A wheel rotates with an angular acceleration given by `alpha =4at^3 - 3 bt^2`, where `t` is the time and `a` and `b` are constants. If the wheel has initial angular speed `omega_0`, write the equations for the (a) angular speed (b) angular displacement.

Explanation

We know `d omega = alpha dt `
Integrating both sides , we get
`overset(omega) underset (omega_(0)) int d omega = overset (t) underset (0) alpha dt = underset(t) overset(0) ( 4 at^(3) - 3bt^(2))` or
`omega = omega_(0) +at^(4) - bt^(3)`
we know `d theta = omega dt `
On integrating both the sides , we get
`overset (theta) underset(0)int d theta = overset(t) underset (0) int omega dt = overset (t) underset (0) omega dt = overset (t) underset (0) int (omega_(0) +at^(4) -bt^(3))dt`
or `theta = omega_(0) t + (at^(5))/5 - (bt^(4))/4 `