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1.Explain what happens when we ground any shell and if one surface is charged. Please tell all the variations possible regarding this.

2.How does dynamo work on the basis of electromagnetic induction. Give examples and explain that how do i understand that the particular equation follows lepton law

3. A particle is dropped from a height h. A constant horizontal velocity w is given to the particle. Taking g to be constant everywhere kinetic energy e of the particle with respect to time t is correctly shown? Why the graph is so and tell why not other graphs possible

4. In a projectile motion kinetic energy varies with time as per the standard graph and why not any other type of graph is possible?



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1.    When there is a shell it could be metallic or non-metallic.  The way solutions are worked out depends on this point.

      Since you say one surface is Earthed, it is then metallic.  In a metallic spherical shell, when a surface is grounded, the potential of that surface becomes 0.  The charges present on the surface go inside the Earth or charges from inside Earth flow on to the surface to make it neutral.  Inside a metal substance the electric field is 0 everywhere.  Also the charge enclosed with In a closed surface inside a metal is 0.

    So if there are no other charges present around or inside the conducting spherical shell, then the charges present on the inner surface will flow to Earth.  If there are charges present inside the hollow space, then equal and opposite charges will be retained on the inside surface.
     Let the unit vectors along x and y directions be   i and j .    
Velocity of the particle at time t:   vector v(t) = g t  j + w   i

     KE (t) = 1/2 * m * [ w^2 + g^2 t^2 ] = (m w^2/2) +( m g^2/2) t^2      

This is in the form :    y  -  b = a x^2

   This is a parabola, with symmetry axis being x = 0  ie., y axis.   x is similar to t.   y is similar to KE.  Its vertex is  at  y = b.  So the curve starts at (0,  mw^2/2) and continuously increases upwards in the shape of the outer surface of a cup.  It is convex towards  x axis and concave towards y axis.

     Basically the equation of   y versus x   or x versus y, tells the shape.   ============

     A dynamo is a gadget that converts mechanical energy into electrical energy. A simple example is the dynamo attached to the wheel of a bicycle or motor cycle.  The dynamo consists of a cylindrical metallic container that is attached on one side to a smaller cylinder.  This smaller cylinder is allowed to be in strong contact with a wheel. This wheel is rotated mechanically.  The smaller cylinder which is free to rotate about its axis rotates and rolls on the wheel without sliding.

    There is a strong permanent magnet fixed inside the metallic container.  The surfaces of North and South poles face each other and are in the shape of a cylindrical curved surface.  Between the magnetic poles and the rotor there is an air gap.

    When the wheel rotates, the axle in the dynamo is caused to rotate.  The axle is in the shape of a thin cylinder shaft called rotor.  The rotor is made of ferromagnetic material to concentrate magnetic field lines and increase electromagnetic induction.  Around the two planar faces of the rotor shaft a copper wire is wound in many turns.  This coil is called armature coil and is in the shape of a rectangle.  The two ends of the coil are joined to a small resistance bulb.

       The rotary motion of axle is used to induce electricity in a circuit by Faraday’s electromagnetic induction principle.  As per Faraday’s electromagnetic induction principle, the change in the magnetic flux that passes through the coil produces EMF between the ends of the coil.  Flux changes at the rate proportional to the speed of rotation, area of cross section of the coil, magnetic field strength and the number of turns in the coil.  Induced Emf is proportional to the rate of change of flux.

      The circuit has a bulb connected to it.  The voltage across the bulb will be emf generated – drop across coil resistance.   As long the wheel rotates and axle of the dynamo moves, there is current flowing in the circuit.  Electricity generated is used in the circuit instantaneously and not stored.

 4.  In a projectile motion:    Let A = angle of projection.

Velocity vector v(t) = (u sin A - g t) j + u cosA   i
So kinetic energy KE(t)
      =  1/2* m * [ (usinA - gt)^2 + (u cosA)^2 ]              
      = m/2 * [u^2+ g^2 t^2 - 2 u g t sinA ]

    The graph is like :         y - b =  (x - a)^2

      The graph will be a parabola like a cup open upwards.   t = u sin A /g is the symmetric axis (vertical).   KE is initially u^2 at t =0 (on the vertical axis), decreases in a sloping curve to u^2 Cos^2 A at t = u sinA/t .  Then KE increases in a symmetric fashion  to u^2 at  t = 2 u sinA/g.  

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