A blender with blades of radius 3.2 cm has a switch that flips between two setti

Question

A blender with blades of radius 3.2 cm has a switch that flips between two settings, slow and fast- these settings correspond to angular velocities of +420 rad/s and +548 rad/s, respectively. If the blender is on the slow setting when the switch is flipped, the blades will steadily accelerate at a constant angular rate of +2816 rad/s^2 to reach the fast angular velocity. a) How many revolutions do the blades make in the time it takes them to accelerate from slow to fast? b) What is the time it takes for the blades to accelerate from slow to fast? c) For a point on the tip of the blade, calculate the magnitude and direction of the total acceleration immediately before- and immediately after the switch is flipped from slow to fast. d) Sketch the angular speed vs time graph for the blades as they switch from slow to fast, and show how to figure out the answers to (a) and (b) above from the plot.

Explanation / Answer

slow angular speed is wo = 420 rad/sec

fast angular speed is w = 548 rad/sec

angular accelaration is alpha = 2816 rad/s^2

a) using linematic equations

w^2 - wo^2 = 2*alpha*theta

548^2 - 420^2 = 2*2816*theta

theta = 22 rad

no.of revolutions is 22/(2*pi) = 3.5 rev = 3 complete revolutions

b) using w = wo+(alpha*t)

548 = 420+(2816*t)

t = 0.045 sec

c) immediately before

a = r*w^2 = 0.032*420^2 = 5645 m/s^2

towards tip of the fixed end of the blade

immediately after

a = r*w^2 = 0.032*548^2 = 9610 m/s^2

towards tip of the fixed end of the blade

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