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### The velocity of a small ball of mass M and density d ....

The velocity of a small ball of mass M and density d, when dropped in a container filled with glycerin becomes constant after some time. If the density of glycerin is $\frac {d}{2}$, then the viscous force acting on the ball will be:

(1) $\frac {Mg}{2}$
(2) $Mg$
(3) $\frac {3}{2} Mg$
(4) $2Mg$

Solution

When velocity is constant, net force = 0.

$\therefore B+F=Mg$

Since density of glycerin is $\frac {1}{2}$ of the density of the ball, the buoyant force is half of the weight of the ball.

$B=\frac {Mg}{2}$

Now, $\frac {Mg}{2}+F=Mg$

$\therefore F=\frac {Mg}{2}$

### Sum of the coefficients in the expansion of $(x+y)^n$ ....
If the sum of the coefficients in the expansion of $(x+y)^n$ is 4096, then the greatest coefficient in the expansion is _ _ _ _ . Solution $C_0 + C_1 + C_2 + C_3 + ......................... + C_n =4096$ $\therefore 2^n = 4096 =2^{12}$ $\Rightarrow n = 12$ Greatest coefficient = ${}^{12}{C_6} = 924$