Let's solve the problem **step-by-step** as requested. We'll address all three parts: (a), (b), and (c). --- ## **Given Data** - **Outside diameter** \( d_o = 10. \) in \( \Rightarrow r_o = 5. \) in - **Inside diameter** \( d_i = 2. \) in \( \Rightarrow r_i = 1. \) in - **Density** \( \rho = .284\, \mathrm{lbm/in^3} \) - **Poisson's ratio** \( \nu = .3 \) - **Yield strength** \( S_y = 130\, \text{kpsi} = 130,000\, \text{psi} \) - **Factor of safety** = 1.5 --- ## **Step 1: Unit Conversion** ### **Density (from lbm/in³ to lbf·s²/in⁴)** We need density in force units (consistent with psi for stress). Recall: \[ 1\, \text{lbf} = 32.174\, \text{lbm} \cdot \text{ft/s}^2 \] \[ 1\, \text{lbm} = \frac{1}{32.174}\, \text{lbf} \cdot \text{s}^2/\text{ft} \] But since everything is in inches, convert feet to inches: \[ 1\, \text{lbf} = 32.174\, \text{lbm} \cdot 12\, \text{in/s}^2 = 386.09\, \text{lbm} \cdot \text{in/s}^2 \] So, \[ 1\, \text{lbm/in}^3 = \frac{1}{386.09}\, \text{lbf} \cdot \text{s}^2/\text{in}^4 \] Thus, \[ \rho = .284 \times \frac{1}{386.09} = .000736\, \text{lbf} \cdot \text{s}^2/\text{in}^4 \] --- ## **Step 2: Stress Equations for Rotating Disk** For a disk with a hole: \[ \sigma_t(r) = A + \frac{B}{r^2} - C r^2 \] \[ \sigma_r(r) = A - \frac{B}{r^2} - D r^2 \] But the standard equations for **maximum tangential and radial stresses** at the inner radius \( r_i \) are: \[ \sigma_{t,\text{max}} = \frac{3 + \nu}{8} \rho \omega^2 (r_o^2 + r_i^2) + \frac{1 - \nu}{4} \rho \omega^2 \frac{r_o^2 r_i^2}{r_o^2 - r_i^2}\ln\left(\frac{r_o}{r_i}\right) \] \[ \sigma_{r,\text{max}} = \text{[at inner or outer radius, similar formula]} \] But for exam purposes, **maximum tangential stress at the inner radius** is usually the critical value. --- ## **Step 3: Calculate Angular Velocity (\(\omega\))** \[ \omega = 2\pi N \] Where \(N\) is in revolutions per second. ### For \(N = 10,000\) rpm: \[ N = 10,000\, \text{rpm} = \frac{10,000}{60} = 166.67\, \text{rev/sec} \] \[ \omega = 2\pi \times 166.67 = 1047.2\, \text{rad/sec} \] ### For \(N = 20,000\) rpm: \[ N = 20,000\, \text{rpm} = 333.33\, \text{rev/sec} \] \[ \omega = 2\pi \times 333.33 = 2094.4\, \text{rad/sec} \] --- ## **Step 4: Plug Values into the Formula** ### **Let’s use the formula for maximum tangential stress at inner radius:** \[ \sigma_{t,\text{max}} = \frac{3 + \nu}{8} \rho \omega^2 (r_o^2 + r_i^2) - \frac{1 + 3\nu}{8} \rho \omega^2 r_i^2 \left(1 + \frac{r_o^2}{r_i^2}\right) \] But more commonly used (simplified) for inner radius: \[ \sigma_{t,\text{max}} = \rho \omega^2 r_i^2 \left[\frac{1+\nu}{2}\left(\frac{r_o^2 + r_i^2}{r_o^2 - r_i^2}\right)\right] \] Let’s use this. ### **Plug in numbers:** - \(\nu = .3\) - \(\rho = .000736\, \text{lbf} \cdot \text{s}^2/\text{in}^4\) - \(r_o = 5.\) in - \(r_i = 1.\) in \[ \frac{r_o^2 + r_i^2}{r_o^2 - r_i^2} = \frac{25 + 1}{25 - 1} = \frac{26}{24} = 1.0833 \] So, \[ \sigma_{t,\text{max}} = .000736 \cdot \omega^2 \cdot (1)^2 \cdot \left[\frac{1+.3}{2} \times 1.0833 \right] \] \[ = .000736 \cdot \omega^2 \cdot .65 \cdot 1.0833 \] \[ = .000736 \cdot \omega^2 \cdot .7041 \] \[ = .0005176 \cdot \omega^2 \] --- ## **Step 5: Calculate for Each Case** ### **[a] At 10,000 rpm (\(\omega = 1047.2\) rad/sec):** \[ \sigma_{t,\text{max}} = .0005176 \times (1047.2)^2 \] \[ = .0005176 \times 109700 \] \[ = 568\, \text{psi} \] --- ### **[b] At 20,000 rpm (\(\omega = 2094.4\) rad/sec):** \[ \sigma_{t,\text{max}} = .0005176 \times (2094.4)^2 \] \[ = .0005176 \times 438800 \] \[ = 2272\, \text{psi} \] --- ### **[c] Maximum Speed before Failure** #### **Maximum allowable stress (with factor of safety):** \[ \text{Allowable stress} = \frac{S_y}{\text{FOS}} = \frac{130,000}{1.5} = 86,667\, \text{psi} \] Set \(\sigma_{t,\text{max}} = 86,667\): \[ 86,667 = .0005176 \omega^2 \] \[ \omega^2 = \frac{86,667}{.0005176} = 167,467,000 \] \[ \omega = \sqrt{167,467,000} = 12,942\, \text{rad/sec} \] Convert to rpm: \[ N = \frac{\omega}{2\pi} = \frac{12,942}{2\pi} = 2,061\, \text{rev/sec} = 123,660\, \text{rpm} \] --- ## **Summary of Answers** ### **[a] At 10,000 rpm:** - **Max tangential stress:** \( \boxed{568\, \text{psi}} \) ### **[b] At 20,000 rpm:** - **Max tangential stress:** \( \boxed{2,272\, \text{psi}} \) ### **[c] Maximum Safe Speed:** - **Max safe speed (with FOS 1.5):** \( \boxed{123,660\, \text{rpm}} \) --- ### **Note:** - The **maximum radial stress** at the inner radius is less than the tangential; if requested, a similar equation can be used. - The **factor of safety** was correctly applied in (c). - **Always check units** as per cautionary note. --- If you need the detailed radial stress calculation or further clarification, let me know!