Lesson Worksheet: Equilibrium of a Rigid Body under Coplanar Couples Mathematics

Complete the following: If the couples and are in equilibrium, where , then .

The moments, and , of two couples satisfy the equation . Which of the following is therefore true?

is a square. The forces , 2 N, , and 2 N act along the directions ,,, and respectively. Find the value of for the system to be in an equilibrium state.

is a rod having a length of 90 cm and a negligible weight. It is suspended horizontally by a pin at its midpoint. Two forces, each is of a magnitude 7.5 N, are acting at its ends as shown in the figure. It is also pulled by a string, whose tension is 25 N, in a direction making an angle of with the rod from point . If a force is acting on the rod at point so that the rod is in a horizontal equilibrium position, find the magnitude of , its direction , and the length of .

is a rod having a length of 50 cm and a negligible weight. Two coplanar pairs of forces are acting on the rod as shown in the figure. The first couple consists of two forces acting perpendicularly to the rod, each of magnitude 2 kg-wt, and the second couple consists of two forces, each of magnitude . Determine the value of that makes the rod in equilibrium.

is a uniform rod with length 6 cm. It is free to rotate about a smooth nail in a small hole in the rod at a point between and , where . The rod is in equilibrium, laying horizontally, under the action of two forces, each of magnitude 8 N, acting at either end at an angle of with the rod as shown in the figure below. Find the weight of the rod and the magnitude of the reaction of the nail .

is a regular hexagon having a side length of 18 cm. Forces of magnitudes 18, 7, 18, and 7 newtons are acting along , , , and respectively. Two other forces, each of magnitude newtons, are acting along and . Find the value of , given that the system is in equilibrium.

is a rectangle, in which and . Forces of magnitudes , 14 N, , and 14 N are acting along , , , and respectively. If this system of forces is in equilibrium, find and .

is a rod of negligible weight and length 54 cm. It is suspended horizontally by a pin at its midpoint. Forces of magnitude N act on each end, one of them vertically upward at and the other vertically downward at . The rod is pulled by a string, attached to it at point , inclined at an angle of to . The tension in the string has a magnitude of 192 N. The rod is kept in horizontal equilibrium by a fourth force acting on the rod at point with an angle of to . Assuming that there is no reaction at the pin, find the magnitude of and the length of .