The Rotational Variables — Question 7

Question 7

A shared angle origin does not alter angular rates

Two encoders observe the same fixed-axis rotation. Encoder A uses $\theta_A=0.4+3t-t^2$ in SI units. Encoder B has a zero shifted by $\pi/2$ but keeps the same positive sense.

1. Write $\theta_B$ and compare both angular velocities and accelerations.

2. If B also reverses its positive sense, how do the three signed quantities transform?

Question 7 - Solution

A shared angle origin does not alter angular rates

(a) Derivation and calculation

With B’s zero at A angle $\pi/2$ and the same positive sense, $$\boxed{\theta_B(t)=0.4+3t-t^2-\frac\pi2}.$$ $$\omega_B=\frac{d\theta_B}{dt}=\omega_A=\boxed{3-2t},\qquad \alpha_B=\alpha_A=\boxed{-2}.$$ Rates are in rad/s and rad/s$^2$, respectively.

(b) Result and interpretation

Reversing B’s positive sense also multiplies its coordinate by $-1$: $$\boxed{\theta_B=-(\theta_A-\pi/2),\quad \omega_B=-\omega_A=-3+2t,\quad\alpha_B=-\alpha_A=+2}.$$ A constant origin shift disappears under differentiation; a sense reversal changes every signed rate.

Physical check

Both conventions predict the same stopping time $t=1.5$ s.