1. **State the problem:** We need to find the twist per length (in degrees per inch) of a shaft transmitting 0.42 hp at 1147 rpm, with given dimensions and shear modulus. 2. **Given data:** - Power, $P = 0.42$ hp - Rotational speed, $N = 1147$ rpm - Ratio of inner diameter to outer diameter, $\frac{d_i}{d_o} = 0.86$ - Shaft thickness, $t = 0.036$ in - Shear modulus, $G = 3800$ ksi 3. **Find:** Twist per length $\theta/L$ in degrees per inch. 4. **Formulas and important rules:** - Power transmitted by a shaft: $$P = \frac{2 \pi N T}{33000}$$ where $T$ is torque in lb-in, $N$ in rpm, and $P$ in hp. - Torque $T$ can be found by rearranging: $$T = \frac{33000 P}{2 \pi N}$$ - Polar moment of inertia for hollow circular shaft: $$J = \frac{\pi}{32} (d_o^4 - d_i^4)$$ - Twist per length formula: $$\frac{\theta}{L} = \frac{T}{G J}$$ where $\theta$ in radians per inch. - Convert radians to degrees: $$1 \text{ rad} = \frac{180}{\pi} \text{ degrees}$$ 5. **Calculate outer diameter $d_o$ and inner diameter $d_i$:** Since thickness $t = \frac{d_o - d_i}{2}$, then: $$d_o - d_i = 2t = 2 \times 0.036 = 0.072 \text{ in}$$ Given $\frac{d_i}{d_o} = 0.86$, so $d_i = 0.86 d_o$. Substitute: $$d_o - 0.86 d_o = 0.072 \Rightarrow 0.14 d_o = 0.072 \Rightarrow d_o = \frac{0.072}{0.14} = 0.5143 \text{ in}$$ Then: $$d_i = 0.86 \times 0.5143 = 0.4423 \text{ in}$$ 6. **Calculate torque $T$:** $$T = \frac{33000 \times 0.42}{2 \pi \times 1147} = \frac{13860}{7207.96} = 1.922 \text{ lb-in}$$ 7. **Calculate polar moment of inertia $J$:** $$J = \frac{\pi}{32} (d_o^4 - d_i^4) = \frac{3.1416}{32} (0.5143^4 - 0.4423^4)$$ Calculate powers: $$0.5143^4 = 0.0699, \quad 0.4423^4 = 0.0383$$ So: $$J = 0.09817 \times (0.0699 - 0.0383) = 0.09817 \times 0.0316 = 0.0031 \text{ in}^4$$ 8. **Calculate twist per length in radians/inch:** $$\frac{\theta}{L} = \frac{T}{G J} = \frac{1.922}{3800 \times 1000 \times 0.0031}$$ Note: $G = 3800$ ksi = $3800 \times 1000$ psi $$\frac{\theta}{L} = \frac{1.922}{11,780,000} = 1.632 \times 10^{-7} \text{ radians/in}$$ 9. **Convert twist per length to degrees/inch:** $$\frac{\theta}{L} = 1.632 \times 10^{-7} \times \frac{180}{\pi} = 9.35 \times 10^{-6} \text{ degrees/in}$$ 10. **Final answer:** The twist per length is approximately **0.00000935 degrees per inch**. Rounded to 3 decimal places in degrees/inch, it is 0.000 degrees/inch (very small).