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-a) Determine the work done by the force F = (xy + 3z)i + (2y2-x2)i + (z-2y)k intaking a particle from x=0 to x = 1 along a curve defined by the equations:x2 = 2y;2x3 = 32 |
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Answer» GIVEN : F = (xy + 3z)i + (2y2-x2)i + (z-2y)k PARTICLE moves from x=0 to x = 1 along a curve defined by the equations: x2 = 2y; 2x3 = 32 SOLUTION : ◆Work done W=∫F. DL where F - Force and dl - displacement and are vectors. ◆Since the particle is taken from. x= 0 to 1, in x direction, dl= dx i^ ◆(i^ , j^, k^ - unit vector along x ,y and z respectively.) ◆As, F = (xy + 3z)i^ + (2y2-x2)j^ + (z-2y)k^ W = ∫F.dl ◆Substituting values, = [(xy + 3z)i^ + (2y2-x2)j^ + (z-2y)k^ ]. dx i^ ◆W=( xy + 3z).dx --(1) ◆Given, x^2 = 2y , y = x^2 /2 , 2x^3 = 3z , z = 2x^3 / 3 ◆Substituting in equation (1) F.dl= x^3/2 + 2x^3 = 5x³/2 ◆Integrating along x from 0 to 1 W = ∫F.dl = 5/8. ANSWER : W = ∫F.dl = 5/8. |
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