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The 165-lb flywheel has a radius of gyration about its shaft axis of [tex]k = 20 \text{ in.}[/tex] and is subjected to the torque [tex]M = 8(1 - e^{-t}) \text{ lb-ft}[/tex], where [tex]t[/tex] is in seconds. If the flywheel is at rest at time [tex]t = 0[/tex], determine its angular velocity [tex]\omega[/tex] at [tex]t = 3 \text{ sec}[/tex].

Answer :

Final answer:

To determine the angular velocity of the flywheel at t = 3 seconds, calculate the angular acceleration and integrate it to find the change in angular velocity.

Explanation:

To determine the angular velocity of the flywheel at t = 3 seconds, we need to find the angular acceleration first. The torque, M, can be written as M = Iα, where I is the moment of inertia and α is the angular acceleration. From the given torque equation, M = 8(1−e^-t) lb-ft, we can solve for α by rearranging the equation as α=(1/I)dM/dt.

To find ω at t = 3 seconds, we can integrate α with respect to t from 0 to 3 seconds to find the change in angular velocity. ω = ∫(0 to 3)αdt.

Finally, substitute the values into the equation to calculate the angular velocity ω at t = 3 seconds.

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