rigid body es defined in the study of mechanics? 11) Given vectors r and is state the type of results obtained for r. Fand r x 1?? (2 marks) iii) Given equation of force field as F=(2xy+z^3)i+x^3i+3xz^2 k determine whether it is conservative or not (6 marks) (2 marks)

i) A rigid body in mechanics is defined as an idealized solid object in which the distance between any two given points remains constant regardless of external forces or moments applied to it. This means that a rigid body does not deform under stress.<br /><br />ii) Given vectors \(\mathbf{r}\) and \(\mathbf{F}\), the results obtained for \(\mathbf{r} \cdot \mathbf{F}\) (dot product) is a scalar, and for \(\mathbf{r} \times \mathbf{F}\) (cross product) is a vector.<br /><br />iii) To determine whether the force field \(\mathbf{F} = (2xy + z^3)\mathbf{i} + x^3\mathbf{j} + 3xz^2\mathbf{k}\) is conservative, we need to check if the curl of \(\mathbf{F}\) is zero. The curl of a vector field \(\mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k}\) is given by:<br /><br />\[<br />\nabla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right)\mathbf{i} - \left( \frac{\partial R}{\partial x} - \frac{\partial P}{\partial z} \right)\mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right)\mathbf{k}<br />\]<br /><br />For \(\mathbf{F} = (2xy + z^3)\mathbf{i} + x^3\mathbf{j} + 3xz^2\mathbf{k}\):<br /><br />- \(P = 2xy + z^3\)<br />- \(Q = x^3\)<br />- \(R = 3xz^2\)<br /><br />Calculate each component of the curl:<br /><br />1. \(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} = 0 - 0 = 0\)<br />2. \(\frac{\partial R}{\partial x} - \frac{\partial P}{\partial z} = 3z^2 - 3z^2 = 0\)<br />3. \(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 3x^2 - 2x = 3x^2 - 2x\)<br /><br />Since the third component is not zero (\(3x^2 - 2x \neq 0\)), the curl of \(\mathbf{F}\) is not zero, indicating that the force field is not conservative.