A fan blade rotates with angular velocity given by ω z (t) = g - bt 2 , where g = 5.00 rad/s and b = 0.800 rad/s 3 . Calculate the angular acceleration as a function of time.

Welcome back everybody. We are given the formula for angular velocity as a function of time for some rotating particle. And it is given as follows, C. Which is a constant equal to 4.2 -3. Another constant equal to 0.25 times our time squared. And we need to find the angular acceleration as a function of time. Well this is going to be just the derivative of our angular velocity with respect to time. So let's go ahead and take the derivative of this equation right here, term by term derivative of 4.2, it's a constant. So it's just gonna be zero minus. We have this term that too is gonna come down in front times 0.5 Times are T. Which gives us that our angular acceleration as a function of time equal to negative 0. radiance per second cubed times time corresponding to our answer choice of Thank you all so much for watching. Hope this video helped. We will see you all in the next one.

Ch 09: Rotation of Rigid Bodies

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 09: Rotation of Rigid Bodies

Problem 4a

Chapter 9, Problem 4a

A fan blade rotates with angular velocity given by ω_z(t) = g - bt², where g = 5.00 rad/s and b = 0.800 rad/s³. Calculate the angular acceleration as a function of time.

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Start by recalling the relationship between angular velocity (ω) and angular acceleration (α). Angular acceleration is the time derivative of angular velocity: α(t) = dω(t)/dt.

Substitute the given expression for angular velocity, ω_z(t) = g - bt^2, into the formula for angular acceleration. This gives α(t) = d/dt [g - bt^2].

Differentiate the expression term by term. The derivative of g (a constant) is 0, and the derivative of -bt^2 is -2bt. Thus, α(t) = -2bt.

Substitute the given value of b = 0.800 rad/s^3 into the expression for angular acceleration. This gives α(t) = -2(0.800)t.

Simplify the expression to obtain the final formula for angular acceleration as a function of time: α(t) = -1.60t rad/s^2.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Angular Velocity

Angular velocity is a measure of how quickly an object rotates around an axis, expressed in radians per second (rad/s). In this context, the angular velocity function ω_z(t) describes how the speed of the fan blade's rotation changes over time, influenced by constants g and b.

Angular Acceleration

Angular acceleration is the rate of change of angular velocity over time, typically denoted as α. It can be calculated by taking the derivative of the angular velocity function with respect to time. In this case, finding α as a function of time will provide insight into how the rotation speed of the fan blade is changing.

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. It is used to determine rates of change, such as how angular velocity changes over time to yield angular acceleration. Understanding differentiation is crucial for solving the problem of calculating angular acceleration from the given angular velocity function.

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Textbook Question

The angular velocity of a flywheel obeys the equation ω_z(t) = A + Bt², where t is in seconds and A and B are constants having numerical values 2.75 (for A) and 1.50 (for B). What is the angular acceleration of the wheel at (i) t = 0 and (ii) t = 5.00 s?

A fan blade rotates with angular velocity given by ω_z(t) = g - bt², where g = 5.00 rad/s and b = 0.800 rad/s³. Calculate the instantaneous angular acceleration α_z at t = 3.00 s and the average angular acceleration α_av-z for the time interval t = 0 to t = 3.00 s. How do these two quantities compare? If they are different, why?

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Textbook Question

The angle θ through which a disk drive turns is given by θ(t) = a + bt - ct³, where a, b, and c are constants, t is in seconds, and θ is in radians. When t = 0, θ = π/4 rad and the angular velocity is 2.00 rad/s. When t = 1.50 s, the angular acceleration is 1.25 rad/s². Find a, b, and c, including their units.

A fan blade rotates with angular velocity given by ωz(t) = g - bt2, where g = 5.00 rad/s and b = 0.800 rad/s3. Calculate the angular acceleration as a function of time.

Verified video answer for a similar problem:

Key Concepts

Angular Velocity

Angular Acceleration

Differentiation

A fan blade rotates with angular velocity given by ω_z(t) = g - bt², where g = 5.00 rad/s and b = 0.800 rad/s³. Calculate the angular acceleration as a function of time.