Question

As steam is slowly injected into a turbine, the angular acceleration of the rotor is observed to increase linearly with the time t. Know that the rotor starts from rest at t = 0 and that after 10 s the rotor has completed 20 revolutions. Determine the angular velocity at t = 24 s.

Answer 1

Answer:

217.04 rad/s

Explanation:

We are given that

Initial velocity,$\omega_0=0$

t=10 s

Number of rev=20

We have to find the angular velocity at t=24 s

$\theta_1=2\pi\times 20=40\pi rad$

$\alpha=kt$

$\frac{d\omega}{dt}=kt$

$\int d\omega=\int_{0}^{t} ktdt$

$\omega=\frac{kt^2}{2}$

$\frac{d\theta}{dt}=\frac{kt^2}{2}$

$\int d\theta=\int_{0}^{t}\frac{kt^2}{2} dt$

$\theta=\frac{kt^3}{6}$

Substitute the values

$40\pi=\frac{k(10)^3}{6}$

$k=\frac{40\pi\times 6}{(10)^3}=\frac{24\pi}{100}$

Substitute the value of k and t=24

$\omega=\frac{24\pi\times (24)^2}{2\times 100}=217.04 rad/s$