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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: enter image description here

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

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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.

For the acceleration it's a bit harder to visualize, but the principle is exactly the same.

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