CALC A thin, uniform rod has length L and mass M . A small uniform sphere of mass ?m is placed a distance x from one end of the rod, along the axis of the rod (?Fig. E13.34?). (a) Calculate the gravitational potential energy of the rod–sphere system. Take the potential energy to be zero when the rod and sphere are infinitely far apart. Show that your answer reduces to the expected result when x is much larger than L. (?Hint: Use the power series expansion for ln (1 + x) given in Appendix B.) (b) Use Fx = -dU/dx to find the magnitude and direction of the gravitational force exerted on the sphere by the rod (see Section 7.4). Show that your answer reduces to the expected result when x is much larger than L.

NEWTON: THE MIND THAT FOUND THE FUTURE ● different laws had to apply on earth and in the heavens ● science begins from theories or hypotheses that you have to prove ● in newton’s time, the gravity was in the apple ● all forces (including gravity) are between two objects, acting equally on both ○ newton’s 3rd law ● the natural path of a speeding object is a straight line, constant speed ○ uniform motion ● newton’s 1st law ● according to Halley, the force of gravity varies inversely with the square of the distance ● newton’s law of universal gravitation = everything attracts everything else ○ force is directly proportional to the two masses ○ force is inversely proportional to the separation of their ce