Answer:
a)
, b) ![\omega = 0\,\frac{rad}{s}](https://tex.z-dn.net/?f=%5Comega%20%3D%200%5C%2C%5Cfrac%7Brad%7D%7Bs%7D)
Explanation:
The magnitude of torque is a form of moment, that is, a product of force and lever arm (distance), and force is the product of mass and acceleration for rotating systems with constant mass. That is:
![\tau = F \cdot r](https://tex.z-dn.net/?f=%5Ctau%20%3D%20F%20%5Ccdot%20r)
![\tau = m\cdot a \cdot r](https://tex.z-dn.net/?f=%5Ctau%20%3D%20m%5Ccdot%20a%20%5Ccdot%20r)
![\tau = m \cdot \alpha \cdot r^{2}](https://tex.z-dn.net/?f=%5Ctau%20%3D%20m%20%5Ccdot%20%5Calpha%20%5Ccdot%20r%5E%7B2%7D)
Where
is the angular acceleration, which is constant as torque is constant. Angular deceleration experimented by the unpowered flywheel is:
![\alpha = \frac{170\,\frac{rad}{s} - 200\,\frac{rad}{s} }{10\,s}](https://tex.z-dn.net/?f=%5Calpha%20%3D%20%5Cfrac%7B170%5C%2C%5Cfrac%7Brad%7D%7Bs%7D%20-%20200%5C%2C%5Cfrac%7Brad%7D%7Bs%7D%20%7D%7B10%5C%2Cs%7D)
![\alpha = -3\,\frac{rad}{s^{2}}](https://tex.z-dn.net/?f=%5Calpha%20%3D%20-3%5C%2C%5Cfrac%7Brad%7D%7Bs%5E%7B2%7D%7D)
Now, angular velocities of the unpowered flywheel at 50 seconds and 100 seconds are, respectively:
a) t = 50 s.
![\omega = 200\,\frac{rad}{s} - \left(3\,\frac{rad}{s^{2}} \right) \cdot (50\,s)](https://tex.z-dn.net/?f=%5Comega%20%3D%20200%5C%2C%5Cfrac%7Brad%7D%7Bs%7D%20-%20%5Cleft%283%5C%2C%5Cfrac%7Brad%7D%7Bs%5E%7B2%7D%7D%20%5Cright%29%20%5Ccdot%20%2850%5C%2Cs%29)
![\omega = 50\,\frac{rad}{s}](https://tex.z-dn.net/?f=%5Comega%20%3D%2050%5C%2C%5Cfrac%7Brad%7D%7Bs%7D)
b) t = 100 s.
Given that friction is of reactive nature. Frictional torque works on the unpowered flywheel until angular velocity is reduced to zero, whose instant is:
![t = \frac{0\,\frac{rad}{s}-200\,\frac{rad}{s} }{\left(-3\,\frac{rad}{s^{2}} \right)}](https://tex.z-dn.net/?f=t%20%3D%20%5Cfrac%7B0%5C%2C%5Cfrac%7Brad%7D%7Bs%7D-200%5C%2C%5Cfrac%7Brad%7D%7Bs%7D%20%7D%7B%5Cleft%28-3%5C%2C%5Cfrac%7Brad%7D%7Bs%5E%7B2%7D%7D%20%5Cright%29%7D)
![t = 66.667\,s](https://tex.z-dn.net/?f=t%20%3D%2066.667%5C%2Cs)
Since
, then the angular velocity is equal to zero. Therefore:
![\omega = 0\,\frac{rad}{s}](https://tex.z-dn.net/?f=%5Comega%20%3D%200%5C%2C%5Cfrac%7Brad%7D%7Bs%7D)