Rolling as Translation and Rotation Combined — Question 9

Question 9

A radius mismatch creates two incompatible no-slip demands

A rigid spool rests on its outer radius $R=0.30\,\mathrm m$ and has a cord leaving horizontally from the top of an inner hub of radius $r=0.10\,\mathrm m$. It rolls right on stationary ground at center speed $V=0.60\,\mathrm{m/s}$; the cord does not slip on the hub.

1. Find cord speed.

2. Could the same motion occur if the cord end were held fixed?

Question 9 - Solution

A radius mismatch creates two incompatible no-slip demands

(a) Derivation and calculation

Ground rolling imposes $$V+\omega R=0\Rightarrow\omega=-.60/.30=-2.00\ \mathrm{rad/s}.$$ At the top hub point $\vec r=r\hat y$, no cord slip requires $$v_{\mathrm{cord}}=V-\omega r=.60-(-2)(.10) =\boxed{.800\ \mathrm{m/s}\ \text{right}}.$$

(b) Result and interpretation

A fixed cord would add the constraint $$V-\omega r=0.$$ Subtract this from $V+\omega R=0$: $$\omega(R+r)=0\Rightarrow\boxed{\omega=0,\quad V=0}.$$ The proposed nonzero motion cannot satisfy both fixed-cord and fixed-ground no-slip constraints.

Physical check

The hub contact velocity exceeds the center velocity because both contributions point right.