The non-inertial frame can be converted into an inertial frame by the introduction of the simple inertial force "-ma" where "m" is the mass of any object under consideration and "a" is the uniform acceleration of the simple non-inertial frame.

Inertial forces are body forces that act equally per unit mass on all bits of a body.

So if a body does NOT resist inertial forces, it suffers NO deformation/strain.

In the classical limit, one can view the introduction of inertial forces as way of generalizing Newton's laws of motion to non-inertial frames.

You do NOT introduce inertial forces at all if you are using an inertial frame right from the beginning of an analysis.

However, the perspective of the conversion of non-inertial frames to inertial frames is extremely useful in many cases. For important example, rotating frame which we are consideriong in this file.

Note, the conversion of an non-inertial frames to an inertial frame is NOT just a trick. The equivalence principle, axiom of general relativity, implies that almost all physical laws are referenced to inertial frames whether they are simple inertial frames or converted inertial frames. Thus, there is a fundamental likeness of all inertial frames.

General relativity itself is NOT referenced to inertial frames and, in fact, tells us what they are.

One can quibble about whether there are other physical laws NOT referenced to inertial frames, but yours truly thinks the quibbling is a matter of perspective or may amount to saying you are NOT using inertial frames in some definitional sense when effectively you are using them.

However, by convention, a rotating frame is considered one non-inertial frame. It certainly is one reference frame.

But first note that to avoid tedious and unenlightening generality, we will limit our discussion to rotating frames where the rotation axis does NOT have axial precession relative to the observable universe and is at rest relative to an EXTERNAL inertial frame. We also limit ourselves mostly to rotating frames that have constant angular velocities. These limitations can all be relaxed if one needs to.

An extreme example of the kind of rotating frame we are NOT discussing is one rotating relative to another rotating frame, but NOT rotating relative to the observable universe. Such tricky cases have their interest, but are finicky to discuss.

If the ball is given an initial velocity relative to the outside world, it just slides in a straight line relative at contant velocity relative to the outside world since no net force acts on it.

If the ball is given an initial velocity relative to the outside world, (which defines an inertial frame), it moves in a straight line at a constant speed, and so is unaccelerated. It is in uniform linear motion. By Newton's 2nd law of motion (AKA F=ma), there is NO ORDINARY net force on the ball as aforesaid.

Recall Newton's laws of motion are referenced to inertial frames although this essential fact is often omitted in high-school presentations.

But, as aforesaid, we can convert non-inertial frames to inertial frame by introducing inertial forces.

1. The centrifugal force which is easy to understand. It's just the force that tries to throw you off carnival centrifuges. From the outside EXTERNAL inertial frame of the ground, you are just trying to move in a straight line at a constant speed in accordance to Newton's 1st law of motion and the force on your back by the carnival centrifuge is needed to accelerate you with the rotating frame of the carnival centrifuge. In general, if you try to stay in one place in the rotating frame, you have to exert real forces to hang on to the rotating frame and stay at rest relative to it.

2. The Coriolis force is trickier to understand. It is a velocity-relative-to-the-rotating frame-dependent force. It depends linearly on the relative relative velocity. If you are NOT moving relative to the rotating frame, the Coriolis force is zero. The Coriolis force is important in weather phenomena in the rotating frame of the Earth's rotation. It's the cause of the vortex motion of cyclones (see Wikipedia: Cyclone: Structure) and anticyclones (see Wikipedia: Anticyclone: Structure). See more explication of cyclones and anticyclones in Mechanics file: coriolis_force.html. The Coriolis force also affects long-range artillery ballistics. We do NOT ordinarily notice the Coriolis force of the Earth's rotation on smaller scales than weather and long-range artillery ballistics. However, on human size scale, Coriolis force causes the behavior of a Foucault pendulum. For an explication of the Foucault pendulum, see Mechanics file: pendulum_foucault.html.

3. The Euler force which arises if the rotation of a rotating frame is accelerating.

The main reason the ball follows a curved path relative to the rotating reference frame is the Coriolis force since the ball has velocity relative to the rotating frame.