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Motion in Straight Line

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Physics Part I

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Physics

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CBSE Gujarat Board Haryana Board

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Two identical charged spheres suspended froma  common point by two massless strings of lengths l, are initially at a distance d (d<< l) apart because of their mutual repulsion. The charges begin to leak from both the spheres at a constant rate. As a result, the spheres approach each other with a velcoity v. Then, v varies as a function of the distance x between the sphere, as

  • vproportional tox

  • straight v space proportional to space straight x to the power of begin inline style bevelled fraction numerator negative 1 over denominator 2 end fraction end style end exponent
  • vproportional tox-1

  • straight v space proportional to space straight x space to the power of begin inline style bevelled 1 half end style end exponent

B.

straight v space proportional to space straight x to the power of begin inline style bevelled fraction numerator negative 1 over denominator 2 end fraction end style end exponent

Two identical charged spheres suspended from a common point by two massless strings of length L.

In increment space ABC,
tan space theta space equals space fraction numerator F over denominator m g end fraction

rightwards double arrow space straight F over mg space equals space tan space straight theta space... space left parenthesis straight i right parenthesis
The charge begins to leak from both the sphere at a constant rate. As a result, the spheres approach each other with velocity v.

Therefore, equation (i) can be rewritten as,
fraction numerator Kq squared over denominator straight X squared mg end fraction space equals space fraction numerator X divided by 2 over denominator square root of l squared minus begin display style x squared over 4 end style end root end fraction

rightwards double arrow space fraction numerator Kq squared over denominator straight X squared mg end fraction space equals space fraction numerator straight x over denominator 2 straight l end fraction
rightwards double arrow space straight q squared space proportional to space straight x cubed
rightwards double arrow space straight q space proportional to space straight x to the power of begin inline style bevelled 3 over 2 end style end exponent
rightwards double arrow space dq over dt space proportional to space fraction numerator straight d space left parenthesis straight x to the power of bevelled 3 over 2 end exponent right parenthesis over denominator dx end fraction. space dx over dt
rightwards double arrow space dq over dt space proportional to space straight x to the power of begin inline style bevelled 1 half end style end exponent
rightwards double arrow space straight v space proportional to space 1 over straight x to the power of bevelled 1 half end exponent
rightwards double arrow space straight v space proportional to space straight x to the power of negative 1 divided by 2 end exponent

Two identical charged spheres suspended from a common point by two massless strings of length L.

In increment space ABC,
tan space theta space equals space fraction numerator F over denominator m g end fraction

rightwards double arrow space straight F over mg space equals space tan space straight theta space... space left parenthesis straight i right parenthesis
The charge begins to leak from both the sphere at a constant rate. As a result, the spheres approach each other with velocity v.

Therefore, equation (i) can be rewritten as,
fraction numerator Kq squared over denominator straight X squared mg end fraction space equals space fraction numerator X divided by 2 over denominator square root of l squared minus begin display style x squared over 4 end style end root end fraction

rightwards double arrow space fraction numerator Kq squared over denominator straight X squared mg end fraction space equals space fraction numerator straight x over denominator 2 straight l end fraction
rightwards double arrow space straight q squared space proportional to space straight x cubed
rightwards double arrow space straight q space proportional to space straight x to the power of begin inline style bevelled 3 over 2 end style end exponent
rightwards double arrow space dq over dt space proportional to space fraction numerator straight d space left parenthesis straight x to the power of bevelled 3 over 2 end exponent right parenthesis over denominator dx end fraction. space dx over dt
rightwards double arrow space dq over dt space proportional to space straight x to the power of begin inline style bevelled 1 half end style end exponent
rightwards double arrow space straight v space proportional to space 1 over straight x to the power of bevelled 1 half end exponent
rightwards double arrow space straight v space proportional to space straight x to the power of negative 1 divided by 2 end exponent

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