Question

A water wheel with a radius of 12 feet spins at a rate of 1 revolution in 1.5 minutes. Find the angular velocity of the water wheel in radians per minute:

Answer 1

Explanation:

there are per definition 2pi radians in the circumference of a circle (any circle, in the same way as every circle has 360°).

1 radian = the angle of the length of 1 radius on the circumference (full arc) of the circle.

because the definition is

360° = 2pi × radians

and the circumference of a circle is

2pi × radius

so, the actual length of the radius is for this kind of problem irrelevant, as every circle has the same number of radius units (radians) on its circumference.

when we are asking about how many radius units (radians) the wheel turns per minute, the size of the wheel is irrelevant.

FYI : but if we would ask e.g. about the speed of a single point on the circumference when turning with a specific number of radians/minute, then the length of the radius is important.

therefore,

1 rotation = 2pi radians

1 rotation / 1.5 minutes = 2pi radians / 1.5 minutes

to get radians / 1 minute we need to find the factor to go from 1.5 to 1.

remember, we need to multiply both, the numerator and the denominator, by the same factor to keep the overall value of the fraction or ratio the same.

to go from 1.5 to 1 we need to multiply by

1/1.5 = 1 / 3/2 = 2/3

and we get

2pi radians / 3/2 minutes × 2/3 / 2/3 =

= (2pi×2/3 radians) / (3/2 × 2/3 minutes) =

= 4pi/3 radians / 1 minute =

= 4.188790205... radians/minute

Answer 2

Angular velocity (in radians per minute) = Rotational speed (in revolutions per minute) * 2 * π

where π (pi) =3.14159.

the speed of rotation is 1 revolution in 1.5 minutes, or 1/1.5 = 0.66667 revolutions per minute.

Angular velocity = 0.66667 * 2 * π ≈ 4.18879 radians per minute

So the angular speed of the water wheel is about 4.18879 radians per minute.

Explanation: