Cylindrical workpieces of diameter 60 mm and length 400 mm are machined on a lathe at a cutting speed of 25 m min^-1 and a feed of 0.2 mm rev^-1. The Taylor’s tool life parameters 𝐶 and 𝑛 for this setup are 75 and 0.25, respectively. The tool changing time is 3 minutes. With a labor and overhead cost of ₹ 5 per minute, the tool changing cost per piece is ₹ _______ (rounded off to 2 decimal places).

The correct answer is 2.79 (or ₹2.79).

Here is the step-by-step derivation to calculate the tool changing cost per piece:

1. Given Data:
Diameter of the workpiece, D = 60 mm
Length of the workpiece, L = 400 mm
Cutting speed, V = 25 m min^-1
Feed rate, f = 0.2 mm rev^-1
Taylor's tool life parameters: exponent n = 0.25, constant C = 75
Tool changing time, T_c = 3 minutes
Labor and overhead cost, C_o = ₹ 5 per minute

2. Step 1: Calculate the Tool Life (T)
Taylor's tool life equation is given by:

$$V\cdot T^n=C$$

Substitute the given values into the equation:

$$25\cdot T^{0.25}=75$$

$$T^{0.25}=\frac{75}{25}=3$$

$$T=3^4=81\text{minutes}$$

3. Step 2: Calculate the Machining Time per Piece (t_m)
First, we find the spindle rotational speed (N) in rev min^-1:

$$N=\frac{1000\cdot V}{\pi \cdot D}$$

$$N=\frac{1000\cdot 25}{\pi \cdot 60}\approx 132.63\text{rev min}{\text{−1}}^{}$$

Now, compute the machining time (t_m) for a single piece:

$$t_m=\frac{L}{f\cdot N}=\frac{\pi \cdot D\cdot L}{1000\cdot V\cdot f}$$

$$t_m=\frac{\pi \cdot 60\cdot 400}{1000\cdot 25\cdot 0.2}=4.8\pi \approx 15.08\text{minutes}$$

4. Step 3: Calculate the Tool Changing Cost per Piece
The fraction of tool life consumed per piece is given by:

$$\text{Fraction of tool life per piece}=\frac{t_m}{T}$$

The cost incurred to perform one full tool change is:

$$\text{Cost per tool change}=T_c\cdot C_o=3\cdot 5=\text{₹}15$$

Therefore, the tool changing cost allocated per piece is:

$$\text{Tool changing cost per piece}=\left(\frac{t_m}{T}\right)\cdot T_c\cdot C_o$$

$$\text{Tool changing cost per piece}=\frac{15.08}{81}\cdot 15\approx 2.7925$$

Rounding off to two decimal places, the tool changing cost per piece is ₹ 2.79.