Vectors and the Laws of Physics — Question 4

Question 4

Turning the apparatus is different from turning the axes

A physical arrow initially has components $(5,0)$ m. In experiment I, the arrow is held fixed while the axes rotate $60^\circ$ counterclockwise. In experiment II, the axes stay fixed while the arrow physically rotates $60^\circ$ counterclockwise.

1. Find the reported components after each experiment.

2. If both the arrow and the axes rotate together by $60^\circ$, what components are reported?

3. Identify what is common and what differs in the three situations.

Question 4 - Solution

Turning the apparatus is different from turning the axes

Here $Q_\phi$ transforms components when axes turn counterclockwise by $\phi$; $R_\phi=Q_{-\phi}$ rotates the arrow in fixed axes. An unsubscripted $Q$ denotes the same component transformation, and $I$ is the identity.

(a) Reverse the relative angle for an axis rotation.

$$\boxed{\vec A_I=(5/2,-5\sqrt3/2),\quad\vec A_{II}=(5/2,+5\sqrt3/2)}\quad\mathrm m$$ In I the fixed arrow is at $-60^\circ$ relative to the new horizontal axis, so its components are $$(5/2,-5\sqrt3/2)\ \mathrm m.$$

In II the arrow is at $+60^\circ$ in the fixed basis, giving the pair $$(5/2,+5\sqrt3/2)\ \mathrm m.$$

(b) Rotate the reference and object together.

$$Q_{60}R_{60}\vec A=I\vec A=(5,0)\,\mathrm m$$ The arrow remains aligned with the new positive horizontal axis, so the components are still $(5,0)$ m. Its physical direction in the room has nevertheless changed by $60^\circ$.

(c) Separate invariant size from orientation.

$$|\vec A_I|=|\vec A_{II}|=|\vec A_{\mathrm{joint}}|=5\,\mathrm m,\qquad\theta'=\theta_{\mathrm{physical}}-\phi_{\mathrm{axes}}$$ All three arrows have magnitude 5 m. I changes only the description, II changes the physical arrow, and the joint rotation changes the physical arrow while preserving its components relative to the moving axes. Equal or unequal components alone cannot identify the physical change without knowing the frame.