Relating the Linear and Angular Variables — Question 9

Question 9

A tape speed law changes when the winding radius grows

A tape winds without slipping onto a reel whose effective radius is $R(t)=0.10+0.002t\,\mathrm m$. The reel maintains constant $\omega=5\,\mathrm{rad/s}$. The incoming tape speed is constrained by $v=R\omega$.

1. Find the tape speed and its time derivative at $t=10\,\mathrm s$.

2. Why would identifying the tape's speed derivative with $R\alpha$ fail?

Question 9 - Solution

A tape speed law changes when the winding radius grows

(a) Derivation and calculation

At the specified time, $$R(10)=0.10+0.002(10)=0.120\ \mathrm m.$$ $$\boxed{v(10)=R(10)\omega=0.120(5)=0.600\ \mathrm{m/s}}.$$ Apply the product rule: $$\boxed{\dot v=\dot R\omega+R\alpha=0.002(5)+0=0.010\ \mathrm{m/s^2}}.$$

(b) Result and interpretation

The two derivatives represent different constraints: $$\left.\frac{d(R\omega)}{dt}\right|_{R\ \mathrm{fixed}}=R\alpha, \qquad\frac{d(R(t)\omega(t))}{dt}=R\alpha+\dot R\omega.$$ Successive tape elements meet an increasing effective radius; the incoming speed is not the speed of a single fixed-radius material point.

Physical check

At constant angular speed, each added layer increases the incoming tape speed.