Question

After the fuel is turned off, the rotor of a jet engine decelerates under the action of air friction, which depends on the square of its speed, and constant bearing friction. Thus the deceleration is written as b + c * w^2, where b and c are constants and w is the angular velocity of the rotor. Determine the time t required for the rotor to come to rest from a speed w_0.

Thanks in advance.

Answer 1

dw/dt = -b - c w^2
Solve by separation of variables

dt = -dw/(b + c w^2)
= -(dw/c)/[(b/c) + w^2]

Integrate the left side from t=0 to T while the right side goes from w_o to 0

The term on the right integrates to an arctangent function.

T = (1/c)*sqrt(c/b)[tan^-1(sqrt(c/b)w_o)) - tan^-1(0]
sqrt(1/bc)*tan^-1{sqrt[(c/b)*w_o]}

No guarantees on that. Try it yourself