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Ch 12: Rotation of a Rigid Body
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoKnight Calc 5th EditionCh 12: Rotation of a Rigid BodyProblema 25
Capitolo 12, Problema 25

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?

Guida verificata passo dopo passo
1
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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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.
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Percorso guidato
11:47
Intro to Moment of Inertia

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.
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Percorso guidato
08:55
Net Torque & Sign of 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.
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Percorso guidato
06:18
Intro to Angular Momentum
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