In the arrangement shown the rod is freely pivoted at point O and is in contact with the equilateral triangular block which can slide on the horizontal frictionless ground without toppling. As the block is given a constant speed v forward, the rod rotates about O. Find magnitude of angular velocity of rod in rad/s at the instant when θ=30°. (friction is absent at O) [Take v=20 m/s, a=1 m]
Physics · Mixed Chapter
JEE Mains

Question

In the arrangement shown the rod is freely pivoted at point O and is in contact with the equilateral triangular block which can slide on the horizontal frictionless ground without toppling. As the block is given a constant speed v forward, the rod rotates about O. Find magnitude of angular velocity of rod in rad/s at the instant when θ=30°. (friction is absent at O) [Take v=20 m/s, a=1 m]

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1: From the figure the equilateral triangle has base length 2a/√3, so its height is H=(√3/2)(2a/√3)=a. Step 2: If the top vertex at height a contacts the pivoted rod, its horizontal position x from pivot O obeys tanθ=a/x, or x=a cotθ. Step 3: The block translates horizontally with speed |dx/dt|=v. Differentiate x=a cotθ: v=a csc²θ |dθ/dt|. Step 4: Hence |ω|=v sin²θ/a=20×(1/2)²/1=5 rad/s.