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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

  1. 3.01 m s to the power of negative 1 end exponent    
  2. 4.01 m s to the power of negative 1 end exponent    
  3. 8.2 m s to the power of negative 1 end exponent    
  4. 3.96 m s to the power of negative 1 end exponent    

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 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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