Why is Tangential Speed unaffected by a change in Centripetal Force? When we rotate a mass in circular motion the centripetal force due to which the object moves in a circular path . But when we started to move our hand more fastly then we observe that tangential speed is also increasing so why we say that tangential speed is not effected by centripetal force?
Why does tangential speed do change when there is increase in centripetal force?
 A: When you move your hand you move the center of the circle so the motion is not purely circular anymore. The tension in the string will have a component tangent to the instantaneous trajectory. The motion is circular only if you keep the rotation center fixed. Then the speed will slowly decerase due to frictions. To maintain the speed constant (and not just to increase it) you move the other end of the string (the one in your hand) so that the direction of the string is not exactly normal to the instantaneous tangent to the trajectory.
A: If we make an object move faster along the same circular path, we must apply a tangential force component to give it a tangential acceleration. But because the object is moving faster it has to be supplied with a greater  inward radial (centripetal) force component if it is to keep moving along the same path.
So the increase in tangential speed is related to the increase in centripetal force, but it is a tangential force, not the change in centripetal force, that physically causes the change in tangential speed.
Example: a puck on a flat air-table is travelling at speed $v_1$ in a circular path (of radius $r_1$), kept on that path by a string attached to the puck, passing through a small frictionless hole in the table at the centre of the circle and held by a hand underneath the table.
We can indeed make the puck go faster by pulling the cord with a slightly greater force than $mv_1\,^2/r_1$. Surely that means that increasing the centripetal force makes the car go faster? No. Increasing the pull shortens the horizontal part of the string and makes the puck go (faster than $v_1$) in a path of decreasing distance from the hole. But while the string is being shortened, the puck is spiralling inwards, rather than going in a circular path. The pull (approximately $mv^2/r$) of the string isn't at right angles to the puck's velocity, but there is a component ($m\frac{v^2}r \times\frac{-dr}{v\ dt}$) of the pull parallel to the velocity, and this is what makes the puck go faster!
[For completeness... If we equate the tangential force, $m\frac{v^2}r \times\frac{-dr}{v\ dt}$, to $m\frac{dv}{dt}$, simplify, separate variables and integrate between limits we get
$$v_2 r_2=v_1 r_1.$$ This can be deduced (much more easily) from the conservation of angular momentum, showing the 'tangential force' treatment to be correct.]
A: $$W = \int \vec{F} \cdot \vec{dr}$$
$$W = \int [\vec{F} \cdot \vec{v}] dt$$
The centripetal  force is perpendicular to the velocity in circular motion.
$$\vec{F} \cdot \vec{v} = 0$$
Thus
$$W=0$$
meaning no work is done, thus its speed is constant
Rotating our hand does not provide a force purely in the centripetal direction, we move our hand either side of the center of rotation causing a component that DOES do work on the object, hence why it speeds up.
A: 
why we say that tangential speed is not effected by centripetal force

Not true. It does affect tangential speed. Tangential speed in terms of vector algebra:
$$\vec{v} = \vec {\Omega} \times \vec {r}.$$
$\vec{\Omega}$ is the angular velocity vector. Centripetal force produces thge angular velocity vector, which in turn, by the vector cross product, changes tangential velocity at every time moment. Velocity is vector; it has magnitude AND direction. If just direction changes, it means that the overall velocity vector changes too—i.e. the velocity vector can be split into magnitude and direction (unit vectors) parts:
$$ \vec {v} = v_x\cdot \vec {\imath} + v_y\cdot \vec{\jmath} + v_z\cdot \vec{k} $$
Where $\vec{\imath}$, $\vec{\jmath}$, and $\vec{k}$ are the $x$, $y$, and $z$ Cartesian axis unit directions.
