4) A planet that has a hollow core consists a uniform spherical shell with mass M, outer radius R, and inner radius R/2. Let r be the distance from the center of the planet. a. Derive an expression for the gravitational force on a small mass as a function of r when it is moving outside the planet. b. Find the period T of a satellite if it orbits very near the surface of the planet. c. Derive an expression for the gravitational force on a small mass as a function of r when it is moving inside the planet. d. Given the potential energy of an object is described by the formula: U(x,y) = a(x² + y³) + bxy, where a and b are positive constants. Find the x component of the applied force.

4) A planet that has a hollow core consists a uniform spherical shell with mass M, outer radius R, and inner radius R/2. Let r be the distance from the center of the planet. a. Derive an expression for the gravitational force on a small mass as a function of r when it is moving outside the planet. b. Find the period T of a satellite if it orbits very near the surface of the planet. c. Derive an expression for the gravitational force on a small mass as a function of r when it is moving inside the planet. d. Given the potential energy of an object is described by the formula: U(x,y) = a(x² + y³) + bxy, where a and b are positive constants. Find the x component of the applied force.