Aheavy rope, 50 ft long, weighs 0.5 lb/ft and hangs over the edge of a building 120 ft high. how much work is done (in ft-lb) in pulling half the rope to the top of the building?

The formula used in calculating for values of work is given as:

W = F d

However on the above formula, the force is constant but i this case, it is changing. Therefore we make use of the general form with the integral notation: 
W = ∫F(x) dx

with limits of x from 0 to 25 since only half of the rope is be pulled

The equation for F (x) is equal to the product of mass density and length x:
F(x) = 0.5x 
Substituting this into the integral work equation:
W = 0.5∫x dx

W = 0.5(0.5) [x2^2 – x1^2]                  ---> x from 0 to 25

W = 0.25 [25^2 -0]

W = 156.25 ft * lbs 

An  atom  (ours, not carroll's) consists of a positively charged  nucleus, surrounded by one or more negatively charged particles called  electrons. the positive charges equal the negative charges, so the atom has no overall charge; it is electrically neutral. most of an atom's mass is in its nucleus; the mass of an electron is only 1/1836 the mass of the lightest nucleus, that of hydrogen. although the nucleus is heavy, it is quite small compared with the overall size of an atom. the radius of a typical atom is around 0.1 to 0.25 nanometres (nm), whereas the radius of a nucleus is about 10-6  nm.  *  if an atom were enlarged to the size of the earth, its nucleus would be only 200 feet in diameter and could easily rest inside a small football stadium.read more on -