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.

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3.

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.
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2.

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.

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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.