In the figure shown a uniform rod of mass $m$ and length l is hinged. The rod is released when the rod makes angle $θ = 6 0^{\circ}$ with the vertical.

Question

In the figure shown a uniform rod of mass m and length l is hinged. The rod is released when the rod makes angle θ = 6 0 ∘ with the vertical.

TestHub · Accepted Answer

(A) τ = m g ( 2 ℓ ​ ) sin 6 0 ∘ I = 3 1 ​ m ℓ 2 α = I τ ​ = ( 1/3 ) m ℓ 2 m g ( ℓ /2 ) ( 3 ​ /2 ) ​ = 4 ℓ 3 3 ​ g ​ ​ (B) a CM ​ = α ( 2 ℓ ​ ) = 8 3 3 ​ g ​ Using R − m g = m a CM ​ : R 2 = ( m g ) 2 + ( m a CM ​ ) 2 R = m g 1 + ( 8 3 3 ​ ​ ) 2 ​ = 8 19 ​ ​ m g ​ (C) m g ( 2 ℓ ​ ) ( 1 − cos 6 0 ∘ ) = 2 1 ​ I ω 2 m g ( 4 ℓ ​ ) = 2 1 ​ ( 3 1 ​ m ℓ 2 ) ω 2 ω = 2 ℓ 3 g ​ ​ ​ (D) a CM ​ = ω 2 ( 2 ℓ ​ ) = 4 3 g ​ R − m g = m a CM ​ R = m g + 4 3 ​ m g = 4 7 ​ m g ​

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