How does Inertial force arise?

Question

Consider the following scenario$-$

A ball sits on the floor of a bus, which was originally at rest w.r.t the ground. Suddenly it accelerates forward, and we observe the ball moving backwards. Well, originally the ball does not move, it is only the bus that moves forward. Why doesn't the ball simply accelerate with the bus? A common answer might be $-$

The body tries to maintain it's inertia of rest.

In this case, my question would be, how is this force generated that tries to oppose the motion of the ball with the bus?

P.S.- Please consider frictional force too, since I feel like I know the frictionless case.

Answer 1

This is called an inertial force. And it is due solely to inertia.

It is not a real force that opposes the forward motion of the ball. It is actually the fact that the object has inertia. And what happens is that an observer inside the bus will see the ball accelerate backward if the bus accelerates forwards, and this is due to this inertial force.

I know you stated that you are looking for another explanation apart from inertia. The reality is that there is no other explanation.

Inertia is a property of objects that have mass. And indeed Newton’s first law of motion states that inertia is the property whereby an object moving with constant velocity will remain in constant velocity or remain stationary, unless it is acted upon by an external force.

This means that when the bus begins moving, the object will remain stationary (to an observer outside of the bus). There is no force that causes this, or causes inertia. Rather, inertia is a natural property of the ball and all material objects.

There is only one case where the ball will move forward with the bus, and that is in the presence of friction. This is the only possibility for such a situation.

Why objects have inertia is really a philosophical question and I do not presume to have the answer, and in fact I believe no one has the answer. Newton was able to quantity it, and Einstein took it further by making the marvellous discovery that inertial mass was equivalent to gravitational mass.

We may have the answer as to what is the cause for inertia sometime in the future. It was about 300 hundred years ago when Newton formulated the law of universal gravitation. Then in the early 20th century Einstein successfully showed how spacetime curvature explains Newton’s apparent “mass attracting mass”.

Answer 2

Actually, you don't have to include inertia for an intuitive explanation.

Imagine an astronaut up in outer space. He is working on the space station from the outside, floating about in free space. He then let's go of his wrench.

What would you intuitively expect the wrench to do? You would expect it to just hang there, right? Because there are no forces acting on it. Only when he grabs it again will it move, since he now applies a pulling force in it.

Our intuitive understanding is that things should not start moving unless acted upon by a force. This is one of the key ideas of Newton's first law.

Now hold on to this intuitive understanding and bring it into your bus scenario. A ball is lying still on the bus floor. Then the bus accelerates due to its engine forcing it forwards. But nothing is accelerating the ball yet.

So, the bus moves forwards underneath the ball and there's no reason to expect the ball to move along - not unless a force pulls in the ball, that is. And we do have a force pulling in the ball. That would be friction.

Friction pulls only in the bottom point, though, so only that point is accelerated forwards, not the whole ball. Because that point is connected to all other points in the ball, but slightly off of the centre-of-mass, pulling in this lowest point causes the ball to spin.

And there you have it. No force is pushing the ball backwards. The fact that it seems to move backwards is just an illusion - it is rather the bus which is moving forwards underneath it. And the forwards pull in the bottom-most point causes the ball to rotate (to roll).

That's it. There is no force "generated" that opposes acceleration of an object. The fact that the object resist acceleration is the expected intuitive default - that has then been named inertia and is represented (in linear cases) by what we call mass. Inertial forces or Euler forces which are typical names for the apparent "force" that pushes the ball backwards are merely illusions and are just invented to explain the seemingly backwards push from the non-inertial reference frame. Centrifugal forces are in this same category, just in the circular-motion case. From an inertial frame these forces do not exist - they are what is called pseudo-forces.

You mention frictionless cases and friction cases in the last sentence of the question. Note that for a frictionless case, the exact same explanation applies just without any rolling/spinning of the ball. The ball would simply remain where it is in space not being influenced by the fact that the bus is moving forwards underneath it.

Answer 3

Because the rolling resistance of the ball is less than the force of acceleration transferred to the ball from the floor of the bus

Answer 4

@joseph h

inertia is a natural property of the ball and all material objects

This may be seen from a different perspective. There are confusing uses of mass in which it can be interpreted as an emergent behavior (like temperature) or arguably fundamental, such as its association with the Higgs particle. It is not clear, at least to me, that we understand what mass is and its manifestation as inertia, although we do use the concept successfully to describe the dynamics of events.