Question

a ballon with m is descending down with an acceleration a how much mass should be removed from it so that it starts moving up with an acceleration a?.

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Answer 1

Question:- A balloon with m is descending  down with  an acceleration a. How much mass should be removed fromm it so that it starts moving up with an acceleration a ? [AIPMT - 2014]

Answer:  2ma/(g + a)

Explanation:

Case:1

In the first case balloon is moving downward with acceleration a.

Let the suppose that, due of air resistance and pressure of atmosphere force, there is force R, which is working in upward direction and forcing  balloon to move upward. But because of having a mass m, the balloon is moving downward.

Force Acting upward = R

Force Acting Downward = mg

Here mg is greater than R, That's why balloon is moving downward with acceleration a.

Resultant force = ma

We have,

mg - R = ma             ...........................I)

Case:-2

Let that we remove mass x from balloon. Now balloon has started moving upward with same acceleration a

Mass of Balloon = m - x

Force acting downward = (m - x)g

Force acting upward = R (Will be same)

Resultant force = (m - x)a

Here  R is greater than (m - x)g,

We have,

R - (m -x)g = (m - x)a     ......................II)

We need to eliminate R from equation I) and II)

So, Adding both equations,

(mg - R) + [R - (m - x)g] = ma + (m - x)a

mg -R + R -mg + xg = ma + ma - ax

xg + ax = 2ma

x(a + g) = 2ma

x = 2ma/(g + a)

Hence Option 1) is correct.

Answer 2

Answer:

$\huge{\pink{\green{\sf{\therefore X=\frac{2ma}{g+a}}}}}$

Step-by-step explanation:

$\huge{\pink{\green{\underline{\red{\sf{SOLUTION-}}}}}}$

• In the given question information given about a balloon is descending with mass m.

• We have to find the minimum weight that can be remove from ballon so that the balloon goes up with acceration a.

• So, we remove mass x from the mass of balloon.

• Value of g (gravity) is more than value of acceleration.

$\underline \bold{Given : } \\ \implies Mass \: of \: balloon = m \\ \implies Force \: acting \: upward = f \\ \implies Downward \: force = mg \\ \implies a < g \\ \implies Let \: remove \: mass = x\\ \\ \bold{ Removing \: x \: mass } \\ \bold{So \: downward \: force = xg }\\ \\ \underline \bold{To \: Find : } \\ \implies Extract \: mass = x$

• According to given question :

$\bold {Case \:1} \\ \bold{When \: balloon \: weight \: is \: m}\\ \implies mg - f = ma - - - - - (1) \\ \\ \bold {Case \:2} \\ \bold{when \: x \: mass \: remove \: from \: balloon} \\ \implies f - mg - xg = ma -xa \\ \implies f - (m - x)g = (m - x)a - - - - - (2) \\ \\ \bold {Adding \: (1) \: and \: (2)} \\ \implies mg - f + f - (m - x)g = ma + (m - x)a \\ \implies mg - mg + xg = ma + ma - xa\\ \implies xg + ax = 2ma \\ \implies x(g + a) = 2ma \\ \bold{\implies x = \frac{2ma}{g + a} }$