A $50\text{ kg}$ wooden crate rests on a flat horizontal concrete floor. The coefficient of static friction between the wood and concrete is $\mu_s = 0.400$. A worker pushes the crate horizontally. What is the minimum horizontal force required to start the crate moving?

A block of mass $m$ is placed on an adjustable inclined plane. As the angle of the incline $\theta$ is slowly increased, the block begins to slip when the angle reaches $30.0^\circ$. Determine the coefficient of static friction $\mu_s$ between the block and the plane.

A tall refrigerator of height $H = 2.00\text{ m}$ and width $W = 0.800\text{ m}$ rests on a floor with $\mu_s = 0.600$. Its center of gravity is exactly in the middle. A person pushes horizontally at the very top edge of the fridge. Will the refrigerator slip first, or tip over first?

A block weighing $100\text{ N}$ is resting on a rough horizontal plane. The coefficient of static friction is $\mu_s = 0.300$. A pulling force $P$ is applied to the block at an angle of $20.0^\circ$ above the horizontal. Find the magnitude of $P$ required to cause impending motion.

A $200\text{ N}$ block rests on a $40.0^\circ$ incline. The coefficient of static friction is $\mu_s = 0.250$. What is the minimum force $P$ applied parallel to the incline required to prevent the block from sliding down?

Using the same block from Example 5 ($200\text{ N}$ on a $40.0^\circ$ incline, $\mu_s = 0.250$), what is the minimum force $P$ applied parallel to the incline required to cause impending motion up the incline?

A rope is wrapped once around a fixed cylindrical post (one full turn, $\beta = 2\pi$ radians). A worker wants to hold a $1000\text{ N}$ load suspended from one end of the rope. If the coefficient of static friction between the rope and the post is $\mu_s = 0.300$, what minimum force must the worker apply to the other end of the rope?

A $500\text{ N}$ block rests on a horizontal surface. A small wedge with an angle of $\alpha = 10.0^\circ$ is driven under it to lift it. The coefficient of static friction is $\mu_s = 0.200$ at all contacting surfaces (wedge-floor, wedge-block, block-wall). What horizontal force $P$ must be applied to the wedge to start lifting the block? Assume the block is constrained vertically by a frictionless wall (to simplify, let's just consider friction at the wedge). Correction: Let's consider friction at the block-wall interface too for a complete analysis.

Let $\phi_s = \tan^{-1}(0.200) = 11.31^\circ$ be the friction angle.

In many structural applications, the goal is to prevent motion entirely. Discuss the difference between static and kinetic friction and why engineers typically design for a state well below impending motion.

Consider a tall, freestanding bookcase loaded heavily at the top. Explain the mechanism that determines whether it will slide or tip over when subjected to a lateral earthquake load.

When a large ship is moored to a dock, the mooring lines are often wrapped around a bollard. Explain how belt friction allows a single person to hold a massive ship in place.

Wedges are simple machines commonly used in construction to lift heavy loads short distances or to level machinery. Explain the role of friction in making a wedge "self-locking."