Determine the cm of a rod assuming its linear mass density λ (its mass per unit length) varies linearly from λ = λ0 at the left end to double that value, λ = 2λ0

Center Of Mass lies at 5L/9 from O as shown.

Explanation:

The Center of mass of any object is given by

where (x,y) is location of point with mass dm

Now since rod is a linear object we have it's y co-ordinates as zero

Thus

The mass of an element 'dx' at a distance 'x' from O can be written as

Thus we have

Solving with limits form (0,L) we get

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.

What formula  this one  (x_/