Question

A thin but rigid semicircular wire frame of radius r is hinged at O and can rotate in its own vertical plane. A smooth peg P starts from O and moves horizontally with constant speed v₀, lifting the frame upward as shown in the figure. Find the angular velocity ω of the frame when its diameter makes an angle of 60^∘ with the vertical. 

 

• A. (v₀)/(r)
• B. (v₀)/(2r)
• C. (2v₀)/(r)
• D. v₀ r

Hint:

For rigid bodies rotating about a hinge: v = ω r_⊥ Always take the perpendicular distance from the axis of rotation.

Answer 1

The Correct Option is B

Step 1: The peg moves horizontally with speed v₀ while remaining in contact with the semicircular frame. 

Step 2: The instantaneous velocity of the contact point on the frame is equal to the velocity of the peg. 

Step 3: At the given position, the perpendicular distance of point P from the hinge O is: OP_⊥ = 2r sin 60^∘ 

Step 4: Using the relation for rotational motion: v₀ = ω × OP_⊥ v₀ = ω × (2r sin 60^∘) ⟹ ω = (v₀)/(2r)