剛體平衡：练习题

A uniform beam of weight \(100 \text{ N}\) and length \(4 \text{ m}\) is supported horizontally by two vertical pillars at its ends, \(A\) and \(B\). A block of weight \(400 \text{ N}\) is placed on the beam at a distance of \(1 \text{ m}\) from end \(A\). Calculate the magnitude of the vertical reaction force at pillar \(B\).

A uniform square lamina \(ABCD\) of side \(2a\) has a triangular portion \(OAB\) removed, where \(O\) is the center of the square. Find the distance of the center of mass of the remaining portion from the side \(CD\).

A uniform ladder of length \(2L\) and weight \(W\) rests in equilibrium with its upper end against a smooth vertical wall and its lower end on rough horizontal ground. The ladder makes an angle \(\theta\) with the horizontal. If the coefficient of friction between the ladder and the ground is \(\mu\), find the minimum value of \(\tan \theta\) for which the ladder does not slip.

A uniform rod \(AB\) of weight \(W\) and length \(2L\) is smoothly hinged at \(A\) to a vertical wall. It is held in equilibrium at an angle of \(60^\circ\) to the downward vertical by a horizontal force \(F\) applied at \(B\). Find the magnitude of \(F\).

A uniform solid cone has base radius \(r\) and height \(h\). It is placed with its base on a rough plane which is slowly tilted. Given that the coefficient of friction is large enough to prevent sliding, find the maximum ratio \(\frac{h}{r}\) such that the cone can be tilted to an angle of \(30^\circ\) without toppling.

A uniform rod \(AB\) of weight \(60\text{ N}\) and length \(2\text{ m}\) is pivoted at end \(A\). Calculate the moment of the rod's weight about the pivot \(A\) when the rod is held in a horizontal position.

A uniform L-shaped lamina is formed by joining a rectangular piece of size \(6\text{ cm} \times 2\text{ cm}\) to another rectangular piece of size \(2\text{ cm} \times 4\text{ cm}\) such that the total height is \(6\text{ cm}\) and total width is \(4\text{ cm}\). Determine the distance of the centre of mass from the longer vertical edge.

A uniform rod \(AB\) of length \(2a\) and weight \(W\) is in equilibrium with end \(A\) against a smooth vertical wall and leaning over a smooth peg \(C\) at a distance \(d\) from the wall. If the rod makes an angle \(\theta\) with the wall, express \(d\) in terms of \(a\) and \(\theta\).

A uniform L-shaped lamina is formed from a single piece of metal. It consists of two rectangles: rectangle A with dimensions \(4 \text{ cm} \times 1 \text{ cm}\) and rectangle B with dimensions \(1 \text{ cm} \times 3 \text{ cm}\), joined such that the total width is \(4 \text{ cm}\) and total height is \(4 \text{ cm}\).

Find the distance of the centre of mass of the lamina from the longer external edge of rectangle A.

A uniform solid cube of side length \(a\) is placed on a rough plane inclined at an angle \(\theta\) to the horizontal. The coefficient of friction between the cube and the plane is \(\mu = 0.6\). The angle \(\theta\) is gradually increased.

Determine whether the cube will topple or slide first as \(\theta\) increases.