Conservation of Angular Momentum — Question 6

Question 6

A radial pull changes tangential speed without supplying torque

A $1\,\mathrm{kg}$ bead slides on a frictionless horizontal table, attached to a cord through a small central hole. Initially it moves tangentially at radius $2\,\mathrm m$ and speed $3\,\mathrm{m/s}$. The cord is pulled until the radius is $1\,\mathrm m$; radial speed is zero at both endpoints.

1. Find final tangential speed.

2. Find work done by the pull and explain why zero torque does not imply zero work.

Question 6 - Solution

A radial pull changes tangential speed without supplying torque

(a) Derivation and calculation

The cord force is radial, so its torque about the hole vanishes: $$\vec\tau=\vec r\times(-T\hat e_r)=0\Rightarrow mr_iv_i=mr_fv_f.$$ Thus $$\boxed{v_f=\frac{r_i}{r_f}v_i=\frac21(3)=6.00\ \mathrm{m/s}}.$$

(b) Result and interpretation

Radial speed is zero at both endpoints, so the work is the change in tangential kinetic energy: $$\boxed{W=\Delta K=\tfrac12(1)(6^2-3^2)=13.5\ \mathrm J}.$$ During inward motion, $$\mathcal P_T=\vec T\cdot\vec v=-T\dot r>0\quad(\dot r<0).$$ Zero torque does not imply zero work: radial tension has a zero moment but can act along the radial displacement.

Physical check

The final angular speed is $v_f/r_f=6$ rad/s, while $mr v$ stays at 6 kg m$^2$/s.