(a) Explanation Given information: The force ( F ) is F = ( x i + y j + z k ) ( x 2 + y 2 + z 2 ) 3 / 2 . Calculation: Write the potential function associated with force F: F = ( x i + y j + z k ) ( x 2 + y 2 + z 2 ) 3 / 2 Calculate the force F in ‘ x ’ component using the relation: F x = x ( x 2 + y 2 + z 2 ) 3 / 2 Partial differentiate the equation with respect to ‘ y ’. ∂ F x ∂ y = ∂ ∂ y ( x ( x 2 + y 2 + z 2 ) 3 / 2 ) = x ( − 3 2 ) ( 2 y ) ( x 2 + y 2 + z 2 ) 3 / 2 (1) Calculate the force F in ‘ y ’ component using the relation: F y = y ( x 2 + y 2 + z 2 ) 3 / 2 Partial differentiate the equation with respect to ‘ x ’. ∂ F y ∂ x = ∂ ∂ x ( y ( x 2 + y 2 + z 2 ) 3 / 2 ) = y ( − 3 2 ) ( 2 x ) ( x 2 + y 2 + z 2 ) 3 / 2 (2) The equation (1) and (2) are equal. Thus, ∂ F x ∂ y = ∂ F y ∂ y . Consider the condition 2. Calculate the force F in ‘ x ’ component using the relation: F x = x ( x 2 + y 2 + z 2 ) 3 / 2 Partial differentiate the equation with respect to ‘ z ’. ∂ F x ∂ z = ∂ ∂ z ( x ( x 2 + y 2 + z 2 ) 3 / 2 ) = x ( − 3 2 ) ( 2 z ) ( x 2 + y 2 + z 2 ) 3 / 2 (3) Calculate the force F in ‘ z ’ component using the relation: F z = z ( x 2 + y 2 + z 2 ) 3 / 2 Partial differentiate the equation with respect to ‘ x ’ (b) To determine Prove the potential function V ( x , y , z ) associated with F .

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 13.84P

(a)

To determine

Prove the force F(x,y,z) is conservative.

(b)

To determine

Prove the potential function V(x,y,z) associated with F.

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Chapter 13.2, Problem 13.83P

Chapter 13.2, Problem 13.85P

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Prove the force F ( x , y , z ) is conservative.