Answer:
a
The number of radians turned by the wheel in 2s is [tex]\theta= 8\ radians[/tex]
b
The angular acceleration is [tex]\alpha =14 rad/s^2[/tex]
Explanation:
The angular velocity is given as
[tex]w(t) = (2.00 \ rda/s^2)t + (1.00 rad /s^4)t^3[/tex]
Now generally the integral of angular velocity gives angular displacement
So integrating the equation of angular velocity through the limit 0 to 2 will gives us the angular displacement for 2 sec
This is mathematically evaluated as
[tex]\theta(t ) = \int\limits^2_0 {2t + t^3} \, dt[/tex]
[tex]= [\frac{2t^2}{2} + \frac{t^4}{4}] \left\{ 2} \atop {0}} \right.[/tex]
[tex]= [\frac{2(2^2)}{2} + \frac{2^4}{4}] - 0[/tex]
[tex]= 4 +4[/tex]
[tex]\theta= 8\ radians[/tex]
Now generally the derivative of angular velocity gives angular acceleration
So the value of the derivative of angular velocity equation at t= 2 gives us the angular acceleration
This is mathematically evaluated as
[tex]\frac{dw}{dt} = \alpha (t) = 2 + 3t^2[/tex]
so at t=2
[tex]\alpha (2) = 2 +3(2)^2[/tex]
[tex]\alpha =14 rad/s^2[/tex]
