Multiple-Concept Example 7 explores the approach taken in problems such as this
ID: 2182028 • Letter: M
Question
Multiple-Concept Example 7 explores the approach taken in problems such as this one. The blades of a ceiling fan have a radius of 0.375 m and are rotating about a fixed axis with an angular velocity of +1.78 rad/s. When the switch on the fan is turned to a higher speed, the blades acquire an angular acceleration of +1.93 rad/s2. After 0.625 s have elapsed since the switch was reset, what is (a) the total acceleration (in m/s2) of a point on the tip of a blade and (b) the angle between the total acceleration and the centripetal acceleration (See Figure 8.13b)?
Multiple-Concept Example 7 explores the approach taken in problems such as this one. The blades of a ceiling fan have a radius of 0.375 m and are rotating about a fixed axis with an angular velocity of +1.78 rad/s. When the switch on the fan is turned to a higher speed, the blades acquire an angular acceleration of +1.93 rad/s2. After 0.625 s have elapsed since the switch was reset, what is (a) the total acceleration (in m/s2) of a point on the tip of a blade and (b) the angle between the total acceleration and the centripetal acceleration (See Figure 8.13b)?Explanation / Answer
You're given angular acceleration and angular velocity and and you want to know the total acceleration at the tip fo the blade. The total acceleration is composed of tangential acceleration and the centripetal acceleration. The tangential acceleration (a_t) is a*R and the centripetal acceleration (a_c) is ?^2*R=(?0+a*t)^2*R (since you need to add a*t to the original angular velocity) sqrt( (1.51*0.361)^2+ ( ( ( (1.87+1.51*0.492)^2)*0.361)^2) ) By pythagoras theorem: a_total=sqrt(a_t^2+a_c^2)=sqrt( (a*R)^2 + ( (?0+a*t)^2*R)^2) ) =2.52423479 m*s^-2 (but you better plug in yourself and check the result) b) The total acceleration is the hypotenuse here, the centripetal acceleration is the ordinate. We want to find the adjacent angle, so: centripetal acceleration/total acceleration=cos f => => f=arccos(centripetal acceleration/total acceleration)=arccos((?0+a*t)^2*R/(a_tot… = =arccos(1.2623809/2.52423479)= 1.04707701 rad= 59.9930935 degrees
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