Ch. 10 - Rotational Motion

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 10 - Rotational Motion

Problem 26b

Chapter 10, Problem 26b

The angular acceleration of a wheel, as a function of time, is α = 4.2 t² ― 9.0 t , where α is in rad/s² and t in seconds. If the wheel starts from rest (θ = 0 , ω = 0, at t = 0), determine a formula for the angular position θ, both as a function of time.

Verified step by step guidance

Start by recalling the relationship between angular acceleration (α), angular velocity (ω), and angular position (θ). Angular acceleration is the derivative of angular velocity with respect to time, and angular velocity is the derivative of angular position with respect to time. Mathematically: α = dω/dt and ω = dθ/dt.

Integrate the given angular acceleration function α = 4.2t² - 9.0t with respect to time to find the angular velocity ω(t). Use the initial condition ω(0) = 0 to determine the constant of integration.

After finding ω(t), integrate it with respect to time to find the angular position θ(t). Use the initial condition θ(0) = 0 to determine the constant of integration.

Write the final expression for θ(t) after performing the integrations and applying the initial conditions. Ensure that the units are consistent throughout the process.

Verify the derived formula for θ(t) by checking its consistency with the given initial conditions (θ = 0 and ω = 0 at t = 0) and the relationship between α, ω, and θ.

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Key Concepts

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

Angular Acceleration

Angular acceleration (α) is the rate of change of angular velocity over time. It is measured in radians per second squared (rad/s²). In this problem, the angular acceleration is given as a function of time, indicating how it varies as the wheel spins. Understanding this concept is crucial for determining how the wheel's motion evolves over time.

Kinematics of Rotational Motion

The kinematics of rotational motion describes the relationships between angular displacement, angular velocity, and angular acceleration. Similar to linear motion, these quantities are related through equations of motion. For a wheel starting from rest, the angular position can be derived by integrating the angular velocity, which itself is obtained by integrating the angular acceleration over time.

Integration in Physics

Integration is a mathematical process used to find the total accumulation of a quantity. In the context of this problem, integrating the angular acceleration function will yield the angular velocity function, and further integration will provide the angular position function. This process is essential for transitioning from acceleration to velocity and then to position in rotational dynamics.

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