Machining: Taylor's Tool Life Equation Analysis
The problem involves Taylor's tool life equation, which describes the relationship between cutting speed (V) and tool life (T) as:
$VT^n = C$
where 'n' is the exponent and 'C' is a constant.
Applying the Given Conditions
Let the initial cutting speed be $V_1$ and the initial tool life be $T_1$. The equation is:
$V_1 T_1^n = C \quad (1)$
According to the problem:
- The cutting speed is doubled: $V_2 = 2V_1$
- The tool life becomes one-eighth: $T_2 = \frac{1}{8}T_1$
The equation for the new conditions is:
$V_2 T_2^n = C \quad (2)$
Calculating the Exponent 'n'
Equating the two conditions (since C is constant):
$V_1 T_1^n = V_2 T_2^n$
Substitute the values of $V_2$ and $T_2$:
$V_1 T_1^n = (2V_1) \left(\frac{1}{8}T_1\right)^n$
Simplify the equation:
$V_1 T_1^n = 2V_1 \frac{1}{8^n} T_1^n$
Divide both sides by $V_1 T_1^n$ (assuming $V_1 \neq 0$ and $T_1 \neq 0$):
$1 = 2 \times \frac{1}{8^n}$
$1 = \frac{2}{8^n}$
Rearrange to solve for $8^n$:
$8^n = 2$
Express 8 as a power of 2 ($8 = 2^3$):
$(2^3)^n = 2^1$
$2^{3n} = 2^1$
Equate the exponents:
$3n = 1$
Solve for 'n':
$n = \frac{1}{3}$
Therefore, the exponent 'n' in Taylor's tool life equation is $\frac{1}{3}$.
Important Questions from Miscellaneous
- Which of the following relationships is/are not true?
(A). Most probable velocity = $\sqrt{\frac{2RT}{M}}$
(B). PV = $\frac{3}{2}kT$
(C). Compressibility factor Z = $\frac{pV}{nRT}$
(D). Average kinetic energy of gas = $\frac{1}{2}kT$
Choose the correct answer from the options given below - Match List-I with List-II
List-I List-II Electronic Configuration First Ionisation energy (kJ mol$^{-1}$) (A). ns$^2$ (I). 2100 (B). ns$^2$np$^1$ (II). 1400 (C). ns$^2$np$^3$ (III). 800 (D). ns$^2$np$^6$ (IV). 900
Choose the correct answer from the options given below: - The shielding constant of a 2p electron (calculated using Slater's rules) is
- Match List-I with List-II
List-I List-II Spectroscopy Property (A). Raman (I). Polarizability (B). FTIR (II). Dipole Moment (C). UV-Visible (III). Absorbance (D). NMR (IV). Spin
Choose the correct answer from the options given below: - The structure of protein comprises of:
(A). Primary structure of protein is associated with amino acids
(B). Secondary structure of protein is associated to peptides
(C). Tertiary structure of protein is associated with polypeptide chains
(D). Quaternary structure of protein is associated with polypeptide chains
Choose the correct answer from the options given below:
