Question

1. a rotating fan blade has a tip speed of 18 m/s and a rotation period of 0.60 s.

a) what is the radius of the fan blade?
b) what is the centripetal acceleration at the tip?

Explanation:

Step1: Recall tip speed formula

The tip speed $v$ of a rotating object is given by $v = \frac{2\pi r}{T}$, where $r$ is the radius and $T$ is the rotation period. Rearranged for $r$: $r = \frac{vT}{2\pi}$

Step2: Calculate blade radius

Substitute $v=18\ \text{m/s}$, $T=0.60\ \text{s}$:
$r = \frac{18 \times 0.60}{2\pi} = \frac{10.8}{2\pi} \approx 1.72\ \text{m}$

Step3: Recall centripetal acceleration formula

Centripetal acceleration $a_c$ is $a_c = \frac{v^2}{r}$

Step4: Calculate centripetal acceleration

Substitute $v=18\ \text{m/s}$, $r \approx 1.72\ \text{m}$:
$a_c = \frac{18^2}{1.72} = \frac{324}{1.72} \approx 188.37\ \text{m/s}^2$

Answer:

a) $\approx 1.72\ \text{m}$
b) $\approx 188.4\ \text{m/s}^2$