Old time mills

The radius of the wheel ( r ) is 3 m.

The period of the water wheel is the time required to cover the entire circumference of the wheel.

The circumference of the wheel = 2 * pi * r

                                                   = 2 * 3.14  * 3

                                                   = 18.84 m

Speed of the river = .5 m/s

Period of the wheel = time required to cover the cirumference at .5 m/s

                         = 18.81 / .5

                         = 37.68 seconds

As the water wheel spins, the bottoms of the paddles trace out a circle with radius $3,$ so every paddle travels exactly $2\pi r = 2\pi \cdot 3 = 6\pi \; \text{m}$ in one period. The speed of the river (and so the speed of the paddles) is $0.5 \; \text{m/s},$ so the period of the wheel is $\dfrac{6\pi \; \text{m}}{0.5 \; \text{m/s}} = 12\pi \; \text{sec},$ which is about $\boxed{37.7}\; \text{sec}.$

The speed of water flowing is will be used to rotate the wheel. So liner speed of Wheel = Speed of Water flowing

Here, liner speed of Wheel = Vc, Speed of Water flowing = Vw

So Vc = Vw

In one revolution The Wheel will rotate 360 degree or 2 pi radian.

From the defination of radian angle, angle = s/r

or, s = angle x r , angle = 2 pi , or, s = 2 x pi x r
,pi = 3.1416 , r = 3, s = 2 x 3.1416 x 3 = 18.85 Now, Vc = Vw = 0.5 vc = s/t t = s/ vc t = 18.8 / 0.5= 37.6 seconds

omega(angular velocity)=tangential velocity/radius=0.5/3=0.1667 omega * time=theta(in radians) here,theta = 2 * pi time =2 * pi/0.1667=37.7

The key to this problem is knowing that the rate at which the wheel turns is equal to the rate of the flowing water, namely $0.5 m/s$ . Thus, the time it takes for the water wheel is equal to the length of its circumference divided by $0.5 m/s$ . So, all we need to do now is plug and chug: $C = 2 \pi r= 6\pi \rightarrow P = \frac {6 \pi}{0.5} = 12 \pi = 37.7$ reported with $3$ significant figures.

The radius of the wheel is 3m, so the circumfrence is 6π. You multiply the circumfrence by the reciprocal of the river speed to find the time it takes to complete one rotation. So 12π = 37.699