# Why do we need to convert revolutions into radians before we multiply with meters?

When we are calculating tangential acceleration, I notice that we have to convert the $$\alpha$$ that was in unit of rev/min^2 into the unit of rad/min^2 before we are able to multiply this value with a value of unit $$m$$, such as in the following picture:

Why is this unit conversion necessary? Why can't we multiply rev/min^2 with $$m$$?

It's because the circumference of a circle with radius $$r$$ is $$2\pi r$$. It's easier to see with velocities, so I'll stick to that for now. If you want to know the tangential velocity of a particle rotating around some point, you have to divide the distance it travels by the time it takes to do that. At $$1\,\mathrm{rpm}$$, the distance traveled in $$1\,\mathrm{min}$$ is exactly the circumference, which is $$2\pi r$$. At $$2\,\mathrm{rpm}$$, we travel around the circumference twice per minute, so $$4\pi r$$. With units, and angular velocity $$\omega$$, that is
$$v_\mathrm{T}~\mathrm{[m/min] = \omega~\mathrm{[rev/min]}} \cdot 2\pi\,r~\mathrm{[m]}$$
So the $$2\pi$$ really comes from calculating the circumference using the radius. Radians already have this feature contained in their definition. The distance traveled by something having moved $$1\,\mathrm{rad}$$ along a circle of radius $$r$$ is exactly $$r$$ meters. That makes using radians often very convenient.
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