Newton's Second Law for Rotation — Question 4

Question 4

A compound drum uses different moment arms for its cords

A fixed compound drum has inertia $0.50\,\mathrm{kg\,m^2}$ and coaxial radii $0.30$ and $0.20\,\mathrm m$. Tangential cord forces of $12$ and $8\,\mathrm N$ respectively produce positive and negative torques.

1. Find its angular acceleration and each rim's signed tangential acceleration.

2. Can one subtract the cord forces before multiplying by a single radius?

Question 4 - Solution

A compound drum uses different moment arms for its cords

(a) Derivation and calculation

Each cord force must be multiplied by its own radius: $$\tau_{\mathrm{net}}=12(0.30)-8(0.20)=2.0\ \mathrm{N\,m}.$$ $$\boxed{\alpha=\frac{2.0}{0.50}=4.0\ \mathrm{rad/s^2}},\qquad \boxed{a_{t,o}=0.30(4)=1.20},\quad\boxed{a_{t,i}=0.20(4)=0.80\ \mathrm{m/s^2}}.$$ These tangential components use the positive local angular tangent at each rim.

(b) Result and interpretation

The correct moment difference is $$F_oR_o-F_iR_i\ne(F_o-F_i)R\quad\text{for an arbitrary common }R.$$ There is no shared application radius for the two forces, so subtracting forces first loses the lever-arm information.

Physical check

The larger radius has the larger tangential acceleration even though both rims share one angular acceleration.