(a) Explanation Given information: A thin homogeneous wire is bent into the shape of an isosceles triangle of sides are b , b and 1.6 b. Calculation: Show the position of centroid and distance as in Figure (1). Write the equation for mass moment of inertia ( I ¯ ) : I ¯ = m r 2 Here, r is distance of each particle from the axis of rotation. Calculate the expression for mass moment of inertia ( I A ¯ ) using mass by length ratio at point A : I A = 2 [ 1 3 ( m 1 + 1 + 1.6 ) b 2 ] + ( 1.6 m 1 + 1 + 1.6 ) [ ( 1.6 b ) 2 12 + ( 0.6 b ) 2 ] Here, 2 [ 1 3 ( m 1 + 1 + 1.6 ) b 2 ] represents the moment of inertia of two inclined wires. Modify the above equation, I A = 2 [ 1 3 ( m 1 + 1 + 1.6 ) b 2 ] + ( 1.6 m 1 + 1 + 1.6 ) [ ( 1.6 b ) 2 12 + ( 0.6 b ) 2 ] = 5 27 m b 2 + 172 675 m b 2 = 11 25 m b 2 Calculate the centroid equation ( y ¯ ) about y axis: y ¯ = ( b + b ) ( 0.6 b 2 ) b + b + 1.6 b = 0.6 b 3.6 b = b 6 Calculate the distance equation AG as below A G = 0.6 b − y ¯ Substitute b 6 for y ¯ . A G = 0.6 b − b 6 = 13 30 b The external forces in the system are force due mass of the thin wire and the effective restoring couple is I ¯ θ ¨ . Take moment about A in the system for external forces. ( ∑ M A ) e x t e r n a l = − m g ( A G ) θ Substitute 13 30 b for AG . ( ∑ M A ) e x t e r n a l = − m g ( 13 30 b ) θ = − 13 m g b 30 θ Take moment about A in the system for effective forces (b) To determine The period ( τ n ) of small oscillation if the wire is suspended from point B .

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ISBN: 9781264095032

Author: BEER

Publisher: MCGRAW-HILL HIGHER EDUCATION

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Chapter 19.2, Problem 19.51P

(a)

To determine

The period (τn) of small oscillation if the wire is suspended from point A.

(b)

To determine

The period (τn) of small oscillation if the wire is suspended from point B.

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Chapter 19.2, Problem 19.50P

Chapter 19.2, Problem 19.52P

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The period ( τ n ) of small oscillation if the wire is suspended from point A .

Solution Summary: The author explains the period of small oscillation if the wire is suspended from point A and the expression for mass moment of inertia.