Question

At 45° latitude, the gravitational acceleration as a function of elevation z above sea level is given by g = a − bz , where a = 9.807 m/s2 and b = 3.32 × 10−6 s−2 . Determine the height above sea level where the weight of an object will decrease by 0.3 percent. Answer: 8862 m

Answer 1

8861.75 m approximately 8862 m

Step-by-step explanation:

We need to remember Newton's 2nd Law which says that the force experienced by an object is proportional to his acceleration and that the constant of proportionality between those two vectors correspond to the mass of the object.

`F=ma` for the weight of an object (which is a force) we have that the acceleration experienced by that object is equal to the gravitational acceleration, obtaining that
`W = mg`

For simplicity we work with
`g =9.807`
`(m)/(s^(2))` despiting the effect of the height above sea level. In this problem, we've been asked by the height above sea level that makes the weight of an object 0.30% more lighter.

In accord with the formula
`g = a-bz` the "normal" or "standard" weight of an object is given by
`W = mg = ma` when
`z = 0`, so we need to find the value of
`z` that makes
`W = m(a-bz) = 0.997ma` meaning that the original weight decrease by a 0.30%, so now we operate...

`m(a-bz) = 0.997ma` now we group like terms on the same sides
`ma(1-0.997) = mbz` we cancel equal tems on both sides and obtain that
`z = (a)/(b) (0.003) = (9.807 (m)/(s^(2) ) )/(3.32*10^(-6) s^(-2) ) (0.003) = 8861.75 m`