(B) Let $F$ be the upthrust (buoyant force) acting on the balloon.When the balloon is descending with acceleration $\alpha$,the net force equation is:

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$\left[ \frac{2\alpha}{\alpha + g} \right] M$

(B) Let $F$ be the upthrust (buoyant force) acting on the balloon.When the balloon is descending with acceleration $\alpha$,the net force equation is:$Mg - F = M\alpha$ --- $(i)$When a mass $m$ is released,the new mass of the balloon becomes $(M - m)$. It starts rising with the same acceleration $\alpha$. The net force equation is:$F - (M - m)g = (M - m)\alpha$ --- $(ii)$From equation $(i)$,we have $F = Mg - M\alpha = M(g - \alpha)$.Substitute this value of $F$ into equation $(ii)$:$M(g - \alpha) - (M - m)g = (M - m)\alpha$$Mg - M\alpha - Mg + mg = M\alpha - m\alpha$$mg + m\alpha = 2M\alpha$$m(g + \alpha) = 2M\alpha$$m = \left[ \frac{2\alpha}{\alpha + g} \right] M$

Class 11 Physics Newton's Laws of Motion and Friction Mix Examples-Newton's Laws of Motion and Friction

Medium

$A$ balloon of mass $M$ is descending at a constant acceleration $\alpha$. When a mass $m$ is released from the balloon,it starts rising with the same acceleration $\alpha$. Assuming that its volume does not change,what is the value of $m$?

A
$\left[ \frac{\alpha}{\alpha + g} \right] M$
B
$\left[ \frac{2\alpha}{\alpha + g} \right] M$
C
$\left[ \frac{\alpha + g}{\alpha} \right] M$
D
$\left[ \frac{\alpha + g}{2\alpha} \right] M$

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Class 11

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Chapter

Newton's Laws of Motion and Friction

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Mix Examples-Newton's Laws of Motion and Friction

$A$ cubical box of side $a$ sitting on a rough table-top is pushed horizontally with a gradually increasing force until the box moves. If the force is applied at a height from the table top which is greater than a critical height $H$,the box topples first. If it is applied at a height less than $H$,the box starts sliding first. Then,the coefficient of friction between the box and the table top is

MediumKVPY 2009

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Can the work done by a frictional force be positive or negative?

Medium

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$A$ bird weighs $2 \ kg$ and is inside a closed cage of $1 \ kg$. If it starts flying,then what is the weight of the bird and cage assembly in $kg$?

Easy

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On the horizontal surface of a truck,a block of mass $1 \; kg$ is placed $(\mu = 0.6)$. If the truck is moving with an acceleration of $5 \; m/s^2$,then the frictional force on the block will be: (in $; N$)

MediumAIPMT 2001

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Column $II$ shows five systems in which two objects are labelled as $X$ and $Y$. Also in each case a point $P$ is shown. Column $I$ gives some statements about $X$ and/or $Y$. Match these statements to the appropriate system$(s)$ from Column $II$.

Column $I$ Column $II$

$(A)$ The force exerted by $X$ on $Y$ has a magnitude $Mg$. $(p)$ Block $Y$ of mass $M$ on a fixed inclined plane $X$,slides on it with a constant velocity.

$(B)$ The gravitational potential energy of $X$ is continuously increasing. $(q)$ Two ring magnets $Y$ and $Z$,each of mass $M$,are kept in a frictionless vertical plastic stand. $Y$ rests on base $X$ and $Z$ hangs in equilibrium. The system is in a lift moving up with constant velocity.

$(C)$ Mechanical energy of the system $X+Y$ is continuously decreasing. $(r)$ $A$ pulley $Y$ of mass $m_0$ is fixed to a table $X$. $A$ block of mass $M$ hangs from a string over the pulley,fixed at $P$. The system is in a lift moving down with constant velocity.

$(D)$ The torque of the weight of $Y$ about point $P$ is zero. $(s)$ $A$ sphere $Y$ of mass $M$ is released in a non-viscous liquid $X$ and moves down.

$(t)$ $A$ sphere $Y$ of mass $M$ is falling with terminal velocity in a viscous liquid $X$.

Column $I$	Column $II$
$(A)$ The force exerted by $X$ on $Y$ has a magnitude $Mg$.	$(p)$ Block $Y$ of mass $M$ on a fixed inclined plane $X$,slides on it with a constant velocity.
$(B)$ The gravitational potential energy of $X$ is continuously increasing.	$(q)$ Two ring magnets $Y$ and $Z$,each of mass $M$,are kept in a frictionless vertical plastic stand. $Y$ rests on base $X$ and $Z$ hangs in equilibrium. The system is in a lift moving up with constant velocity.
$(C)$ Mechanical energy of the system $X+Y$ is continuously decreasing.	$(r)$ $A$ pulley $Y$ of mass $m_0$ is fixed to a table $X$. $A$ block of mass $M$ hangs from a string over the pulley,fixed at $P$. The system is in a lift moving down with constant velocity.
$(D)$ The torque of the weight of $Y$ about point $P$ is zero.	$(s)$ $A$ sphere $Y$ of mass $M$ is released in a non-viscous liquid $X$ and moves down.
	$(t)$ $A$ sphere $Y$ of mass $M$ is falling with terminal velocity in a viscous liquid $X$.

IIT 2009

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$A$ balloon of mass $M$ is descending at a constant acceleration $\alpha$. When a mass $m$ is released from the balloon,it starts rising with the same acceleration $\alpha$. Assuming that its volume does not change,what is the value of $m$?

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Can the work done by a frictional force be positive or negative?

$A$ bird weighs $2 \ kg$ and is inside a closed cage of $1 \ kg$. If it starts flying,then what is the weight of the bird and cage assembly in $kg$?

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