Physics-
General
Easy

Question

A particle of mass m is initially situated at the point P inside a hemispherical surface of radius r as shown in figure. A horizontal acceleration of magnitudea subscript 0 end subscriptis suddenly produced on the particle in the horizontal direction. If gravitational acceleration is neglected, the time taken by particle to touch the sphere again is

  1. square root of fraction numerator 4 r blank s i n alpha over denominator a subscript 0 end subscript end fraction end root  
  2. square root of fraction numerator 4 r blank t a n alpha over denominator a subscript 0 end subscript end fraction end root  
  3. square root of fraction numerator 4 r blank c o s alpha over denominator a subscript 0 end subscript end fraction end root  
  4. None of these  

The correct answer is: square root of fraction numerator 4 r blank c o s alpha over denominator a subscript 0 end subscript end fraction end root


    Let the particle touches the sphere t the point A.
    Let P A equals 1
    therefore P B equals fraction numerator l over denominator 2 end fraction
    In increment O P B comma cos invisible function application alpha equals fraction numerator P B over denominator r end fraction

    therefore P B equals r cos invisible function application a
    or fraction numerator l over denominator 2 end fraction equals r cos invisible function application a
    therefore l equals 2 r cos invisible function application alpha
    B u t blank l equals fraction numerator 1 over denominator 2 end fraction a subscript 0 end subscript t to the power of 2 end exponent
    therefore blank t equals square root of open parentheses fraction numerator 2 l over denominator a subscript 0 end subscript end fraction close parentheses end root equals square root of open parentheses fraction numerator 2 cross times 2 r cos invisible function application a over denominator a subscript 0 end subscript end fraction close parentheses end root equals square root of open parentheses fraction numerator 4 blank r cos invisible function application a over denominator a subscript 0 end subscript end fraction close parentheses end root

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