. A slender rod with length Lhas a mass per unit length that varies with distance from the left end, where 푥=0, according to 푑푚/푑푥=훾푥, where 훾has units of kg/m2.a) Calculate the total massMof the rod in terms of 훾and L. b) Use the equation 퐼=∫푟=푑푚to calculate the moment of inertia of the rod for an axis at the left end, perpendicular to the rod. Express your resultinterms of Mand L.(Hint: Revise the problem-solving section 9)c) Assume that the rodwithlength퐿=0.4푚andmass푀=3.0푘푔is rotating about theaxis as in part b. The angle through which the rod has turned varies with time according to휃(푡)=(1.1푟푎푑푠⁄)푡+(6.3푟푎푑푠=⁄)푡=.Wh at is the kinetic energy of the rod when it hasrotated0.1 rev? (Note: 1푟푒푣=2휋푟푎푑)

Neptune is the eighth planet from the sun. it was the first planet to get its existence predicted by  mathematical calculations  before it was actually seen through a telescope on sept. 23, 1846. irregularities in the orbit of  uranus  led french astronomer alexis bouvard to suggest that the gravitational pull from another celestial body might be responsible. german astronomer johann galle then relied on subsequent calculations to   spot neptune via telescope. previously, astronomer galileo galilei sketched the planet, but he mistook it for a star due to its slow motion. in accordance with all the other planets seen in the sky, this new world was given a name from greek and roman mythology — neptune, the roman god of the sea.