R \text { the moment of inertia, about an axis through its centre, is } I=\frac{1}{2} M R^{2} \text {. } A 80-kg man stands still on the rim of an initially stationary circular carousel of radius 3 m and mass 200 kg. The man begins to walk at 1 ms-1 (with respect to the ground) along the rim.What is his angular velocity, w? What is his angular momentum? What is the angular velocity of the carousel? How fast is the man walking with respect to the carousel? When orbiting a body of mass M at a distance r the speed is given by v = √GM/r. Using this,find a relation between r and the orbital period P. A geostationary satellite orbits the Earth in the same time the Earth rotates once. What is the radius of its orbit? [The Earth's mass is 6 x 1024 kg.] An enemy projectile is launched from Earth's surface to collide with the satellite. If the mass of the missile is 10 kg, what is the minimum energy needed to achieve this? [Earth's radius is 6380 km.]satellite If a replacement satellite has a mass of 50 kg, what is the energy needed to place a new satellin that orbit?