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Ch 09: Rotation of Rigid Bodies
Young & Freedman Calc - University Physics 14th Edition
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoYoung & Freedman Calc 14th EditionCh 09: Rotation of Rigid BodiesProblema 4a
Capitolo 9, Problema 4a

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.

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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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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.
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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.
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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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