Awheel is rotating about an axis that is in the z-direction. the angular velocity ωz is −6.00rad/s at t=0, increases linearly with time and is +8.00rad/s at t=7.00s. we have taken counterclockwise rotation to be positive. q1: is the angular acceleration during this time interval positive or negative? q2: find the duration of the time interval when the speed of the wheel is increasing?

Q1)
The angular acceleration of the wheel is constant (because the angular velocity is increasing linearly), therefore we can calculate it as the difference between the final and initial angular speed divided by the duration of the motion:


In our problem, ,  and , therefore the angular acceleration is

and its positive sign tells us that the angular acceleration is positive: the angular velocity is increasing during the motion.

q2) 
the duration of the time interval when the speed of the wheel is increasing corresponds to the duration of the time interval when the angular velocity is positive, so we have to calculate the time instant when the angular velocity is zero: .
Since we know the acceleration, we can calculate this time by using

and by substituting :


This means that after the instant t=3.00 s the angular velocity of the wheel becomes positive, and this means also that its speed starts to increase. Therefore, the duration of the time interval when the speed of the wheel increases is

Alinear relationship would look like a straight line on the graph. it could have a positive negative or 0 slope. so it could be a line potentially going in any direction but it must be perfectly straight.