Question 66

The Correct Answer is A ### 1. Explanation for Option A ("3/8 inch") being Correct This problem requires calculating the total tailstock offset needed to achieve a specified taper rate over a given workpiece length, using the tailstock offset method (turning between centers). **Formula for Tailstock Offset:** The amount of tailstock offset ($O$) required is calculated using the following relationship: $$O = \frac{Taper_{rate} \times L_{total}}{2}$$ Where: * $Taper_{rate}$ is the required taper per unit length (expressed as Taper Per Foot, or TPF). * $L_{total}$ is the total length of the workpiece being turned (in inches). * The division by 2 is necessary because the tailstock offset only changes the radius of the cut, while the taper rate is specified as the change in diameter. **Applying the Values:** 1. **Given Taper Rate ($Taper_{rate}$):** 1/2 inch per foot (1/2 TPF). 2. **Given Workpiece Length ($L_{total}$):** 18 inches. First, convert the workpiece length into feet: $$L_{total} = \frac{18 \text{ inches}}{12 \text{ inches/foot}} = 1.5 \text{ feet}$$ Now, calculate the offset ($O$): $$O = \frac{(1/2 \text{ in/ft}) \times (1.5 \text{ ft})}{2}$$ $$O = \frac{0.5 \times 1.5}{2}$$ $$O = \frac{0.75}{2}$$ $$O = 0.375 \text{ inches}$$ Since $0.375 \text{ inches}$ is equal to $3/8 \text{ inch}$, the required tailstock offset is **3/8 inch**. --- *(Alternative calculation using inches directly):* If we use the workpiece length in inches, we must divide the Taper Per Foot by 12 to get the Taper Per Inch (TPI) before multiplying by the length: $$O = \frac{(Taper_{TPF} / 12) \times L_{total}}{2}$$ $$O = \frac{(0.5 / 12) \times 18}{2}$$ $$O = \frac{(0.04166...) \times 18}{2}$$ $$O = \frac{0.75}{2}$$ $$O = 0.375 \text{ inches } (3/8 \text{ inch})$$ ### 2. Explanation of Why Other Options Are Incorrect **B) 1/2 inch:** This value (1/2 inch) is the required taper rate *per foot* (TPF). It incorrectly ignores the length of the workpiece (1.5 feet) and the need to divide by 2 (offset is radial). If the offset were simply $Taper_{rate} / 2$, the result would be 1/4 inch. **C) 3/4 inch:** This value ($0.75$ inches) represents the total diameter change (taper) over the entire 18-inch length of the workpiece. $$Taper_{total} = Taper_{rate} \times L_{feet} = 0.5 \times 1.5 = 0.75 \text{ inches}$$ The tailstock offset must only be half of this total diameter change. **D) 7/8 inch:** This option is significantly larger than the required offset (3/8 inch) and does not correspond to any standard calculation derived from the given input values.