3D Torque Equilibrium Analysis

00:00 6 marks

A uniform beam of length \(L = 5\ m\) is fixed at one end at the origin with the beam lying along the x-axis. Two forces are applied as follows:

– Force \(F_1 = 20\ N\) is applied at the free end (point \((5, 0, 0)\)) making an angle of \(30^\circ\) above the horizontal in the vertical (x-y) plane.
– Force \(F_2 = 15\ N\) is applied at a point 3 m from the fixed end (point \((3, 0, 0)\)) in the negative y-direction (i.e. \(\vec{F_2} = (0, -15, 0)\)).

Using the vector cross product \(\vec{\tau} = \vec{r} \times \vec{F}\), determine the net torque about the fixed end. In your answer, provide the vector representations, show all calculations, and discuss the condition that must be met for complete equilibrium in this 3D mechanical setting.

Hint Hint

Hint

Break the problem into steps: first, represent the position and force vectors; second, compute each torque using \(\vec{\tau} = \vec{r} \times \vec{F}\); finally, sum the torques and consider equilibrium conditions.

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Answer Answer

Complete Answer

The net torque is \(\vec{\tau}_{net} = (0, 0, 5)\ N\cdot m\). For equilibrium, the net torque must be zero, so an additional counteracting torque of \((0, 0, -5)\ N\cdot m\) is required.

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