It takes a fan 0.8 seconds to reach its final angular speed of 4 revolutions per

Question

It takes a fan 0.8 seconds to reach its final angular speed of 4 revolutions per second. The fan is 20 cm in radius. Assuming that the magnitude of the acceleration is constant:
A.) determine the equation that gives the speed of the blade tip as a function of time for the time interval of 0 to 0.8 seconds. That is, let V=Ct, where C is a constant. Determined the value of the constant, C.
B.) what is the instantaneous speed of the tip of the fan blade of 0.5 seconds after the fan is turned on?
C.) what are the magnitudes of the tangential and radial accelerations of the blade tip at 0.5 second?
D.) what is the resultant total acceleration vector at t=0.5 second? Give magnitude and angle. Denote this on a diagram, being sure to indicate the direction of rotation of the fan on the diagram! It takes a fan 0.8 seconds to reach its final angular speed of 4 revolutions per second. The fan is 20 cm in radius. Assuming that the magnitude of the acceleration is constant:
A.) determine the equation that gives the speed of the blade tip as a function of time for the time interval of 0 to 0.8 seconds. That is, let V=Ct, where C is a constant. Determined the value of the constant, C.
B.) what is the instantaneous speed of the tip of the fan blade of 0.5 seconds after the fan is turned on?
C.) what are the magnitudes of the tangential and radial accelerations of the blade tip at 0.5 second?
D.) what is the resultant total acceleration vector at t=0.5 second? Give magnitude and angle. Denote this on a diagram, being sure to indicate the direction of rotation of the fan on the diagram! It takes a fan 0.8 seconds to reach its final angular speed of 4 revolutions per second. The fan is 20 cm in radius. Assuming that the magnitude of the acceleration is constant:
A.) determine the equation that gives the speed of the blade tip as a function of time for the time interval of 0 to 0.8 seconds. That is, let V=Ct, where C is a constant. Determined the value of the constant, C.
B.) what is the instantaneous speed of the tip of the fan blade of 0.5 seconds after the fan is turned on?
C.) what are the magnitudes of the tangential and radial accelerations of the blade tip at 0.5 second?
D.) what is the resultant total acceleration vector at t=0.5 second? Give magnitude and angle. Denote this on a diagram, being sure to indicate the direction of rotation of the fan on the diagram!

Explanation / Answer

here ,

final angular speed , wf = 4 * 2pi /s

wf = 25.13 rad/s

initial angular speed , wi - 0 rad/s

a) angular acceleration = (wf - wi)/t

angular acceleration = (25.13 - 0)/.8

angular acceleration = 31.4 rad/s^2

Now, using first equation of motion

v = angular acceleration * R * t

v = 31.4 * 0.20 * t

v = 6.28 * t

the speed of blades is 6.28 *t

b)

at t = 0.5 s

v = 6.28 * 0.5

v = 3.14 m/s

c)

magnitude of tangential accelertion ,

at = angular acceleration * r

at = 31.4 * 0.20

at = 6.28 m/s^2

radial acceleration = v^2/r = 3.14^2/0.20

radial acceleration = 49.3 m/s^2

D)

resultant acceleration = sqrt(6.28^2 + 49.3^2) at arctan(49.3/6.28)

resultant acceleration = 49.7 m/s^2 at 82.7 degree from the direction of motion of tip

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