Rigid Body Balance

A uniform plank of length $L$ and mass $M = 6.0$ kg rests on two supports at its ends, A and B. An object of mass $m = 4.0$ kg is placed on the plank at a distance of $L/4$ from end A. Take $g = 10$ m/s².

A ladder AB of length $L = 4.0$ m and weight $G = 100$ N leans against a smooth vertical wall. Its center of mass C is located a distance $d = 1.5$ m from its lower end B. The ladder is in static equilibrium, making an angle $\theta = 20^\circ$ with the wall.

A non-uniform rod AB of length L = 2.0 m and weight G = 100 N is suspended horizontally by two ropes and is in static equilibrium. One rope at end A makes an angle of 37° with the vertical, and the other rope at end B makes an angle of 53° with the vertical. (Use sin 37° = 0.6).

A uniform rod AB, weighing $W_{rod} = 100$ N, has its center of gravity at its midpoint C. End A is connected to a wall by a hinge, allowing free rotation. End B is connected to the wall by a horizontal rope BE. The rod makes an angle of $60^\circ$ with the vertical wall, as shown in the diagram. An object weighing $W_{obj} = 300$ N is hung at point D, where the distance BD is one-third of the rod's length AB.

Three people are carrying a uniform heavy beam. One person lifts one end, and the other two people use a massless crossbar to lift the beam.

To roll a cylinder over an obstacle, it is known that the cylinder's radius is $R$, its weight is $G$, and the obstacle's height is $h$ (as shown in Figure).

A trapdoor on a ceiling is a square board with a side length of $1.00$ m and a weight of $100$ N. One of its sides is connected to the ceiling with hinges, while the opposite side is secured by a latch. It is known that the door's center of gravity is located on the same horizontal plane as the door's center but is displaced by $10$ cm towards the hinged side.

Two uniform rods are attached to a wall with hinges, and their other ends are joined by a bolt, as shown in the figure. The horizontal rod has a weight $W_1 = 120$ N and a length $L = 2.0$ m. The slanted rod has a weight $W_2 = 150$ N. The vertical distance between the two hinges on the wall is $h = 1.5$ m.

A uniform cubical box filled with sand weighs $W = 1000$ N. A horizontal force $F$ is applied at the top of the box to make it tip over without sliding.

Two objects with masses $m_1 = 1.0$ kg and $m_2 = 2.0$ kg are placed at the ends of a light rod of length $l = 1.20$ m. The rod is supported by a fulcrum as shown in the figure. Take $g = 10$ m/s².