Converting Between Linear & Rotational Video Lessons

Concept

# Problem: A fan blade rotates with angular velocity given by ωz(t)= γ − β t2, where γ = 4.85 rad/s and β = 0.780 rad/s3.a- Calculate the angular acceleration as a function of time t in terms of β and γ. Express your answer in terms of some or all of the variables β, γ, and t.b- Calculate the instantaneous angular acceleration αz at t0 = 3.3 s. c- Calculate the average angular acceleration αav−z for the time interval t=0 to t0 = 3.3 s.

###### FREE Expert Solution

a-

Angular acceleration:

$\overline{){\mathbf{a}}{\mathbf{=}}\frac{\mathbf{d}{\mathbf{\omega }}_{\mathbf{z}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}}{\mathbf{d}\mathbf{t}}}$

$\mathbit{a}\mathbf{=}\frac{\mathbf{d}}{\mathbf{d}\mathbf{t}}\mathbf{\left(}\mathbit{\gamma }\mathbf{-}\mathbit{\beta }{\mathbit{t}}^{\mathbf{2}}\mathbf{\right)}$

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###### Problem Details

A fan blade rotates with angular velocity given by ωz(t)= γβ t2, where γ = 4.85 rad/s and β = 0.780 rad/s3.

a- Calculate the angular acceleration as a function of time t in terms of β and γ. Express your answer in terms of some or all of the variables β, γ, and t.

b- Calculate the instantaneous angular acceleration αz at t0 = 3.3 s.

c- Calculate the average angular acceleration αav−z for the time interval t=0 to t0 = 3.3 s.