(a) Explanation Given information: The weight of the collar ( W ) is 8 oz . The radius of the semicircle ( r ) is 1.5 ft . The height of the point A to point C ( x ) is 7.5 ft . The acceleration due to gravity ( g ) is 32.2 ft / s 2 . The velocity at the point A is v 0 . Calculation: Consider the loop 1. Convert the unit of weight of the collar ( W ) from oz to lb using the relation: W = W oz 16 lb Here, W oz is the mass of the collar in oz . Substitute 8 oz for W oz . W = 8 16 = 0.5 lb Calculate the mass of the collar ( m ) using the formula: W = m g m = W g Substitute 0.5 lb for W and 32.2 ft / s 2 for g . m = 0.5 32.2 = 0.01553 lb ⋅ s 2 / ft Assume the datum is at position A . Show the free body diagram of the point B with the forces acting as in Figure (1). Here, the point B has small velocity when the force exerted by the tube on package is zero. Consider the point A . Calculate the kinetic energy in the position A ( T A ) using the formula: T A = 1 2 m v 0 2 Substitute 0.01553 lb ⋅ s 2 / ft for m . T A = 1 2 ( 0.01553 lb ⋅ s 2 / ft ) v 0 2 = 0.007765 v 0 2 Here, the potential energy in the position A ( V A ) is zero. Consider the point B . Calculate the velocity at the point B ( v B ) by considering the force equilibrium condition: ∑ F = 0 m g − m v B 2 r = 0 m g = m v B 2 r g = v B 2 r Substitute 32.2 ft / s 2 for g and 1.5 ft for r . 32.2 = v B 2 1.5 v B 2 = 32.2 × 1.5 v B 2 = 48.3 ft / s v B = 48.3 v B = 6.9498 ft / s Calculate the kinetic energy in the position B ( T B ) using the formula: T B = 1 2 m v B 2 Here, v B is the velocity at the point B . Substitute 0.01553 lb ⋅ s 2 / ft for m and 48.3 ft / s for v B . T B = 1 2 × 0.01553 × 48.3 ft / s = 0 (b) To determine Find the force exerted by the package on the loop just before the package leaves the loop at C ( N C ) .

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 13.74P

(a)

To determine

Find the smallest velocity (v0) for which the package will reach C.

(b)

To determine

Find the force exerted by the package on the loop just before the package leaves the loop at C (NC).

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Chapter 13.2, Problem 13.73P

Chapter 13.2, Problem 13.75P

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Find the smallest velocity ( v 0 ) for which the package will reach C .

Solution Summary: The author calculates the mass of the collar using the formula: cW=mg m=