Exemplo de Physics, 6.1 Angle of Rotation and Angular Velocity

Angle of rotation at a Clock Tower

Problema e solução em inglês, como no livro original.

The clock on a clock tower has a radius of 1.0 m. (a) What angle of rotation does the hour hand of the clock travel through when it moves from 12 p.m. to 3 p.m.? (b) What’s the arc length along the outermost edge of the clock between the hour hand at these two times?

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Estratégia
We can figure out the angle of rotation by multiplying a full revolution ( $2 π$ radians) by the fraction of the 12 hours covered by the hour hand in going from 12 to 3. Once we have the angle of rotation, we can solve for the arc length by rearranging the equation $Δ θ = \frac{Δ s}{r}$ since the radius is given.
Solução
Solution to (a)

In going from 12 to 3, the hour hand covers 1/4 of the 12 hours needed to make a complete revolution. Therefore, the angle between the hour hand at 12 and at 3 is $\frac{1}{4} \times 2 π rad = \frac{π}{2}$ (i.e., 90 degrees).
Solução
Solution to (b)

Rearranging the equation

$Δ θ = \frac{Δ s}{r},$
6.4

we get

$Δ s = r Δ θ .$
6.5

Inserting the known values gives an arc length of

$\begin{matrix} Δ s & = & (1.0 m) (\frac{π}{2} rad) \\ = & 1.6 m \end{matrix}$
6.6

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