50. Multiple-Concept Example 7 explores the approach taken in problems such as t

Question

50. Multiple-Concept Example 7 explores the approach taken in problems such as this one. The blades of a ceiling fan have a radius of 0.380 m and are rotating about a fixed axis with an angular velocity of +1.50 rad/s. When the switch on the fan is turned to a higher speed, the blades acquire an angular acceleration of +2.00 rad/s. After 0.500 s has elapsed since the switch was reset, what is (a) the total accelera- tion (in m/s) of a point on the tip of a blade and (b) the angle between the total acceleration a and the centripetal acceleration ae? (See Figure 8.12b.)

Explanation / Answer

Here ,

radius , r = 0.380 m

angular velocity , w = 1.50 rad/s

angular acceleration ,alpha = 2 rad/s^2

after the time 0.5 s

for the final angular velocity

wf = wi + alpha * t

wf = 1.50 + 2 * 1

wf = 2.5 rad/s

centripetal acceleratio , ac = wf^2 * r

ac = 2.5^2 * 0.380 m/s^2

ac = 2.375 m/s^2

for tangential acceleration

at = alpha * r

at = 1.5 * 0.380

at = 0.57 m/s^2

a) for the total acceleration = sqrt(at^2 + ac^2)

total acceleration = sqrt(2.375^2 + 0.57^2)

total acceleration = 2.44 m/s^2

b)

angle =arctan(at/acceleration)

angle = arctan(0.57/2.44)

angle = 13.14 degree

the angle of total acceleration with centripetal acceleration is 13.14 degree

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