Explanation Given information: The radius ( r ) of the homogeneous wire is 220 mm. The pin is pushed down ( y B ) for 20 mm. The time ( t ) is 8 sec. Calculation: Show the free body diagram and kinetic diagram of the circular wire in Figure (1). Write the expression for total volume ( V ) of the wire bent: V = 2 r + π r Write the expression of the total mass ( m ) of the wire: m = ρ ( V ) Substitute 2 r + π r for V . m = ρ ( 2 r + π r ) = ρ r ( 2 + π ) Here, ρ is the mass per unit length Write the expression for the centroid ( y ¯ ) of the semicircle about y axis: y ¯ = 2 r π Calculate the distance ( c ) using the relation: Take moment about center O for external and effective force before the point B is push down. ∑ M 0 = 0 m g c = 0 + π r ρ ( 2 r π ) g m g c = 2 r 2 ρ g Substitute ρ r ( 2 + π ) for m . ρ r ( 2 + π ) g c = 2 r 2 ρ g c = 2 r ( 2 + π ) Write the expression for the phase angle ( α ) : α = θ ¨ Write the expression for the normal acceleration ( a n ) : a n = c α Substitute θ ¨ for α . a n = c θ ¨ Find the mass ( m s e m i c i r c l e ) of the semicircle: m s e m i c i r c l e = ρ π r Find the mass m l i n e of the line: m l i n e = 2 ρ r Find the density ( ρ ) of the wire bent using the relation: ρ = m ( 2 + π ) r Calculate the moment of inertia ( I 0 ) s e m i c i r c l e of the semicircle using the relation: ( I 0 ) s e m i c i r c l e = m s e m i c i r c l e r 2 Substitute ρ π r for m s e m i c i r c l e . ( I 0 ) semicircle = ρ π r r 2 = ρ π r 3 Calculate the moment of inertia ( I 0 ) l i n e of the line using the relation: ( I 0 ) l i n e = m l i n e ( 2 r ) 2 12 Substitute 2 ρ r for m l i n e . ( I 0 ) l i n e = 2 ρ r × ( 2 r ) 2 12 = 2 3 ρ r 3 Calculate the total moment of inertia ( I 0 ) using the relation: I 0 = ( I 0 ) s e m i c i r c l e + ( I 0 ) l i n e Substitute ρ π r 3 for ( I 0 ) semicircle and 2 3 ρ r 3 for ( I 0 ) line . I 0 = ρ π r 3 + 2 3 ρ r 3 = ρ r 3 ( π + 2 3 ) Substitute m ( 2 + π ) r for ρ . I 0 = m ( 2 + π ) r r 3 ( π + 2 3 ) = m r 2 ( 2 + π ) ( π + 2 3 ) Express the term ( I 0 ) , I 0 = I ¯ + m c 2 The external forces in the system are force due mass of the rod and the effective restoring couple is I ¯ θ ¨ . Take moment about center O of wire bent in the system for external forces after point B pushed. ( ∑ M O ) e x t e r n a l = − m g c sin θ For small oscillation sin θ ≈ θ . ( ∑ M O ) e x t e r n a l = − m g c θ Take moment about center O of wire bent in the system for effective forces after point B pushed

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ISBN: 9781259633133

Author: BEER

Publisher: MCG

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Chapter 19.2, Problem 19.62P

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The magnitude of velocity of B (vB) 8 sec later.

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Chapter 19 Solutions

VECTOR MECH...,STAT.+DYNA.(LL)-W/ACCESS

Ch. 19.1 - A particle moves in simple harmonic motion....Ch. 19.1 - Prob. 19.2P Ch. 19.1 - Prob. 19.3P Ch. 19.1 - Prob. 19.4P Ch. 19.1 - Prob. 19.5P Ch. 19.1 - Prob. 19.6P Ch. 19.1 - Prob. 19.7P Ch. 19.1 - Prob. 19.8P Ch. 19.1 - Prob. 19.9P Ch. 19.1 - Prob. 19.10P

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The magnitude of velocity of B ( v B ) 8 sec later.

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