Question:hard
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
The Correct Option is A, D
Solution and Explanation
Step 1: What This Question Is Really Testing:
This problem is about two competing assumptions Merchant used for the shear stress on the shear plane, and how each assumption changes the constant $C$ in the shear angle formula.
Rather than describing both theories first, this solution goes through the four statements one by one and checks each against the underlying physics.
Step 2: The Underlying Physics to Use:
If the shear stress $\tau_s$ on the shear plane does NOT depend on the normal stress $\sigma_n$, minimizing the cutting energy always gives the same angle relation, $\phi = 45^\circ - \beta/2 + \alpha/2$, so $C = 45^\circ$ is a universal number, true for every material.
If instead $\tau_s$ increases linearly with $\sigma_n$ through a material constant $k$ (that is, $\tau_s = \tau_0 + k\sigma_n$), the minimization instead gives $C = \cot^{-1}(k)/2$, which changes from one material to another because $k$ does.
Step 3: Checking Each Statement:
Statement (A) says $C$ is constant when $\tau_s$ is independent of $\sigma_n$. That matches the first case above exactly ($C = 45^\circ$ always), so (A) is TRUE.
Statement (B) says $C$ is constant when $\tau_s$ depends on $\sigma_n$. But we just saw that in this case $C = \cot^{-1}(k)/2$ changes with the material constant $k$, so $C$ is NOT constant here, making (B) FALSE.
Statement (C) says $C$ depends on material properties when $\tau_s$ is independent of $\sigma_n$. This is the opposite of what happens in the first case, where $C$ stays fixed at $45^\circ$ regardless of material, so (C) is FALSE.
Statement (D) says $C$ depends on material properties when $\tau_s$ is linearly dependent on $\sigma_n$. This is exactly the second case, where $C = \cot^{-1}(k)/2$ varies with the material constant $k$, so (D) is TRUE.
Final Answer:
Going through all four statements individually confirms that only (A) and (D) hold. \[ \boxed{\text{Correct statements: A, D}} \]
This problem is about two competing assumptions Merchant used for the shear stress on the shear plane, and how each assumption changes the constant $C$ in the shear angle formula.
Rather than describing both theories first, this solution goes through the four statements one by one and checks each against the underlying physics.
Step 2: The Underlying Physics to Use:
If the shear stress $\tau_s$ on the shear plane does NOT depend on the normal stress $\sigma_n$, minimizing the cutting energy always gives the same angle relation, $\phi = 45^\circ - \beta/2 + \alpha/2$, so $C = 45^\circ$ is a universal number, true for every material.
If instead $\tau_s$ increases linearly with $\sigma_n$ through a material constant $k$ (that is, $\tau_s = \tau_0 + k\sigma_n$), the minimization instead gives $C = \cot^{-1}(k)/2$, which changes from one material to another because $k$ does.
Step 3: Checking Each Statement:
Statement (A) says $C$ is constant when $\tau_s$ is independent of $\sigma_n$. That matches the first case above exactly ($C = 45^\circ$ always), so (A) is TRUE.
Statement (B) says $C$ is constant when $\tau_s$ depends on $\sigma_n$. But we just saw that in this case $C = \cot^{-1}(k)/2$ changes with the material constant $k$, so $C$ is NOT constant here, making (B) FALSE.
Statement (C) says $C$ depends on material properties when $\tau_s$ is independent of $\sigma_n$. This is the opposite of what happens in the first case, where $C$ stays fixed at $45^\circ$ regardless of material, so (C) is FALSE.
Statement (D) says $C$ depends on material properties when $\tau_s$ is linearly dependent on $\sigma_n$. This is exactly the second case, where $C = \cot^{-1}(k)/2$ varies with the material constant $k$, so (D) is TRUE.
Final Answer:
Going through all four statements individually confirms that only (A) and (D) hold. \[ \boxed{\text{Correct statements: A, D}} \]
Was this answer helpful?
0
