A balloon with mass `m` is descending down with an acceleration a `(where altg)` . How much mass should be removed from it so that it starts moving up with an acceleration a?

Options

A. `(2ma)/(g+a)`
B. `(2ma)/(g-a)`
C. `(ma)/(g+a)`
D. `(ma)/(g-a)`

Answer

A

Explanation

(a) When the balloon is descending down with acceleration a
So, `mg-B=mxxxa`…..(i) (B`rarr` Buoyant force)
Here, we should assume that while removing some mass the volume of balloon and hence buoyant force will not change.
Let the new mass of the balloon is m'.
So, mass removed =m-m'
So, `B-m'g=m'zza`....(ii)
On solving Eqs. (i) nad (ii), we get
` mg-B=mxxa`
`B-m'g=m'xxa`
`implies mg-m'g =ma+m'a`
`implies (mg-ma)=m'(g+a)`
` implies m(g-a)=m'(g+a)`
`therefore m'(m(g-a))/(g+a)`
So, mass removed, `Delta m=m-m'`
`=m[1-((g-a))/((g+a))]=m[((g+a)-(g-a))/(g+a)]`
`=m[(g+a-g+a)/(g+a)]=(2ma)/(g+a)`