Let's solve the problem step by step: --- ## **Problem Statement (Translation & Explanation)** The turbine in the figure rotates around the fixed axis OA. **Given:** - Find the **velocity** of a point with coordinates \((3, 2, 2)\) m. - The angular velocity, \(\omega = 900 \, \text{rad/s}\), but it is **decreasing** at a rate of \(100 \, \text{rad/s}^2\) (so angular acceleration \(\alpha = -100 \, \text{rad/s}^2\)). - **Find:** The **velocity** and **acceleration** of the point. --- ## **Step 1: Velocity of the Point** ### **Formula** The velocity of a point in rotational motion: \[ \vec{v} = \vec{\omega} \times \vec{r} \] Where: - \(\vec{\omega}\) = Angular velocity vector - \(\vec{r}\) = Position vector of the point ### **Identify \(\vec{\omega}\) and \(\vec{r}\):** - **Axis OA** appears to be along the vector from the origin to (7, 4, 4). - Direction vector of OA: \( \langle 7, 4, 4 \rangle \). Let's **normalize** OA for the unit vector direction: \[ |\text{OA}| = \sqrt{7^2 + 4^2 + 4^2} = \sqrt{49 + 16 + 16} = \sqrt{81} = 9 \] Unit vector along OA: \[ \hat{u}_{OA} = \left\langle \frac{7}{9}, \frac{4}{9}, \frac{4}{9} \right\rangle \] So, \[ \vec{\omega} = 900 \cdot \left\langle \frac{7}{9}, \frac{4}{9}, \frac{4}{9} \right\rangle = \langle 700, 400, 400 \rangle \ \text{rad/s} \] \[ \vec{r} = \langle 3, 2, 2 \rangle \ \text{m} \] --- ### **Cross Product Calculation** \[ \vec{v} = \vec{\omega} \times \vec{r} \] Using the determinant method: \[ \vec{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 700 & 400 & 400 \\ 3 & 2 & 2 \\ \end{vmatrix} \] \[ = \mathbf{i} \left(400 \cdot 2 - 400 \cdot 2 \right) - \mathbf{j} \left(700 \cdot 2 - 400 \cdot 3\right) + \mathbf{k} \left(700 \cdot 2 - 400 \cdot 3\right) \] \[ = \mathbf{i} (800 - 800) - \mathbf{j} (1400 - 1200) + \mathbf{k} (1400 - 1200) \] \[ = \mathbf{i} (0) - \mathbf{j} (200) + \mathbf{k} (200) \] \[ = \langle 0, -200, 200 \rangle \ \text{m/s} \] --- #### **Final Velocity:** \[ \boxed{\vec{v} = \langle 0, -200, 200 \rangle \ \text{m/s}} \] --- ## **Step 2: Acceleration of the Point** ### **Formula** \[ \vec{a} = \vec{\alpha} \times \vec{r} + \vec{\omega} \times (\vec{\omega} \times \vec{r}) \] Where: - \(\vec{\alpha}\) = Angular acceleration vector ### **Find \(\vec{\alpha}\):** \[ \vec{\alpha} = -100 \cdot \hat{u}_{OA} = -100 \cdot \left\langle \frac{7}{9}, \frac{4}{9}, \frac{4}{9} \right\rangle = \langle -77.78, -44.44, -44.44 \rangle \ \text{rad/s}^2 \] --- ### **Compute \(\vec{\alpha} \times \vec{r}\):** \[ \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -77.78 & -44.44 & -44.44 \\ 3 & 2 & 2 \\ \end{vmatrix} \] \[ = \mathbf{i}((-44.44) \cdot 2 - (-44.44) \cdot 2) - \mathbf{j}((-77.78) \cdot 2 - (-44.44) \cdot 3) + \mathbf{k}((-77.78) \cdot 2 - (-44.44) \cdot 3) \] \[ = \mathbf{i}( -88.88 + 88.88) - \mathbf{j}( -155.56 + 133.32) + \mathbf{k}( -155.56 + 133.32 ) \] \[ = \mathbf{i}(0) - \mathbf{j}(-22.24) + \mathbf{k}(-22.24) \] \[ = \langle 0, 22.24, -22.24 \rangle \ \text{m/s}^2 \] --- ### **Compute \(\vec{\omega} \times (\vec{\omega} \times \vec{r})\):** First, \(\vec{\omega} \times \vec{r} = \langle 0, -200, 200 \rangle\), as found before. Now, compute \(\vec{\omega} \times (\vec{\omega} \times \vec{r})\): \[ \vec{\omega} \times (\langle 0, -200, 200 \rangle) \] \[ \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 700 & 400 & 400 \\ 0 & -200 & 200 \\ \end{vmatrix} \] \[ = \mathbf{i}(400 \cdot 200 - 400 \cdot (-200)) - \mathbf{j}(700 \cdot 200 - 400 \cdot 0) + \mathbf{k}(700 \cdot (-200) - 400 \cdot 0) \] \[ = \mathbf{i}(80000 + 80000) - \mathbf{j}(140000 - 0) + \mathbf{k}(-140000 - 0) \] \[ = \mathbf{i}(160000) - \mathbf{j}(140000) + \mathbf{k}(-140000) \] \[ = \langle 160000, -140000, -140000 \rangle \ \text{m/s}^2 \] --- ### **Sum Both Parts for Total Acceleration:** \[ \vec{a} = \langle 0, 22.24, -22.24 \rangle + \langle 160000, -140000, -140000 \rangle \] \[ = \langle 160000, -139977.76, -140022.24 \rangle \ \text{m/s}^2 \] --- ## **Final Answers** ### **1. The velocity of the point:** \[ \boxed{\vec{v} = \langle 0, -200, 200 \rangle \ \text{m/s}} \] ### **2. The acceleration of the point:** \[ \boxed{\vec{a} = \langle 160000, -139977.76, -140022.24 \rangle \ \text{m/s}^2} \] --- ### **Let me know if you need a breakdown of any step or a vector diagram!**