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Water in Vertical Circle

Consider a small bucket full of water tied to a string whirled around in vertical circle of radius r without water falling down. At the topmost position when the speed of the inverted bucket is v,

(A) $\frac{{{v^2}}}{r} = g$ necessarily
(B) $\frac{{{v^2}}}{r} < g$ necessarily
(C) $\frac{{{v^2}}}{r} > g$ necessarily
(D) None of the options given

Solution

Consider water as the system. Let, m be the mass of water. The weight mg acts downwards and the normal reaction N also acts downwards.

$mg + N = \frac{{m{v^2}}}{r}$

$\Rightarrow \frac{{m{v^2}}}{r} \ge mg$

$\Rightarrow \frac{{{v^2}}}{r} \ge g$

Hence, Option (D).

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 = ${}^{12}{C_6} = 924$