PX155 - E13 - cylinder rolling down a slope

a cylinder with mass, $m$ , and radius, $r$
where the force acts matters for torque
PX155 - A2 - newton's second law:
- perpendicular to the slope: $N - m g \cos θ = 0. . . [1]$
- parallel to the slope: $m g \sin θ - F_{s} = m a . . . [2]$
- for torque:
  $τ = I \dot{ω}$ about axis through the centre of cylinder, and out of the page
  $F_{s} r = I \dot{ω} . . . [3]$
- rolling without slipping: $v = r ω ⟹ a = r \dot{ω} . . . [4]$
- $[4] \to [3] : F_{s} = \frac{I a}{r^{2}} . . . [5]$
- $[5] \to [2] : m g \sin θ - \frac{I}{r^{2}} a = m a$
$a = \frac{g \sin θ}{1 + \frac{I}{m r^{2}}}$