Inertial force in two different frames

Question

Suppose that a small ball is hung at the roof of a car, through a string. The car is accelerating forward and we see that the ball moves backward giving an angle to the string. In the point of view of an observer inside the car, what balances the horizontal component of the stress of the string? I know that we can say ball moves back due to a pseudo force (since the observer is in a non-inertial frame), but can we call this force as "inertial force"? is that the correct term? Further, according to an observer outside the car, what is the force that causes the ball to move back? is it again inertial force?

A simple explanation would be enough.

Answer 1

In response to your last question:

Further, according to an observer outside the car, what is the force that causes the ball to move back?

There is no force causing the ball to move back. The ball is behind the support point of the string because the support point, attached to the car, accelerated before the ball did, so it ends up moving ahead of the ball until the ball receives enough forward force to accelerate at the same rate as the car. The forward component of force comes from (now) non-vertical string. When the car first begins to accelerate, the ball doesn't because the string is vertical and doesn't exert a forward force. Gradually, the string tilts away from vertical until enough forward force is exerted on the ball to accelerate it along with the car.

If the car stopped accelerating (but continued to move), the ball would again hang vertically, eventually, because the ball would have an acceleration slightly greater than the car until both accelerations become zero, and there would be no forward force from the string.

Answer 2

The string applies a horizontal force to the ball pulling it in the direction of the car and gravity applies a downward force pulling it down.