An object whose moment of inertia is 4.0 kg m² is rotating with angular velocity 0.25 rad/s. It then experiences the torque shown in FIGURE EX12.25. What is the object's angular velocity at t = 3.0s?

검증된 단계별 안내

Step 1: Identify the relationship between torque (τ), angular acceleration (α), and moment of inertia (I). The equation is τ = I * α. Rearrange to find angular acceleration: α = τ / I.

Step 2: Analyze the graph of torque versus time. The torque is 4 Nm from t = 0 to t = 1 s, decreases linearly to 0 Nm from t = 1 s to t = 2 s, and remains 0 Nm from t = 2 s to t = 3 s.

Step 3: Calculate angular acceleration for each time interval using α = τ / I. For t = 0 to t = 1 s, α = 4 Nm / 4.0 kg·m² = 1 rad/s². For t = 1 to t = 2 s, α decreases linearly as torque decreases. For t = 2 to t = 3 s, α = 0 rad/s² since torque is 0.

Step 4: Use the kinematic equation for angular velocity: ω_final = ω_initial + α * Δt. Apply this equation for each time interval sequentially, starting with ω_initial = 0.25 rad/s at t = 0.

Step 5: Integrate angular acceleration over time to find the change in angular velocity for the interval t = 1 to t = 2 s, where α varies linearly. Add this to the angular velocity calculated for the previous interval to find ω at t = 3 s.

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주요 개념

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

Moment of Inertia

Moment of inertia is a measure of an object's resistance to changes in its rotational motion. It depends on the mass distribution relative to the axis of rotation. The greater the moment of inertia, the more torque is required to change the angular velocity of the object. In this case, the moment of inertia is given as 4.0 kg m².

Torque

Torque is a measure of the rotational force applied to an object, causing it to rotate about an axis. It is calculated as the product of the force and the distance from the axis of rotation. The graph provided shows how torque varies with time, which is crucial for determining how the angular velocity of the object changes over time due to the applied torque.

Angular Velocity

Angular velocity is the rate of change of angular displacement and is measured in radians per second (rad/s). It indicates how fast an object is rotating around an axis. In this problem, the initial angular velocity is 0.25 rad/s, and the change in angular velocity over time will be influenced by the torque applied, as shown in the graph.