Bars of 250 mm length and 25 mm diameter are to be turned on a lathe with a feed of 0.2 mm/rev. Each regrinding of the tool costs Rs. 20. The time required for each tool change is 1 min. Tool life equation is given as VT^0.2 = 24 (where cutting speed V is in m/min and tool life T is in min). The optimum tool cost per piece for maximum production rate is Rs. ________ (round off to 2 decimal places)

The correct answer is 26.98.

Let's break down the step-by-step solution to find the optimum tool cost per piece for the maximum production rate:

1. Identify the given parameters:
- Length of the bar to be turned, L = 250 mm
- Diameter of the bar, D = 25 mm
- Feed rate, f = 0.2 mm/rev
- Cost of each tool regrinding, C_g = Rs. 20
- Time required for tool change, T_c = 1 min
- Taylor's tool life equation:

$$V{T}^{0.2}=24$$

Here, n = 0.2 and C = 24.

2. Calculate the optimum tool life for maximum production rate (minimum time per piece):
The formula for optimum tool life (T) for the maximum production rate condition is given by:

$$T=\left(\frac{1-n}{n}\right)\times T_c$$

Substituting n = 0.2 and T_c = 1 min:

$$T=\left(\frac{1-0.2}{0.2}\right)\times 1=4\text{min}$$

3. Calculate the cutting speed (V):
Using Taylor's equation:

$$V=\frac{24}{T^{0.2}}=\frac{24}{4^{0.2}}\approx 18.188\text{m/min}$$

4. Calculate the spindle speed (N):
The relation between cutting speed V and rotational speed N (rpm) is:

$$V=\frac{\pi DN}{1000}$$

where D is in mm. Rearranging to find N:

$$N=\frac{1000\times V}{\pi \times D}=\frac{1000\times 18.188}{\pi \times 25}\approx 231.576\text{rpm}$$

5. Calculate the total machining time per piece (T_m):
The machining time is computed as:

$$T_m=\frac{L}{f\times N}=\frac{250}{0.2\times 231.576}\approx 5.397\text{min}$$

6. Calculate the number of tools required per piece:
The number of tools (or tool consumption rate) per piece is the ratio of the machining time per piece to the tool life:

$$\text{Number of tools per piece}=\frac{T_m}{T}=\frac{5.397}{4}\approx 1.349$$

7. Calculate the optimum tool cost per piece:
The tool cost per piece is determined by multiplying the number of tools used per piece by the cost of one tool regrinding:

$$\text{Optimum tool cost per piece}=1.349\times 20=\text{Rs.}26.98$$