Object rotation about a fixed axis? question about derivatives in this problem?

[nchin]

Object rotation about a fixed axis?? question about derivatives in this problem??

An object rotates about a fixed axis, and the angular position of a reference line on the object is given by θ=0.40e^(2t), where θ is in radians and t is in seconds. Consider a point on the object that is 4.0 cm from the axis of rotation. At t = 0, what are the magnitudes of the point's tangential component of acceleration and radial component of acceleration?

solution is on bottom of page 2 and top of page 3 # HRW 10.25

http://www.nvcc.edu/home/tstantcheva/231files/hrwch10hw.pdf

I have three questions:

1. for part (a), I don't understand how taking the derivative twice gets you alpha? Why is that?

2. for part (b), why does taking the derivative only once gets you the angular velocity (ω)?

3. Also, the angular position which is θ=0.40e^(2t), is that how much the object rotated?

thanks!

 

[haruspex]

Differentiating theta once gives you the rate of change of angle, i.e. the angular velocity; a second time gives you the angular acceleration. For constant radius, the linear acceleration in the tangential direction is given by angular acceleration * radius. (If the radius is also changing there's an r-dot*theta-dot term.)