Question:
In the theory of orthogonal machining, assuming that the shear plane will take up an angle that minimizes energy, the shear angle can be shown to be
\[ \phi = C - \frac{\beta}{2} + \frac{\alpha}{2} \]
where \( \phi \) is the shear angle, \( \alpha \) the rake angle, and \( \beta \) the friction angle.
Which ONE or MORE among the following is/are TRUE?
In the theory of orthogonal machining, assuming that the shear plane will take up an angle that minimizes energy, the shear angle can be shown to be
\[ \phi = C - \frac{\beta}{2} + \frac{\alpha}{2} \]
where \( \phi \) is the shear angle, \( \alpha \) the rake angle, and \( \beta \) the friction angle.
Which ONE or MORE among the following is/are TRUE?
\[ \phi = C - \frac{\beta}{2} + \frac{\alpha}{2} \]
where \( \phi \) is the shear angle, \( \alpha \) the rake angle, and \( \beta \) the friction angle.
Which ONE or MORE among the following is/are TRUE?
Show Hint
Recall Merchant's two theories: constant shear stress gives a fixed C = 45 degrees, shear stress dependent on normal stress gives a material-dependent C.
Updated On: Aug 5, 2026
- \( C \) is constant if the shear stress on the shear plane is independent of the normal stress.
- \( C \) is constant if the shear stress on the shear plane is dependent on the normal stress.
- \( C \) depends on material properties if the shear stress on the shear plane is independent of the normal stress.
- \( C \) depends on material properties if the shear stress on the shear plane is linearly dependent on the normal stress.
Show Solution
Verified By Collegedunia
The Correct Option is A, D
Solution and Explanation
Step 1: Understanding the Question:
The formula given, \( \phi = C - \beta/2 + \alpha/2 \), is the general shear angle relation from the minimum energy theory of orthogonal cutting.
The constant \( C \) is not always the same number, its value depends on what assumption we make about how the shear stress on the shear plane behaves.
We need to decide, for each option, whether \( C \) stays a fixed number or becomes material dependent.
Step 2: Merchant's Original (First) Theory:
Merchant first assumed that the shear strength on the shear plane, \( \tau_s \), does not change with the normal stress \( \sigma_n \) acting on that plane, in other words \( \tau_s \) is independent of \( \sigma_n \).
Minimizing the cutting energy under this assumption gives
\[ \phi = \frac{\pi}{4} - \frac{\beta}{2} + \frac{\alpha}{2} \]
so here \( C = \pi/4 = 45^\circ \), a pure number that never changes.
This means statement (A) is true, and statement (C), which claims \( C \) depends on material properties under this same assumption, is false.
Step 3: Merchant's Modified (Second) Theory:
Because the first theory did not match experiments well, Merchant later assumed the shear stress rises linearly with the normal stress, \( \tau_s = \tau_0 + k\sigma_n \), where \( k \) is a slope that is different for every work material.
Minimizing the energy with this assumption gives
\[ \phi = \frac{\cot^{-1}k}{2} - \frac{\beta}{2} + \frac{\alpha}{2} \]
so now \( C = \dfrac{\cot^{-1}k}{2} \), and since \( k \) is a material property, \( C \) itself becomes material dependent.
This makes statement (D) true, and statement (B), which claims \( C \) stays constant here, false.
Final Answer:
\( C \) is a fixed constant only under the independent-of-normal-stress assumption (A), and becomes material dependent only under the linear-dependence assumption (D). \[ \boxed{\text{A and D are correct}} \]
The formula given, \( \phi = C - \beta/2 + \alpha/2 \), is the general shear angle relation from the minimum energy theory of orthogonal cutting.
The constant \( C \) is not always the same number, its value depends on what assumption we make about how the shear stress on the shear plane behaves.
We need to decide, for each option, whether \( C \) stays a fixed number or becomes material dependent.
Step 2: Merchant's Original (First) Theory:
Merchant first assumed that the shear strength on the shear plane, \( \tau_s \), does not change with the normal stress \( \sigma_n \) acting on that plane, in other words \( \tau_s \) is independent of \( \sigma_n \).
Minimizing the cutting energy under this assumption gives
\[ \phi = \frac{\pi}{4} - \frac{\beta}{2} + \frac{\alpha}{2} \]
so here \( C = \pi/4 = 45^\circ \), a pure number that never changes.
This means statement (A) is true, and statement (C), which claims \( C \) depends on material properties under this same assumption, is false.
Step 3: Merchant's Modified (Second) Theory:
Because the first theory did not match experiments well, Merchant later assumed the shear stress rises linearly with the normal stress, \( \tau_s = \tau_0 + k\sigma_n \), where \( k \) is a slope that is different for every work material.
Minimizing the energy with this assumption gives
\[ \phi = \frac{\cot^{-1}k}{2} - \frac{\beta}{2} + \frac{\alpha}{2} \]
so now \( C = \dfrac{\cot^{-1}k}{2} \), and since \( k \) is a material property, \( C \) itself becomes material dependent.
This makes statement (D) true, and statement (B), which claims \( C \) stays constant here, false.
Final Answer:
\( C \) is a fixed constant only under the independent-of-normal-stress assumption (A), and becomes material dependent only under the linear-dependence assumption (D). \[ \boxed{\text{A and D are correct}} \]
Was this answer helpful?
0
0
Top GATE PI Questions
- In an air-standard Otto cycle, the pressure and temperature of air just before the compression stroke are 200 kPa and 26.85 °C, respectively. The combustion process is assumed to be a constant volume process, where 1.02 MJ/kg heat is added. The cycle efficiency is 50%. The adiabatic index \(γ\) and specific heat at constant volume \(C_v\) can be considered to be constant during the process (corresponding values taken at the mean cycle temperature).
Assuming that the ideal gas law is applicable, \(γ = \frac 43\) and \(c_v= 0.85\ kJ/kg-K\), the maximum pressure (in MPa) reached during the cycle is _____ . (Rounded off to 1 decimal place) - A metallic cylindrical pressure vessel, used to store compressed air in a plant, has 1 m mean radius and 4 mm wall thickness. The maximum allowable normal and shear stresses in the cylindrical portion of the vessel are 100 MPa and 40 MPa, respectively. Considering only these data in the design, the maximum allowable internal gauge pressure (in MPa) of the compressed air is ______ . (Rounded off to 2 decimal places)
- A flat belt drive with pulley of r = 20 cm radius is designed to transmit 6.283 kW power at 600 RPM. In the figure, τ is the corresponding torque. If the coefficient of static friction between the belt and the pulley is 0.3, then the minimum value of the tightening force F (in kN) required to prevent the belt slip is _____ . (Rounded off 2 decimal places)
- Mild steel plates are welded to make butt joints by arc welding with 85% heat transfer efficiency ignoring other losses. The first weld joint is made by selecting arc voltage of 30 V and current of 180 A with a welding speed of 6 mm/s. Using identical plates, a second weld joint is made with the same arc voltage and a welding speed of 8 mm/s. If both the welds have the same heat input, then the welding current (in A) for the second weld joint is _____. (Answer in integer)
- In a single pass cold rolling operation, a flat plate is reduced to a thickness of 3 mm. In this operation, two rolls of diameter 400 mm each are rotating in opposite direction at 300 RPM, and the elastic deflection of these rolls is negligible. The angle of bite is 10°. If the neutral point is present at an angle of 7° from the exit side, then the thickness of the plate (in mm) at the neutral point is _____ . (Rounded off to 1 decimal place)
View More Questions