A balloon with mass m is descending down with an acceleration a (where a <g). How much mass should be removed from it so that it starts moving up with an acceleration a? from Physics Motion in A Plane Class 11 Manipur Board
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A balloon with mass m is descending down with an acceleration a (where a <g). How much mass should be removed from it so that it starts moving up with an acceleration a?

  • fraction numerator 2 ma over denominator straight g plus straight a end fraction
  • fraction numerator 2 ma over denominator straight g minus straight a end fraction
  • fraction numerator ma over denominator straight g plus straight a end fraction
  • fraction numerator ma over denominator straight g minus straight a end fraction

A.

fraction numerator 2 ma over denominator straight g plus straight a end fraction

When the balloon is descending down with acceleration a,


So, mg - B = mx A             ... (i)
where B is the buoyant force.

We assume here that while removing same mass, the volume of balloon and hence buoyant force will not change.

Let, us assume the new mass of the balloon is m'.

So, mass removed is (m-m')

Therefore,

B-m'g = m'x a         ... (ii)

On solving equations (i) and (ii), we have

mg - B = m x a

B - m'g = m' x a

mg - m'g = ma +  m'a

(mg - ma) = m' (g+a) = m (g-a) = m' (g+a)

That is,

straight m apostrophe space equals space fraction numerator straight m space left parenthesis straight g minus straight a right parenthesis over denominator straight g plus straight a end fraction
That is, mass removed is m-m'

equals space straight m space open square brackets fraction numerator 1 minus left parenthesis straight g minus straight a right parenthesis over denominator left parenthesis straight g plus straight a right parenthesis end fraction close square brackets
equals space straight m open square brackets fraction numerator left parenthesis straight g plus straight a right parenthesis minus left parenthesis straight g minus straight a right parenthesis over denominator left parenthesis straight g plus straight a right parenthesis end fraction close square brackets
equals space straight m open square brackets fraction numerator straight g plus straight a minus straight g plus straight a over denominator straight g plus straight a end fraction close square brackets
equals space increment straight m space equals space fraction numerator 2 ma over denominator straight g plus straight a end fraction

When the balloon is descending down with acceleration a,


So, mg - B = mx A             ... (i)
where B is the buoyant force.

We assume here that while removing same mass, the volume of balloon and hence buoyant force will not change.

Let, us assume the new mass of the balloon is m'.

So, mass removed is (m-m')

Therefore,

B-m'g = m'x a         ... (ii)

On solving equations (i) and (ii), we have

mg - B = m x a

B - m'g = m' x a

mg - m'g = ma +  m'a

(mg - ma) = m' (g+a) = m (g-a) = m' (g+a)

That is,

straight m apostrophe space equals space fraction numerator straight m space left parenthesis straight g minus straight a right parenthesis over denominator straight g plus straight a end fraction
That is, mass removed is m-m'

equals space straight m space open square brackets fraction numerator 1 minus left parenthesis straight g minus straight a right parenthesis over denominator left parenthesis straight g plus straight a right parenthesis end fraction close square brackets
equals space straight m open square brackets fraction numerator left parenthesis straight g plus straight a right parenthesis minus left parenthesis straight g minus straight a right parenthesis over denominator left parenthesis straight g plus straight a right parenthesis end fraction close square brackets
equals space straight m open square brackets fraction numerator straight g plus straight a minus straight g plus straight a over denominator straight g plus straight a end fraction close square brackets
equals space increment straight m space equals space fraction numerator 2 ma over denominator straight g plus straight a end fraction

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