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This problem involves two objects, \( X \) and \( Y \), with differing moments of inertia (\( I \) and \( 2I \), respectively) and differing angular momenta over a period of 5 seconds. Based on the graph and given conditions, let's analyze the information provided and calculate the relationship between the angular momenta \( L_X \) and \( L_Y \) for these objects.

### Analysis of Given Information
1. **Moments of Inertia:**
   - Object \( X \): Moment of inertia \( = I \)
   - Object \( Y \): Moment of inertia \( = 2I \)

2. **Average Angular Momentum:**
   - From \( t = 0 \) to \( t = 5 \) s, the average magnitude of the torque for Object \( X \) is given as \( \tau_X \).
   - The average magnitude of the torque for Object \( Y \) (over the same time period) is represented by \( \tau_Y \).

3. **Graph Observations:**
   - The graph shows angular momentum \( L \) (in kg·m²/s) over time for both objects \( X \) and \( Y \).
   - Based on the trends in the graph, the relationship between the slopes or the changes in angular momentum could help establish a direct relation between \( \tau_Y \) and \( \tau_X \).

### Required Calculation
The question asks which of the provided answer options correctly represents the relationship between \( \tau_Y \) and \( \tau_X \).

Let's calculate the rate of change of angular momentum for each object to determine the torque relationships, and we’ll use the graph data to see if one of the options (A-D) aligns with these calculations.

Once I work through this, I'll confirm the correct answer. Would you like a breakdown of each step in this process?

Two objects, X and Y, experience external net torques that vary over a period of 5 seconds. Object X has a moment of inertia I, and Object Y has a moment of inertia 2I. The average value of the magnitude of the external net torque exerted on Object X from time t = 0 to t = 5 s is τx. Similarly, the average value for Object Y is τy. The magnitudes of the angular momenta L of Objects X and Y versus time are shown in the graph. Which of the following expressions correctly relates τy to τx?

Learn how to determine the relationship between torques τy and τx for objects with differing moments of inertia using angular momentum graphs. This problem covers key physics concepts like torque, angular momentum, and moment of inertia. Suitable for advanced high school physics students.

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