By G. Fowles, G. Cassiday
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Extra resources for Analytical Mechanics [SOLUTIONS MANUAL]
29 Transcribe the left-hand side of the following equations into indicial notation and verify that the indicated operations result in the expressions on the right-hand side of the equations for the scalar φ, and vectors u and v. 30 Let the volume V have a bounding surface S with an outward unit normal ni. Let xi be the position vector to any point in the volume or on its surface. Show that (a) ∫ x n dS = δ V S (b) i j ij ∫ ١ (x ⋅ x) ⋅ nˆ dS = 6 V S (c) ∫ λw ⋅ nˆ dS = ∫ w ⋅ grad λ dV , where w = curl v and λ = λ(x).
3, defines the state of stress at that point. By applying Newton’s third law of action and reaction across the cutting plane, we observe that the force exerted by Portion I upon Portion II is equal and opposite to the force of Portion II upon Portion I. Additionally, from the principle of linear momentum (Newton’s second law) we know that the time rate of change of the linear momentum of any portion of a continuum body is equal to the resultant force acting upon that portion. 2-3b) nˆ SI SII i VI ti( n ) dS + ˆ ∫ VII VI VII where SI and SII are the bounding surfaces and VI and VII are the volumes of Portions I and II, respectively.
Additionally, from the principle of linear momentum (Newton’s second law) we know that the time rate of change of the linear momentum of any portion of a continuum body is equal to the resultant force acting upon that portion. 2-3b) nˆ SI SII i VI ti( n ) dS + ˆ ∫ VII VI VII where SI and SII are the bounding surfaces and VI and VII are the volumes of Portions I and II, respectively. Also, bi are the body forces, ρ is the density, and vi is the velocity field for the two portions. We note that SI and SII each contain S* as part of their total areas.