(a) Explanation Given information: The mass of the sphere ( m ) is 0.6 kg . The spring constant ( k ) is 100 N / m . The magnitude of the velocity at A ( v A ) is 20 m / s . The distance between the point O to point A ( r A ) is 0.5 m . The velocity at A is inclined at an angle ( θ ) is 60 ° . The central force ( F ) is 80 r 2 . The acceleration due to gravity ( g ) is 9.81 m / s 2 . Calculation: Show the figure of the elastic cord with the velocity as in Figure (1). The distance from the point O to point B is maximum when the minimum velocity is at B whereas the distance from the point O to point C is minimum when the maximum velocity is at C . Use the principle of conservation of angular momentum at positions A and A ′ . Write the expression for the conservation of angular momentum as follows; r A m v A sin θ = r m m v m r A v A sin θ = r m v m Substitute 20 m / s for v A , 0.5 m for r A and 60 ° for θ . ( 20 m / s ) ( 0.5 m ) sin 60 ° = r m v m v m = 8.66 r m (1) Consider the point A . Calculate the kinetic energy at the point A ( T A ) using the formula: T A = 1 2 m v A 2 Substitute 0.6 kg for m and 20 m / s for v A . T A = 1 2 × ( 0.6 ) × ( 20 2 ) = 120 J Calculate the potential energy at the point A ( V A ) using the formula: V A = ∫ F d r Substitute 80 r 2 for F . V A = ∫ 80 r 2 d r = 80 [ r − 2 + 1 − 2 + 1 ] = 80 ( r − 1 − 1 ) = − 80 r Substitute 0.5 m for r A . V A = − 80 0.5 = − 160 J Consider the point B . Calculate the kinetic energy at the point B ( T B ) using the formula: T B = 1 2 m v m 2 Substitute 0 (b) To determine Find the corresponding maximum ( v max ) and minimum velocity ( v min ) from the sphere to the origin O .

Vector Mechanics for Engineers: Statics and Dynamics

11th Edition

ISBN: 9780073398242

Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek, Phillip J. Cornwell, Brian Self

Publisher: McGraw-Hill Education

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Chapter 13.2, Problem 13.99P

(a)

To determine

Find the maximum (rmax) and minimum distance (rmin) from the sphere to the origin O.

(b)

To determine

Find the corresponding maximum (vmax) and minimum velocity (vmin) from the sphere to the origin O.

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Chapter 13.2, Problem 13.98P

Chapter 13.2, Problem 13.100P

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Find the maximum ( r max ) and minimum distance ( r min ) from the sphere to the origin O .

Solution Summary: The author shows the figure of the elastic cord with the velocity as in Figure (1).

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Find the maximum (rmax) and minimum distance (rmin) from the sphere to the origin O.

Find the corresponding maximum (vmax) and minimum velocity (vmin) from the sphere to the origin O.

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Chapter 13 Solutions