A fan blade rotates with angular velocity given by ωz(t)= γ − β t2. part a calculate the angular acceleration as a function of time. express your answer in terms of the variables β, γ, and t. αz(t) = submitmy answersgive up part b if γ = 5.35 rad/s and β = 0.810 rad/s3 , calculate the instantaneous angular acceleration αz at t = 3.00 s .

Given:

w = z(t)= γ − β*t^2

Differentiating both sides with respect to t, we get:

α = z(t) = -2βt

Given: γ = 5.35 rad/s and β = 0.810 rad/s3

so, For t = 3 sec,

angular acceleration = -2 * 0.810 * 3 = -4.86

Answer Link

johanrusli johanrusli · Answer 2 · 2019-12-12T08:11:37+01:00

a. the angular acceleration = α_z(t) = -2βt

b. instantaneous angular acceleration at t = 3.00 s is -4.86 rad/s²

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Further explanation

Centripetal Acceleration can be formulated as follows:

[tex]\large {\boxed {a = \frac{ v^2 } { R } }[/tex]

a = Centripetal Acceleration ( m/s² )

v = Tangential Speed of Particle ( m/s )

R = Radius of Circular Motion ( m )

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Centripetal Force can be formulated as follows:

[tex]\large {\boxed {F = m \frac{ v^2 } { R } }[/tex]

F = Centripetal Force ( m/s² )

m = mass of Particle ( kg )

v = Tangential Speed of Particle ( m/s )

R = Radius of Circular Motion ( m )

Let us now tackle the problem !

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Given:

ω_z(t)= γ − β t²

Asked:

a. the angular acceleration = α_z(t) = ?

b. the angular acceleration = α_z(3.00) = ?

Solution:

Question a:

[tex]\alpha_z(t) = \frac{d}{dt} \omega_z (t)[/tex]

[tex]\alpha_z(t) = \frac{d}{dt} ( \gamma - \beta t^2 )[/tex]

[tex]\alpha_z(t) = ( 0 - 2\beta t)[/tex]

[tex]\alpha_z(t) = -2\beta t[/tex]

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Question b:

If γ = 5.35 rad/s and β = 0.810 rad/s³ , then at t = 3.00 s :

[tex]\alpha_z(t) = -2\beta t[/tex]

[tex]\alpha_z(3.00) = -2(0.810)(3.00)[/tex]

[tex]\alpha_z(3.00) = -4.86 \texttt{ rad/s}^2[/tex]

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Answer details

Grade: High School

Subject: Physics

Chapter: Circular Motion

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Keywords: Gravity , Unit , Magnitude , Attraction , Distance , Mass , Newton , Law , Gravitational , Constant