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Physics-

General

Easy

Question

Two wires $A C$ and $B C$ are tied at $C$ of small sphere of mass 5 kg, which revolves at a constant speed $v$ in the horizontal circle of radius 1.6 m. The minimum value of $v$ is

3.01 $m s^{- 1}$
4.01 $m s^{- 1}$
8.2 $m s^{- 1}$
3.96 $m s^{- 1}$

The correct answer is: 3.96 $m s to the power of negative 1 end exponent$

From force diagram shown in figure
$T subscript 1 end subscript cos invisible function application 30 degree plus T subscript 2 end subscript cos invisible function application 45 degree equals m g$ (i)
$T subscript 1 end subscript sin invisible function application 30 degree plus T subscript 2 end subscript sin invisible function application 45 degree equals fraction numerator m v to the power of 2 end exponent over denominator r end fraction$ (ii)
After solving Eq. (i) and eq. (ii), we get
$T subscript 1 end subscript equals fraction numerator m g minus fraction numerator m v to the power of 2 end exponent over denominator r end fraction over denominator open parentheses fraction numerator square root of 3 minus 1 over denominator 2 end fraction close parentheses end fraction$
But $T subscript 1 end subscript greater than 0$
$therefore fraction numerator m g minus fraction numerator m v to the power of 2 end exponent over denominator r end fraction over denominator fraction numerator square root of 3 minus 1 over denominator 2 end fraction end fraction greater than 0$

or $m g greater than fraction numerator m v to the power of 2 end exponent over denominator r end fraction$
or $v less than square root of r g end root$
$therefore blank v subscript m a x end subscript equals square root of r g end root equals square root of 1.6 cross times 9.8 end root equals 3.96 blank m s to the power of negative 1 end exponent$

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