Newton's First Law — Question 6

Question 6

A turning cabin fails the free-particle straight-line test

A small free puck travels in a straight line at constant speed relative to an inertial floor. A camera rotates steadily with a platform. Its images show a curved puck track. The puck does not touch the platform.

1. Which observed track directly satisfies the first-law free-motion form?

2. Why is the rotating camera not an inertial frame?

3. Would attaching the puck to the platform describe the same experiment?

Question 6 - Solution

A turning cabin fails the free-particle straight-line test

(a) Use the stipulated free inertial record.

$$\vec r(t)=\vec r_0+\vec v_0t,\quad\boxed{\frac{d^2\vec r}{dt^2}=\vec0}\quad\text{(floor)}$$ The floor record is a straight line at constant speed, the first-law form. Its velocity is unchanged.

(b) Check the axes, not just the origin.

$$\vec r'=Q(t)\vec r:\quad\ddot{\vec r}'=\ddot Q\vec r+2\dot Q\vec v\quad(\ddot{\vec r}=0)$$ The camera axes turn, so coordinate directions vary with time. A free particle need not have constant vector components or a straight track in those coordinates. Rotation requires additional terms when using Newtonian dynamical equations there.

(c) Distinguish observation from interaction.

$$|\vec a_{\mathrm{attached}}|=\omega^2r,\quad |\vec F_{\mathrm{net}}|=m\omega^2r\quad(r>0)$$ Attaching the puck would add a constraint interaction that changes its motion. A coordinate transformation alone does not supply that interaction. The freely moving and attached cases have different physical systems.