A 1 m uniform wooden plank of specific gravity 0.5 is hinged to the bottom of a water tank filled to height 0.5 m. If nonzero equilibrium inclination to the vertical is θ, find 1/cos²θ.
Physics · Mixed Chapter
JEE Main Practice

Question

A 1 m uniform wooden plank of specific gravity 0.5 is hinged to the bottom of a water tank filled to height 0.5 m. If nonzero equilibrium inclination to the vertical is θ, find 1/cos²θ.

G1 Physics · Accepted Answer

Step 1: If submerged plank length is s, its vertical projection equals water depth: s cosθ=0.5 ⇒s=0.5/cosθ. Step 2: Weight of whole plank is W=(0.5ρ_water)Ag×1 and acts at distance 0.5 m from hinge. Step 3: Upthrust is ρ_waterAg·s and acts at s/2 from hinge. Step 4: Taking moments about hinge: ρgA s(s/2) sinθ =0.5ρgA·1(1/2)sinθ ⇒s²=0.5. Step 5: (0.5/cosθ)²=0.5 ⇒1/cos²θ=2.