A force F = 2 i + x j + y k (in N) with y > 0 acts on a particle, producing a displacement s = i - 2 j + 2 k (in m). The work done by the force is zero and the magnitude of the force is 15 N. If this force is applied at a point whose position vector is r = i + j (in m), the magnitude of the torque produced about the origin is N N m. The value of N is _____ .

Step-by-step solution

The work done by the force is zero, so the dot product of force and displacement is zero: W = F s = 0 (2)(1) + (x)(-2) + (y)(2) = 0 2 - 2x + 2y = 0 x - y = 1 x = y + 1 The magnitude of the force is 15 N: | F | = 2^2 + x^2 + y^2 = 15 4 + x^2 + y^2 = 225 x^2 + y^2 = 221 Substitute x = y + 1 into the magnitude equation: (y + 1)^2 + y^2 = 221 y^2 + 2y + 1 + y^2 = 221 2y^2 + 2y - 220 = 0 y^2 + y - 110 = 0 (y + 11)(y - 10) = 0 Since y > 0 , we have y = 10 . Then, x = 10 + 1 = 11 . The force vector is F = 2 i + 11 j + 10