Answer:
a) The tangential component of acceleration at the edge of the motor at
is -1.075 meters per square second.
b) The electric motor will take approximately 3.963 seconds to decrease its angular velocity by 75 %.
Explanation:
The angular aceleration of the electric motor (
), measured in radians per square second, as a function of time (
), measured in seconds, is determined by the following formula:
(1)
The function for the angular velocity of the electric motor (
), measured in radians per second, is found by integration:
(2)
Where
is the initial angular velocity, measured in radians per second.
a) The tangential component of aceleration (
), measured in meters per square second, is defined by the following formula:
(3)
Where
is the radius of the electric motor, measured in meters.
If we know that
,
and
, then the tangential component of the acceleration at the edge of the motor is:
![a_{t} = (7.165\times 10^{-2}\,m)\cdot (-10)\cdot (1.5\,s)](https://tex.z-dn.net/?f=a_%7Bt%7D%20%3D%20%287.165%5Ctimes%2010%5E%7B-2%7D%5C%2Cm%29%5Ccdot%20%28-10%29%5Ccdot%20%281.5%5C%2Cs%29)
![a_{t} = -1.075\, \frac{m}{s^{2}}](https://tex.z-dn.net/?f=a_%7Bt%7D%20%3D%20-1.075%5C%2C%20%5Cfrac%7Bm%7D%7Bs%5E%7B2%7D%7D)
The tangential component of acceleration at the edge of the motor at
is -1.075 meters per square second.
b) If we know that
and
, then the time needed is:
![26.180\,\frac{rad}{s} = 104.720\,\frac{rad}{s}-5\cdot t^{2}](https://tex.z-dn.net/?f=26.180%5C%2C%5Cfrac%7Brad%7D%7Bs%7D%20%3D%20104.720%5C%2C%5Cfrac%7Brad%7D%7Bs%7D-5%5Ccdot%20t%5E%7B2%7D)
![5\cdot t^{2} = 104.720\,\frac{rad}{s}-26.180\,\frac{rad}{s}](https://tex.z-dn.net/?f=5%5Ccdot%20t%5E%7B2%7D%20%3D%20104.720%5C%2C%5Cfrac%7Brad%7D%7Bs%7D-26.180%5C%2C%5Cfrac%7Brad%7D%7Bs%7D)
![t^{2} = \frac{104.720\,\frac{rad}{s}-26.180\,\frac{rad}{s} }{5}](https://tex.z-dn.net/?f=t%5E%7B2%7D%20%3D%20%5Cfrac%7B104.720%5C%2C%5Cfrac%7Brad%7D%7Bs%7D-26.180%5C%2C%5Cfrac%7Brad%7D%7Bs%7D%20%20%7D%7B5%7D)
![t = \sqrt{\frac{104.720\,\frac{rad}{s}-26.180\,\frac{rad}{s} }{5} }](https://tex.z-dn.net/?f=t%20%3D%20%5Csqrt%7B%5Cfrac%7B104.720%5C%2C%5Cfrac%7Brad%7D%7Bs%7D-26.180%5C%2C%5Cfrac%7Brad%7D%7Bs%7D%20%20%7D%7B5%7D%20%7D)
![t \approx 3.963\,s](https://tex.z-dn.net/?f=t%20%5Capprox%203.963%5C%2Cs)
The electric motor will take approximately 3.963 seconds to decrease its angular velocity by 75 %.