The tape is now wound back into the drum at angular rate omega(t). with what speed will the end of the tape move? (note that our drawing specifies that a positive derivative of x(t)implies motion away from the drum. be careful with your signs! the factthat the tape is being wound back into the drum implies that \omega(t) < 0, and for the end of the tape to move closer to the drum, it must be the case that v(t) < 0.
answer in terms of omega(t) and other given quantities from the problem introduction.
v(t) =

See description

Explanation:

Frist we need to know the longitude of tape which is unwinding. Such relationship can be obtained with arc length. An arc length is the distance bewteen two points in a curve.

The relationship is:

Where is the arc length or distance, theta is the angle that results from the initial point of the measure to the final point, and r is the radius of a circumference.

Now let be length the unwinded tape. Change by and you ge the relationship:

if you unwind the tape by one revolution () you get the perimeter of a cricle , if you unwind it two times then and so on.

Then we have that the derivative of is

so we replace:

the derivative of theta with respect to t is ω(t) by definition:

The result is the relationship between angular velocity and the velocity and tangential velocity at the point r

good looking response

asthenosphere and convection