Question

A hot-air balloon consists of a basket, one passenger, and some cargo. Let the total mass be M. even though there is an upward lift force on the balloon, the balloon is initially accelerating downwards at a rate of g/3. Find the upward lift force in terms of the initial total weight Mg.
(A) 2Mg/3
(B) 3Mg/3
(C) 4Mg/3
(D) 5Mg/3

Answer 1

Hint:Balloon of total mass M when in air, the downward force Mg acts due to its weight. From Newton's third law, every reaction has an equal and opposite reaction. So, there will be an upward force or lift force on the balloon. So, lift force = Mg, when the balloon is at rest but when the balloon is in motion then, $\sum\limits_{}^{} {F = Ma} $.

Complete step by step answer:
The Free body diagram of the problem is shown in fig 1

Let, upward lift force be F.
Motion of the balloon is downwards with a = g/3
So, $\sum\limits_{}^{} {{F_y} = Ma} $
$
Mg - F = Ma \\
\Rightarrow F = M(g - a) \\
\Rightarrow F = M(g - \dfrac{g}{3}) \\
\therefore F = \dfrac{{2Mg}}{3} \\
$
So, upward lift force,\[F = \dfrac{{2Mg}}{3}\]. Thus,the correct answer is (A).

Additional information:
If motion of the balloon is upwards with an acceleration a then, $\sum\limits_{}^{} {{F_y} = Ma} $
$
F - Mg = Ma \\
\Rightarrow F = M(g + a) \\
$
Then uplift force will become $M(g + a)$.

Note:Remember that according to Newton’s law, the net force acting on a body is always equal to the net mass of the body into the acceleration of the body. Also apply the sign convention according to the direction of motion of a body.