How do buoyancy and pseudo forces combine? 
About the diagram:
A tennis ball is attached to a string which is attached to the jar.
The string is mass less.
The density of the ball is less than that of water.
The F.B.D of the tennis ball:

Thus the ball is balanced by all this forces...
Experiment:
If we move the jar towards right then due to pseudo force,the ball will moves towards left,which is a obvious scene to observe,right.
My question:
If the jar moves towards right,as the ball has less density than the water, then the water must make the ball move towards right with it along the jar. I mean the buoyant forces acting from the left side will be now more as compared to right side of the jar, because we are pushing the jar from left to right,also as the density if the water is more thus,the force is also more,as compared to the ball's force.
I know I am thinking wrong somewhere, because it is at last the pseudo force making the ball moves towards the left, by can anyone explain where I am going wrong.
 A: We can understand gravity as equivalent to the ground having an uniform acceleration of $g$ upwards in a region of space without gravity. The effect of an additional horizontal acceleration is to change that (pseudo)acceleration vector.
All passes as, inside the jar, the force of gravity was tilted: the level of the water has an inclination (left side higher that right side for an external observer).
For that new gravity configuration, the ball must also be inclined to the right, keeping normal to the water surface.
A: The acceleration of the jar to the right does indeed create a pseudo force to the left when your reference frame accelerates with the jar.
This pseudo force can be treated exactly like you normally treat gravity. The magnitude will exactly match the magnitude of the acceleration.
The force on the ball directly to the left would therefore be equal to
$$F = m\,a$$
The buoyant force to the right would be:
$$F = \rho_{water} \, V \, a $$
Since the water is more dense than the ball, the buoyant force would be greater and the ball would accelerate to the right.
A: You should just du the experiment! accelerating your jar to the right , the ball will go to the right too, an yes it ist the inertia of water and ball wich act together. You may find it earthier to take a Helium filled ballon in your car and see what it does when you accelerate or brake or drive in a curve.
A: I don't know what you call pseudo forces, the ball has less inertia, than the water, so the water stays left pressure left is greater than pressure right  and the ball moves right relativ to the water.
