d) What is the direction of its angular velocity \( \vec{\omega} \) ? \[ O \hat{i} \quad \bigcirc \cdot \hat{i} \quad \bigcirc \hat{j} \quad \circ-\hat{j} \quad \bigcirc \hat{k} \quad \circ-\hat{k} \] e) What is the direction of its anoular

To determine the direction of the angular velocity vector \( \vec{\omega} \) and address part **e)** of your question, let's explore the concepts step-by-step. ### **d) Direction of Angular Velocity \( \vec{\omega} \)** **Understanding Angular Velocity Direction:** - **Right-Hand Rule:** The direction of the angular velocity vector \( \vec{\omega} \) is determined using the right-hand rule. To apply this: 1. **Curl the fingers** of your right hand in the direction of the object's rotation. 2. **Extend your thumb**; it points in the direction of \( \vec{\omega} \). **Interpreting the Options:** Given the options: \[ O \hat{i} \quad \bigcirc \cdot \hat{i} \quad \bigcirc \hat{j} \quad \circ-\hat{j} \quad \bigcirc \hat{k} \quad \circ-\hat{k} \] These likely represent: - \( O \hat{i} \): No angular velocity in the \( \hat{i} \) (x-axis) direction. - \( \bigcirc \cdot \hat{i} \): \( \vec{\omega} \) in the **positive** \( \hat{i} \) direction (out of the plane following the right-hand rule). - \( \bigcirc \hat{j} \): \( \vec{\omega} \) in the **positive** \( \hat{j} \) direction. - \( \circ -\hat{j} \): \( \vec{\omega} \) in the **negative** \( \hat{j} \) direction. - \( \bigcirc \hat{k} \): \( \vec{\omega} \) in the **positive** \( \hat{k} \) direction. - \( \circ -\hat{k} \): \( \vec{\omega} \) in the **negative** \( \hat{k} \) direction. **Determining the Direction:** - **Identify the Rotation Plane and Direction:** Determine the plane in which the object is rotating (e.g., xy-plane) and whether the rotation is clockwise or counterclockwise. - **Apply the Right-Hand Rule:** For example: - **Counterclockwise Rotation (as viewed from above the z-axis):** \( \vec{\omega} \) points in the **positive \( \hat{k} \)** direction. - **Clockwise Rotation (as viewed from above the z-axis):** \( \vec{\omega} \) points in the **negative \( \hat{k} \)** direction. **Example:** If an object is rotating counterclockwise in the xy-plane, the angular velocity vector \( \vec{\omega} \) points **out of the plane**, which corresponds to the **positive \( \hat{k} \)** direction. Therefore, the correct option would be: \[ \bigcirc \hat{k} \] ### **e) Direction of Its Angular Momentum \( \vec{L} \)** Assuming the typo "anoular" refers to **angular momentum**, here's how to determine its direction: **Relationship Between Angular Velocity and Angular Momentum:** - **Angular Momentum Direction:** The angular momentum vector \( \vec{L} \) of a rigid body is parallel to its angular velocity vector \( \vec{\omega} \). - **Right-Hand Rule Application:** Just like with \( \vec{\omega} \), use the right-hand rule to determine \( \vec{L} \). **Key Points:** - **Rigid Body Rotation:** For a rigid body rotating about a fixed axis, \( \vec{L} \) points in the same direction as \( \vec{\omega} \). - **Magnitude:** \( |\vec{L}| = I \cdot |\vec{\omega}| \), where \( I \) is the moment of inertia. **Example:** Continuing from the previous example where \( \vec{\omega} \) points in the **positive \( \hat{k} \)** direction, the angular momentum \( \vec{L} \) also points in the **positive \( \hat{k} \)** direction. ### **Summary** - **Direction of \( \vec{\omega} \):** Determined by the right-hand rule based on the rotation direction. In the example, it's \( \bigcirc \hat{k} \). - **Direction of \( \vec{L} \):** Parallel to \( \vec{\omega} \), so also \( \bigcirc \hat{k} \) in this case. If you have specific details about the rotation (e.g., axis and rotation direction), feel free to provide them for a more tailored explanation!