Torque

You are challenged to construct a bridge using two identical uniform rectangular blocks, length $24 \mathrm{~cm}$, which overhang a table as indicated in Figure. The lower block overhangs the table by $x \mathrm{~cm}$ and the upper block overhangs the lower block by $6.0 \mathrm{~cm}$. Under what condition will one or both blocks collapse? [5]

A ladder of length $L$ and mass $m$, with a uniform density, rests against a frictionless vertical wall at an angle of $60^{\circ}$ to the wall. The lower end rests on a flat surface with a coefficient of static friction of $\mu_{s}=0.40$. A student with a mass $M=2 m$ attempts to climb the ladder. What fraction of the distance up the ladder will the student have reached when the ladder begins to slip? [5]

A window cleaner's ladder shown in Figure is narrower at the top than the bottom. It has a weight of $350 \mathrm{~N}$ and a length of $5.0 \mathrm{~m}$. When it lies flat on the ground, a force of $80 \mathrm{~N}$ is needed to lift the narrow end off the ground.

(i) How far is the centre of mass from the narrow end?

(ii) What force is required to lift the wide end of the ladder off the ground? [5]

A thin rod is balanced in a smooth hemispherical bowl fixed to a table, touching both the interior and the rim as shown in Fig. 1. The rim of the bowl remains horizontal. Expressed in the simplest form, determine the radius of the bowl $r$ in terms of the length of the $\operatorname{rod} \ell$, and the angle $\theta$ to the horizontal. [6]

A uniform plank of mass $m$ stands on a smooth floor and leans against a smooth wall at an angle $\alpha$ to the horizontal. It is held in place by a horizontal string attached to the bottom of the ladder and to the bottom of the wall, as shown in Fig. 1. What is the tension, $T$, in the string in terms of $m, g$ and $\alpha$ ? [3]

A uniform rod, of length $2 a$, floats partly immersed in a liquid, being supported by a string fastened to one of its ends, the other end of the string being attached to a fixed point A, as in Fig. 8. The density of the liquid is a factor $4 / 3$ times that of the rod. Determine
(i) the fraction of the rod's length that will be submerged, and
(ii) the tension, $T$ in the string. [6]

A light lever in the form of a letter L, with arms AB, BC of lengths $a$, $b$, is smoothly pivoted at B so that it can turn in a vertical plane. Masses $m_a$ and $m_b$ are attached at A and C respectively. In an equilibrium position, the inclination of AB to the vertical is given by angle $\theta$. Obtain an expression for $\theta$ in terms of $m_a$, $m_b$, $a$ and $b$. [4]