question_answer 74)

                    There are three forces \[{{F}_{1}},\,{{F}_{2}}\] and \[{{F}_{3}}\] acting on a body, all acting on a point P on the body. The body is found to move with uniform speed.                 (a) Show that the forces are coplanar.                 (b) Show that the torque acting on the body about any point due to these three forces is zero.

Answer:

                  (a) When body moves with uniform speed, no net force acts on it. That is, the vector sum of all forces acting on the body is zero.                 Thus, \[{{\vec{F}}_{1}}\,+{{\vec{F}}_{2}}\,+{{\vec{F}}_{3}}=0\].                 These forces are represented by the three sides of a triangle in a plane taken in the same order.                 Torque about A, \[\vec{\tau }=\vec{r}\,\times \,{{\vec{F}}_{1}}+\,\vec{r}\times \,{{\vec{F}}_{2}}+\vec{r}\,\times \,{{\vec{F}}_{3}}\]                 \[=\,\vec{r}\times \,({{\vec{F}}_{1}}+{{\vec{F}}_{2}}+{{\vec{F}}_{3}})\]                 Since \[{{\vec{F}}_{1}}+{{\vec{F}}_{2}}+{{\vec{F}}_{3}}=0\]                 \[\therefore \] \[\,\vec{\tau }=0\].