# Step-by-Step Guide to Calculate Mass Moment of Inertia About the \( z \)-Axis Let's break down the problem and solve for the **mass moment of inertia of the bracket about the \( z \)-axis**. --- ## **1. Understanding the System** - **Bracket Assembly:** - **Square Plate:** \( 400\,\text{mm} \times 400\,\text{mm} \times 10\,\text{mm} \) - **Hole:** Centered, radius \( R = 125\,\text{mm} \) - **Truss:** Simple rods (not needed for plate inertia about \( z \)-axis) - **Coordinate Frame:** - Plate lies in \( yz \)-plane. - \( z \)-axis is perpendicular to the plate. --- ## **2. What Is Required?** - **Mass moment of inertia (\( I_z \))** of the entire bracket about the \( z \)-axis. - **Treat:** Plate as thin plate, ignore thickness for rotational inertia. --- ## **3. Formula Used** For a **thin plate** about an axis perpendicular to its plane (here, \( z \)-axis through the center): \[ I_z = \iint_{A} r^2 \, dm \] ### For a Composite Area (Plate with Hole): \[ I_{z,\text{total}} = I_{z,\text{square}} - I_{z,\text{hole}} \] --- ## **4. Mass Moment of Inertia of a Thin Square Plate** For a square plate of side \( a \), mass \( m \): \[ I_{z,\text{square}} = \frac{1}{6} m_{\text{square}} a^2 \] ### **Mass of Square Plate (without hole):** \[ m_{\text{square}} = \rho \, a^2 t \] where - \( \rho = 500\,\mathrm{kg/m^3} \) - \( a = .4\,\mathrm{m} \) - \( t = .01\,\mathrm{m} \) --- ## **5. Mass Moment of Inertia of the Hole (Circular Disk), Subtracted** Treat the hole as "negative mass" (subtract its inertia): - For a thin disk of radius \( R \), mass \( m_\text{hole} \): \[ I_{z,\text{hole}} = \frac{1}{2} m_{\text{hole}} R^2 \] ### **Mass of Hole:** \[ m_{\text{hole}} = \rho \, \pi R^2 t \] where \( R = .125\,\mathrm{m} \). --- ## **6. Plug in the Numbers** ### **(A) Plate Mass:** \[ m_{\text{square}} = 500 \times (.4)^2 \times .01 = 500 \times .16 \times .01 = 8\,\mathrm{kg} \] ### **(B) Hole Mass:** \[ m_{\text{hole}} = 500 \times \pi \times (.125)^2 \times .01 = 500 \times \pi \times .015625 \times .01 \] \[ = 500 \times .049087 \times .01 = 500 \times .00049087 = 2.454\,\mathrm{kg} \] ### **(C) Inertia of Plate:** \[ I_{z,\text{square}} = \frac{1}{6} \times 8 \times (.4)^2 = \frac{1}{6} \times 8 \times .16 = \frac{1}{6} \times 1.28 = .213\,\mathrm{kg\,m^2} \] ### **(D) Inertia of Hole:** \[ I_{z,\text{hole}} = \frac{1}{2} \times 2.454 \times (.125)^2 = .5 \times 2.454 \times .015625 = .5 \times .03834 = .0192\,\mathrm{kg\,m^2} \] --- ## **7. Final Answer** \[ I_{z,\text{total}} = I_{z,\text{square}} - I_{z,\text{hole}} = .213 - .0192 = \boxed{.194\,\mathrm{kg\,m^2}} \] --- ## **8. Notes** - **If asked for the rods/truss:** You'd use the slender rod inertia formula, but for the plate about the \( z \)-axis, only the plate (minus hole) is considered. - **Units:** Always check your units! --- ## **Summary Table** | Component | Mass (kg) | Inertia (kg·m²) | |---------------|-----------|-----------------| | Square plate | 8 | .213 | | Hole (remove) | 2.454 | .0192 | | **Total** | | **.194** | --- **If you need to include the truss, let me know! Otherwise, this is the moment of inertia for the plate with the hole about the \( z \)-axis.**