Equilibrium of Rigid Bodies — MCQ Practice

A uniform ladder of length $L$ and mass $m$ rests against a smooth vertical wall and a rough horizontal floor. The coefficient of static friction between the ladder and the floor is $\mu_s$. If the ladder is at the verge of slipping when it makes an angle $\theta$ with the horizontal, what is the value of $\mu_s$?

A uniform rod of mass $M$ and length $L$ is hinged at one end and kept in horizontal position by a string attached to the free end and to a point vertically above the hinge. If the string makes an angle $\theta$ with the rod, what is the tension in the string?

A uniform beam of mass $m$ and length $L$ is supported by two vertical forces, $F_1$ at one end and $F_2$ at a distance of $L/4$ from the other end. What are the magnitudes of $F_1$ and $F_2$?

Which of the following statements is TRUE for a rigid body in equilibrium?

A uniform rod of mass $m$ and length $L$ is pivoted at its center. A mass $m_1$ is placed at one end, and a mass $m_2$ is placed at a distance $L/4$ from the other end. For the rod to remain in horizontal equilibrium, what is the ratio $m_1/m_2$?

A uniform square plate of side $a$ and mass $M$ is suspended from one of its corners. A mass $m$ is attached to the diagonally opposite corner. For the plate to hang in equilibrium such that the diagonal passing through the suspension point is vertical, what is the ratio $M/m$?