A circular disc of radius R is removed from a bigger circular disc of radius 2R such that the circumferences of the discs coincide. The centre of mass of the new disc is α/R from the centre of the bigger disc.The value of α is

  • 1/3

  • 1/2

  • 1/6

  • 1/6


A.

1/3

In this question distance of centre of mass of new disc is αR not α/R.

negative fraction numerator 3 straight M over denominator 4 end fraction space straight alpha space straight R space plus space straight M over 4 space straight R space equals 0
rightwards double arrow space straight alpha space equals space 1 third
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Consider a uniform square plate of side ‘a’ and mass ‘m’. The moment of inertia of this plate about an axis perpendicular to its plane and passing through one of its corners is

  • (5/6)ma2

  • (1/12)ma2

  • (7/12)ma2

  • (7/12)ma2


D.

(7/12)ma2

straight I space equals straight I subscript cm space plus space straight m open parentheses fraction numerator straight a square root of 2 over denominator 2 end fraction close parentheses squared space equals space ma squared over 6 space plus ma squared over 2
space equals space 2 over 3 ma squared
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Let P(r) = Qr/πR4 be the charge density distribution for a solid sphere of radius R and total charge Q. for a point ‘p’ inside the sphere at distance r1 from the centre of the sphere, the magnitude of electric field is

  • fraction numerator straight Q over denominator 4 πε subscript straight o straight r subscript 1 superscript 2 end fraction
  • 0

  • fraction numerator Qr subscript 1 superscript 2 over denominator 4 πε subscript straight o straight R to the power of 4 end fraction
  • fraction numerator Qr subscript 1 superscript 2 over denominator 4 πε subscript straight o straight R to the power of 4 end fraction

C.

fraction numerator Qr subscript 1 superscript 2 over denominator 4 πε subscript straight o straight R to the power of 4 end fraction fraction numerator straight Q over denominator 4 πε subscript 0 straight r subscript 1 superscript 2 end fraction space integral subscript straight r space equals 0 end subscript superscript straight r subscript 1 end superscript space fraction numerator 4 space πr cubed dr over denominator πR to the power of 4 end fraction
space equals space fraction numerator straight Q over denominator 4 πε subscript 0 end fraction fraction numerator straight r subscript 1 superscript 2 over denominator straight R subscript 4 end fraction
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A thin rod of length ‘L’ is lying along the x-axis with its ends at x = 0 and x = L. Its linear density (mass/length) varies with x as,straight k space open parentheses straight x over straight L close parentheses to the power of straight n where n can be zero or any positive number. If the position xCM of the centre of mass of the rod is plotted against ‘n’, which of the following graphs best approximates the dependence of xCM on n? 


A.

space straight x subscript cm space equals space fraction numerator integral dmx over denominator integral dm end fraction
fraction numerator integral begin display style λdx end style begin display style. end style begin display style straight x end style over denominator integral begin display style dm end style end fraction
space equals space fraction numerator integral begin display style straight k end style begin display style begin display style open parentheses straight x over straight L close parentheses end style squared end style begin display style. end style begin display style xdx end style over denominator straight k open parentheses begin display style straight x over straight L end style close parentheses to the power of straight n space dx end fraction
space equals space open square brackets fraction numerator begin display style fraction numerator kx to the power of straight n plus 2 end exponent over denominator left parenthesis straight n plus 2 right parenthesis straight L to the power of straight n end fraction end style over denominator begin display style fraction numerator kx to the power of straight n plus 1 end exponent over denominator left parenthesis straight n plus 1 right parenthesis straight L to the power of straight n end fraction end style end fraction close square brackets subscript 0 superscript straight L
space equals space open square brackets fraction numerator straight x left parenthesis straight n plus 1 right parenthesis over denominator straight n plus 2 end fraction close square brackets subscript 0 superscript straight L
straight x subscript cm space equals space straight L over 2 comma fraction numerator 2 straight L over denominator 3 end fraction comma fraction numerator 4 straight L over denominator 5 end fraction comma fraction numerator 5 straight L over denominator 6 end fraction comma.....
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A round uniform body of radius R, mass M and moment of inertia ‘I’, rolls down (without slipping) an inclined plane making an angle θ with the horizontal. Then its acceleration is


A.




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