Problem — Disc with an unwinding rope

Problem

A disc ($m_d = 4\;\mathrm{kg}$, $R = 1\;\mathrm{m}$, $I_d = \tfrac{1}{2}m_d R^2$) rotates at $\omega_A = 1\;\mathrm{rad/s}$ about a central pivot. A rope connects the rim to a packet ($m_p = 7\;\mathrm{kg}$) initially at $r_A = 1\;\mathrm{m}$ from the centre. When the rope has unwound, the packet is at $r_B = 2\;\mathrm{m}$. Calculate $\omega_B$.

Solution

Choosing the centre of the disc as the pole, the pivot’s reaction and the (radial) tension produce no torque: angular momentum is conserved. Treating the packet as a point mass at distance $r$: $\left(\tfrac{1}{2}m_d R^2 + m_p r_A^2\right)\omega_A = \left(\tfrac{1}{2}m_d R^2 + m_p r_B^2\right)\omega_B$ With $\tfrac{1}{2}m_d R^2 = 2\;\mathrm{kg\,m^2}$: $(2 + 7\cdot 1^2)\cdot 1 = (2 + 7\cdot 2^2)\,\omega_B \quad\Rightarrow\quad 9 = 30\,\omega_B$ $\ev{\omega_B = 0{,}3\;\mathrm{rad/s}}$ The system slows down: by moving the packet outward the moment of inertia grows from $9$ to $30\;\mathrm{kg\,m^2}$, and $\omega$ drops proportionally.

Links

Topics: Rotational dynamics Concepts: Conservation of angular momentum · Moment of inertia Skills: Conservation of angular momentum Objects: Rotating disc