Finding Work in a Conservative Force Field In Exercises 19-22, (a) show that
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- QI-A) Fund the position, vekocity and the speed of the vector function ( r(t) ) that given bekow at t-2. r(t) = 1i-t'iarrow_forwardWork by a constant force Show that the work done by a con- stant force field F = ai + bj + ck in moving a particle along any path from A to B is W = F•AB.arrow_forwardUse Green's theorem to calculate the work done by the force F(x, y) = xy'i + 3x?yj on a particle that is moving counterclockwise around the triangle with vertices (-3, 0), (0, 0), and (0, 3).arrow_forward
- Use Green's theorem to find the work done by the force field F(xy) - хyі + x²+> on a particle that moves along the path that starts from the point (5,0), transverse the upper semi-circle x² + y? = 25 and return to the staring point along the x-axis. %3Darrow_forward10) Find the work done by the force field F(x,y,z) = zi + x j + y k in moving a particle from the point (3, 0, 0) to the point (0,TT/2,3) along (a) straight line (b) the helix x = 3 cost, y = t, z = 3 sintarrow_forwardProj,u =|u/coso- 1=\u[coso u: V V Scal, u = |u|cose = u: V y. V (a) Find the length of the projection of w onto v given that w = (4, 7) and v = (1, 1). (b) A given force F = i+j- k (in newtons) moves an object from point (4, 7,2) to point (8, 3, 6) (in meters). Determine the work done by the force in moving the object.arrow_forward
- Compute work done performed by the force F = (y cos r – rysin z, ry + r³ + x + x cos r) acting on the object moving along the triangle from (0,0) to (0,5), from (0,5) to (4,3), from (4, 3) to (0,0). Work done o -116.67arrow_forwardSketch and describe the vector field F (x, y) = (-y,2x)arrow_forward38. Motion along a circle Show that the vector-valued function r(t) = (2i + 2j + k) %3D + cos t V2 j) + sin t V2 j + V3 V3 V3 describes the motion of a particle moving in the circle of radius 1 centered at the point (2, 2, 1) and lying in the plane x + y – 2z = 2.arrow_forward
- Exercise III Let (a) o = x²y +xż and (b) ó = x² + y² + z?. Then, respectively attempt to find the directional derivative at • (1,2, 1) in the direction of the vector (2i - 3j+ 4k). • (3,0, 1) in the direction of the vector (i- 3j + 2k).arrow_forward(b) Find the work done by force field F = 3x? î + y²j on a particle when it moves from (0, 0) to (-T, 0) along the curves Cl and C2 in Figure Q3 (b) by solving S. F dr. Based on your calculation, judge whether the force, F is conservative or non- conservative and give your explanation. | -TT C1 0, C2 FIGURE Q3(b)arrow_forward+y -n CI Č2 FIGURE Q3(b)arrow_forward
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