A wheel is turning about an axis through its center with constant angular acceleration. Starting from rest, at t = 0, the wheel turns through 8.20 revolutions in 12.0 s. At t = 12.0 s the kinetic energy of the wheel is 36.0 J. For an axis through its center, what is the moment of inertia of the wheel?

검증된 단계별 안내

Step 1: Convert the number of revolutions into radians. Since one revolution corresponds to \(2\pi\) radians, multiply the given number of revolutions (8.20) by \(2\pi\) to find the total angular displacement \(\theta\) in radians.

Step 2: Use the kinematic equation for rotational motion \(\theta = \omega_0 t + \frac{1}{2} \alpha t^2\), where \(\omega_0\) is the initial angular velocity (0 rad/s, since the wheel starts from rest), \(\alpha\) is the angular acceleration, and \(t\) is the time. Substitute \(\theta\) and \(t\) to solve for \(\alpha\).

Step 3: Calculate the angular velocity \(\omega\) at \(t = 12.0\,\text{s}\) using the equation \(\omega = \omega_0 + \alpha t\). Substitute \(\omega_0 = 0\), \(\alpha\) (from Step 2), and \(t = 12.0\,\text{s}\) to find \(\omega\).

Step 4: Use the rotational kinetic energy formula \(K = \frac{1}{2} I \omega^2\), where \(K\) is the kinetic energy (36.0 J), \(I\) is the moment of inertia, and \(\omega\) is the angular velocity (from Step 3). Rearrange the formula to solve for \(I\).

Step 5: Substitute the values of \(K\) and \(\omega\) into the equation \(I = \frac{2K}{\omega^2}\) to calculate the moment of inertia \(I\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Angular Acceleration

Angular acceleration is the rate of change of angular velocity over time. It is a vector quantity that indicates how quickly an object is rotating faster or slower. In this problem, the wheel experiences constant angular acceleration, which means its angular velocity increases uniformly from rest. This concept is crucial for determining the final angular velocity after a given time period.

Kinetic Energy of Rotation

The kinetic energy of a rotating object is given by the formula KE = 0.5 * I * ω², where I is the moment of inertia and ω is the angular velocity. This relationship shows how the energy of a rotating body depends on both its mass distribution (moment of inertia) and its rotational speed. In this question, knowing the kinetic energy at a specific time allows us to relate it to the moment of inertia once we find the angular velocity.

Moment of Inertia

Moment of inertia is a measure of an object's resistance to changes in its rotational motion, analogous to mass in linear motion. It depends on the mass distribution relative to the axis of rotation. For the wheel in this problem, calculating the moment of inertia is essential to relate the kinetic energy and angular velocity, allowing us to solve for the unknown quantity using the provided data.