What is Mass, momentum, force?

Example 12.11 (Block on an incline)

Block of mass $m$ on a plane inclined at $\alpha$, coefficients $\mu_s$, $\mu_d$. Axes: $x$ down the slope, $y$ along the normal. Weight $(mg\sin\alpha, -mg\cos\alpha)$, reaction $(0, N)$, friction $(-T, 0)$ if sliding down. Along $y$: $N = mg\cos\alpha$. At rest along $x$: $T = mg\sin\alpha \leq \mu_sN = \mu_smg\cos\alpha$, i.e. $\tan\alpha \leq \mu_s$: the block holds up to the angle $\arctan\mu_s$ — a measurement of $\mu_s$ needing no balance. Sliding: $ma = mg\sin\alpha - \mu_dmg\cos\alpha$, so $a = g(\sin\alpha - \mu_d\cos\alpha)$, independent of $m$: $3.2\,\mathrm{m}/\mathrm{s}^{2}$ for $\alpha = 30^\circ$, $\mu_d = 0.2$.

Example 12.12 (The simple pendulum)

Mass $m$ on a string of length $\ell$, angle $\theta$ from the downward vertical. Polar coordinates at the suspension point: $\vect a = -\ell\dot\theta^2\,\vect e_r + \ell\ddot\theta\,\vect e_\theta$; forces: weight $mg(\cos\theta\,\vect e_r - \sin\theta\,\vect e_\theta)$ and tension $-T\vect e_r$. Along $\vect e_\theta$: $m\ell\ddot\theta = -mg\sin\theta$,

which for small angles ($\sin\theta \approx \theta$) is the harmonic equation with $\omega_0 = \sqrt{g/\ell}$: period $T_0 = 2\pi\sqrt{\ell/g}$, the $k = 2\pi$ that Example 1.8 could not supply. Along $\vect e_r$: $T = mg\cos\theta + m\ell\dot\theta^2$ — the string pulls hardest at the bottom, where the speed is greatest.