March 29, 2022

Consider a frame non-inertial frame $K'$, moving WRT an inertial frame $K$, with an acceleration of $\vec A$. Thus from kinematics, we know that

$$ \vec a'=\vec a -\vec A \tag{5.1} $$

Where $\vec a'$ is the acceleration of an object WRT $K'$ and $\vec a$ is the acceleration of the object WRT $K$.

Thus the force on the object measured WRT $K'$ is

$$ \vec F'=m\vec a' \\ ‎\\ \vec F=m(\vec a-\vec A)\tag{5.2,\quad 5.3} $$

If we label $-m\vec A$ as a fictitious force then

$$ \vec F'=\vec F+\vec F_{fict}\tag{5.4} $$

Since $\vec A$ is the same for all objects in a non-inertial frame, we can see that $\vec F_{fict}\propto m$, just like the force of gravity for uniformly accelerating non-inertial frame, that is $d\vec A/dt=0$. However, unlike gravity, there is no physical cause for fictitious forces, they arise simply because of being in a non-inertial frame.

Consider what happens to a helium balloon tied to the floor of an accelerating car. When the car is initially at rest, the balloon is vertical WRT car floor and ground, balanced the force of buoyancy and tension. When the car starts to accelerate WRT ground, the air and balloon are stationary, while the back of the car hits the air next to it, increasing the density at the back of the car. Since the density increases at the back, this increases the buoyancy in the direction of acceleration, which overcomes the effect of acceleration on the balloon and hence the balloon tilts towards the front of the car. If we go into the car’s frame, then there is a fictitious force on the air and the balloon. This again causes the density to increase at the back and the balloon to tilt forwards.

Since the fictitious force has the same effect as gravity in a uniformly accelerating non-inertial frame, for an extended body the fictitious force acts on CM.

The Principle of Equivalence:

Consider being in a rocket ship moving with an acceleration of $\vec g$ in space. If you are holding an object in your hand, and let go of it, then according to you it will hit the back of the ship with the acceleration of $\vec g$. This situation is however identical to being stationary on the earth’s surface and then dropping the object. Thus there is no way to distinguish between the force of gravity and the fictitious force caused by being in a non-inertial frame.