A mass $m = 5 k g$ rests on a turntable, as show in the diagram below. The coefficient of static friction between the mass and table surface is $0.4$ . Another mass $M = 7 k g$ , is attached to a string that passes thorugh a hole at the centre of the turntable and is then attached to $m$ . What are possible angular velocity of the turntable, for which $m$ remains fixed at a distance $1 m$ from the centre of the turntable?

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$3.16 \frac{r a d}{s}$

$4.2 \frac{r a d}{s}$

$4.5 \frac{r a d}{s}$

$2.8 \frac{r a d}{s}$

answer is A, B.

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Detailed Solution

$T = M g T \pm μ m g = m ω^{2} r M g + μ m g = m ω_{m a x}^{2} r ω_{m a x} = \sqrt{\frac{(M + μ m) g}{m r}} = \sqrt{\frac{(7 + 2) \times 10}{5 \times 1}} = 3 \sqrt{2} \frac{r a d}{s} = 4.2 r a d / s ω_{m i n} = \sqrt{\frac{(M - μ m) g}{m r}} = \sqrt{\frac{(7 - 2) \times 10}{5 \times 1}} = \sqrt{10} \frac{r a d}{s} = 3.16 r a d / s$