You are at the shoe store to buy a pair of basketball shoes that have the greatest traction on a specific type of hardwood. To determine the coefficient of static friction. \(\mu\), you place each shoe on a plank of the wood and tilt the plank to an angle \(\theta\), at which the shoe just starts to slide. Obtain an expression for \(\mu\) as a function of \(\theta\).

Imagine trying to push a heavy box across the floor. It's tough to get it moving, right? This resistance to start moving is due to static friction. Static friction acts between surfaces at rest relative to each other, preventing motion until a certain threshold force is exceeded. In our exercise, the shoes on the tilted plank experience static friction which holds them steady until the incline becomes too steep and they begin to slide.

This threshold where the shoe just begins to slide indicates the maximum value of static friction. The force of static friction is defined by the equation:
\( F_f = \mu F_N \), where \( \(mu \)\) represents the coefficient of static friction, and \( F_N \) is the normal force. By finding the point where static friction can no longer counteract the gravitational pull down the incline, we manage to deduce the coefficient \( \(mu \)\) as a tangible measure of traction.

An inclined plane is a flat surface tilted at an angle to the horizontal. It's a classic setup to study components of forces and help us understand how objects behave on slopes, like our shoe experiment. When we talk about an inclined plane, we're considering a fundamental physics problem: how does the angle of the incline affect the forces on an object resting on it?

A steeper angle increases the gravitational force component trying to pull the object down, which is also why one feels more pressure walking up a steeper hill. The unique element of our challenge is finding the angle \( \theta \) at which the static friction gives way and the shoe starts to slide down, which is directly correlated with the angle of the incline.

Gravity is the force that pulls us to the Earth, but when an object is on an incline, this gravitational force isn't just a straight 'downward' pull. It's helpful to break it down into two parts, or components: one that's parallel to the slope (trying to slide the object down the plane) and another that's perpendicular (pushing the object into the slope).

These components are central to our shoe example. They are defined mathematically as:
\( F_{g\parallel} = mg\sin{\theta} \) for the parallel component, and
\( F_{g\perp} = mg\cos{\theta} \) for the perpendicular component. Understanding these components helps us unravel the puzzle of at what angle \( \theta \) the component of gravity along the plane overcomes static friction's grip.

When we stand on the ground, not only are we pulled down by gravity, but we are also pushed up by the ground with a force equal in magnitude and opposite in direction. This support force is called the normal force. It’s 'normal' in the sense that it’s perpendicular to the contact surface.

In the context of our inclined plane situation, the normal force is crucial because it balances out the perpendicular component of the gravitational force. It's represented in equations as:
\( F_N = mg\cos{\theta} \). Importantly, the normal force is tied to the force of static friction, a relationship that forms the core of finding the coefficient of static friction, \( \(mu \)\), for the shoe experiment.