Leaky sandbag A bag of sand originally weighing 144 lb was lifted at a constant rate. As it rose, sand also leaked out at a constant rate. The sand was half gone by the time the bag had been lifted to 18 ft. How much work was done lifting the sand this far? (Neglect the weight of the bag and lifting equipment.)

Explanation:

Let m be the total mass initially and h be the height up-to which it is lifted.

Rate of fall of sand

= m/(2xh) lb/ft

At any height p , mass leaked out = m/(2xh) x p

At certain height p, mass  will  be m - m/(2xh) x p . This mass is lifted by small distance dp

work done

dW = [m - m/(2xh) x p ] g dp

Integrating it from 0 to h , we get the total value of work done

W = ∫[m - m/(2xh) x p ] g dp

= mg ( p - 1 / 2h x p²/2 )

Now taking limit between 0 to h

W = mg ( h - 1/2h x h²/2)

= mg ( h - h / 4 )

3 / 4 x mgh

= .75 x 144 x 32 x 18

= 62208 ft-poundal .

jupiter is the largest planet in our solar system, having a mass and radius that are respectively, 318 and 11.2 times that of earth. suppose that an object falls from rest near the surface of each planet and that the acceleration due to gravity remains constant during the fall. each object falls the same distance before striking the ground. determine the ratio of the time of fall on jupiter to that on earth.

the skeletons of most mammals reach a certain size and then stop growing. however, many animals, including some mammals, keep growing throughout their lives. kangaroos, for example, just keep growing and growing until they die. also, the longer an animal lives the greater the chance it might catch some deadly disease.