Answer:
a) The tangential component of acceleration at the edge of the motor at [tex]t = 1.5\,s[/tex] 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 ([tex]\alpha[/tex]), measured in radians per square second, as a function of time ([tex]t[/tex]), measured in seconds, is determined by the following formula:
[tex]\alpha = -10\cdot t\,\left[\frac{rad}{s^{2}} \right][/tex] (1)
The function for the angular velocity of the electric motor ([tex]\omega[/tex]), measured in radians per second, is found by integration:
[tex]\omega = \omega_{o} - 5\cdot t^{2}\,\left[\frac{rad}{s} \right][/tex] (2)
Where [tex]\omega_{o}[/tex] is the initial angular velocity, measured in radians per second.
a) The tangential component of aceleration ([tex]a_{t}[/tex]), measured in meters per square second, is defined by the following formula:
[tex]a_{t} = R\cdot \alpha[/tex] (3)
Where [tex]R[/tex] is the radius of the electric motor, measured in meters.
If we know that [tex]R = 7.165\times 10^{-2}\,m[/tex], [tex]\alpha = 10\cdot t[/tex] and [tex]t = 1.5\,s[/tex], then the tangential component of the acceleration at the edge of the motor is:
[tex]a_{t} = (7.165\times 10^{-2}\,m)\cdot (-10)\cdot (1.5\,s)[/tex]
[tex]a_{t} = -1.075\, \frac{m}{s^{2}}[/tex]
The tangential component of acceleration at the edge of the motor at [tex]t = 1.5\,s[/tex] is -1.075 meters per square second.
b) If we know that [tex]\omega_{o} = 104.720\,\frac{rad}{s}[/tex] and [tex]\omega = 26.180\,\frac{rad}{s}[/tex], then the time needed is:
[tex]26.180\,\frac{rad}{s} = 104.720\,\frac{rad}{s}-5\cdot t^{2}[/tex]
[tex]5\cdot t^{2} = 104.720\,\frac{rad}{s}-26.180\,\frac{rad}{s}[/tex]
[tex]t^{2} = \frac{104.720\,\frac{rad}{s}-26.180\,\frac{rad}{s} }{5}[/tex]
[tex]t = \sqrt{\frac{104.720\,\frac{rad}{s}-26.180\,\frac{rad}{s} }{5} }[/tex]
[tex]t \approx 3.963\,s[/tex]
The electric motor will take approximately 3.963 seconds to decrease its angular velocity by 75 %.
