Relating the Linear and Angular Variables — Question 5

Question 5

A total acceleration limit sets an allowable radius

A rigid disk has $\omega=6\,\mathrm{rad/s}$ and $\alpha=8\,\mathrm{rad/s^2}$. A mounted sensor must have acceleration magnitude no greater than $10\,\mathrm{m/s^2}$. Its distance from the fixed axis can be chosen.

1. Find the largest allowed radius.

2. Would a radius of $0.30\,\mathrm m$ satisfy the limit?

Question 5 - Solution

A total acceleration limit sets an allowable radius

(a) Derivation and calculation

The fixed-axis acceleration magnitude is $$a^2=(r\omega^2)^2+(r\alpha)^2=r^2(\omega^4+\alpha^2).$$ $$r\sqrt{6^4+8^2}\le10\Rightarrow \boxed{r_{\max}=\frac{10}{\sqrt{1360}}=0.2712\ \mathrm m}.$$

(b) Result and interpretation

At the proposed radius, $$a(0.30)=0.30\sqrt{1360}=\boxed{11.06\ \mathrm{m/s^2}}>10\ \mathrm{m/s^2}.$$ It fails the limit. Scalar addition $r\omega^2+r|\alpha|$ would overestimate the actual vector magnitude.

Physical check

Acceleration is proportional to radius for a fixed angular state, so every smaller radius than the bound is admissible.