GATE 2022 ME (ME1) – Question 50
Two rigid massless rods PR and RQ are joined at frictionless pin-joint R and are resting on ground at P and Q, respectively, as shown in the figure. A vertical force $F$ acts on the pin R as shown. When the included angle $\theta < 90°$, the rods remain in static equilibrium due to Coulomb friction between the rods and ground at locations P and Q. At $\theta = 90°$, impending slip occurs simultaneously at points P and Q. Then the ratio of the coefficient of friction at Q to that at P ($\mu_Q/\mu_P$) is _________ (round off to two decimal places).
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Correct answer: 5.73 to 5.79
Explanation
At $\theta = 90°$ the triangle PRQ is right-angled at R with the sides 5 and 12 and the hypotenuse $PQ = 13$ m along the ground. Each massless rod with a pin at each end carries force only along its length. At P the rod PR makes an angle with the horizontal of $\cos^{-1}\frac{5}{13} = 67.4°$, so the force makes $22.6°$ with the vertical (the normal to the ground). At impending slip, $\mu_P = \tan 22.6° = \frac{5}{12}$. At Q the rod makes $22.6°$ with the horizontal, so the force makes $67.4°$ with the vertical and $\mu_Q = \tan 67.4° = \frac{12}{5}$. The ratio is $\frac{12/5}{5/12} = 5.76$.