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A thin circular ring of mass m and radius R is rotating about its axis with a constant angular velocity ω. Two objects each of mass M are attached gently to the opposite ends of a diameter of the ring. The ring now rotates with an angular velocity ω′

  • fraction numerator ωm over denominator left parenthesis straight m plus 2 straight M right parenthesis end fraction
  • fraction numerator straight omega space left parenthesis straight m space plus space 2 straight M right parenthesis over denominator straight m end fraction
  • fraction numerator straight omega space left parenthesis straight m minus 2 straight M right parenthesis over denominator left parenthesis straight m plus 2 straight M right parenthesis end fraction
  • fraction numerator straight omega space left parenthesis straight m minus 2 straight M right parenthesis over denominator left parenthesis straight m plus 2 straight M right parenthesis end fraction


A.

fraction numerator ωm over denominator left parenthesis straight m plus 2 straight M right parenthesis end fraction straight L subscript straight i space equals space straight L subscript straight f
mR squared straight omega space equals space left parenthesis mR squared space plus space 2 MR squared right parenthesis straight omega apostrophe
straight omega apostrophe space equals space open parentheses fraction numerator mω over denominator straight m plus 2 straight M end fraction close parentheses
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Angular momentum of the particle rotating with a central force is constant due to

  • Constant Force

  • Constant linear momentum

  • Zero Torque

  • Zero Torque


C.

Zero Torque

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Four point masses, each of value m, are placed at the corners of a square ABCD of side A. The moment of inertia through A and parallel to BD is

  • m

  • 2m

  • √m

  • √m


D.

√m

I = 2m (

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The rms value of the electric field of the light coming from the Sun is 720 N/C. The average total energy density of the electromagnetic wave is

  • 3.3 × 10−3 J/m3

  • 4.58 × 10−6 J/m3

  • 6.37 × 10−9 J/m3

  • 6.37 × 10−9 J/m3


B.

4.58 × 10−6 J/m3

straight U subscript av space equals space straight epsilon subscript 0 straight E subscript rms superscript 2 space equals space 4.58 space straight x space 10 to the power of negative 6 end exponent space straight J divided by straight m cubed
straight E subscript rms superscript 2 space equals space 4.58 space straight x space 10 to the power of negative 6 end exponent space straight J divided by straight m cubed
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A force of negative straight F straight k with hat on top acts on O, the origin of the coordinate system. The torque about the point (1, −1) is

  • negative straight F left parenthesis straight i with hat on top space minus straight j with hat on top right parenthesis
  • straight F left parenthesis straight i with hat on top space minus straight j with hat on top right parenthesis
  • negative straight F left parenthesis straight i with hat on top space plus straight j with hat on top right parenthesis
  • negative straight F left parenthesis straight i with hat on top space plus straight j with hat on top right parenthesis

C.

negative straight F left parenthesis straight i with hat on top space plus straight j with hat on top right parenthesis straight tau with rightwards arrow on top space equals space left parenthesis negative straight i with hat on top plus straight j with hat on top right parenthesis space straight x space left parenthesis negative straight F straight k with hat on top right parenthesis
equals negative straight F space left parenthesis straight i with hat on top space plus straight j with hat on top right parenthesis
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