The GATE Grind

GATE 2020 ME (ME2) – Question 42

Metrology and Inspection · Linear and angular measurements, comparators · 2 marks · Multiple choice

Rollers of diameters D₁ and D₂ (mm), D₁ > D₂, measure the internal taper angle in the same V-groove. A height gauge measures H₁ and H₂ (mm) after inserting each roller. Which relation evaluates α as shown?

Diagram for GATE 2020 ME (ME2) question 42
  1. $\sin\alpha=\frac{D_1-D_2}{2(H_1-H_2)-(D_1-D_2)}$
  2. $\cos\alpha=\frac{D_1-D_2}{2(H_1-H_2)-2(D_1-D_2)}$
  3. $\operatorname{cosec}\alpha=\frac{(H_1-H_2)-(D_1-D_2)}{2(D_1-D_2)}$
  4. $\sin\alpha=\frac{H_1-H_2}{D_1-D_2}$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (A) $\sin\alpha=\frac{D_1-D_2}{2(H_1-H_2)-(D_1-D_2)}$

Explanation

A roller centre is r/sinα above the V apex; its top is r(1+cscα) above the apex. Subtract measurements: H1−H2=(D1−D2)(1+cscα)/2. Rearranging gives **sinα=(D1−D2)/[2(H1−H2)−(D1−D2)],A**.