Question

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 \(\theta\)

Short answer

Answer: The expression for the coefficient of static friction is \(\mu = \tan(\theta)\), where \(\theta\) is the angle at which the shoe just starts to slide.

Step-by-step solution

Identifying the forces

There are two main forces acting on the shoe when it is on the inclined plane: gravitational force (weight) and the normal force. The shoe's weight, \(W=mg\), acts vertically downward, while the normal force, \(N\), acts perpendicular to the inclined plane. When the shoe is on the verge of sliding, the static friction force, \(f_s\), is equal to \(\mu N\). This force acts along the inclined plane opposing the motion of the shoe. Let's decompose the shoe's weight into two components: one that is parallel to the inclined plane (\(W_\parallel\)) and one that is perpendicular to the inclined plane (\(W_\perp\)).

Decomposing the weight

To decompose the weight into its components, we will use trigonometry. \(W_\parallel\) would be equal to \(W\sin(\theta)\), while \(W_\perp\) would be equal to \(W\cos(\theta)\). So we have: \(W_\parallel = W\sin(\theta) = mg\sin(\theta)\) \(W_\perp = W\cos(\theta) = mg\cos(\theta)\)

Setting up equilibrium conditions

Since the shoe is in equilibrium, the forces acting on the shoe in the directions parallel and perpendicular to the inclined plane must balance each other out. For the forces along the inclined plane, we have: \(f_s = W_\parallel\) \(\mu N = mg\sin(\theta)\) (1) For the forces perpendicular to the inclined plane, we have: \(N = W_\perp\) \(N = mg\cos(\theta)\) (2)

Solving for \(\mu\)

Now, we will use equations (1) and (2) to solve for \(\mu\). Divide equation (1) by equation (2), we get: \((\mu N) / N = (mg\sin(\theta)) / (mg\cos(\theta))\) After simplifying, we will get: \(\mu = \sin(\theta)/\cos(\theta)\) which can be further simplified as: \(\mu = \tan(\theta)\) So, we found that the coefficient of static friction, \(\mu\), is equal to the tangent of the angle, \(\theta\), at which the shoe just begins to slide on the inclined plane.

Key concepts

Physics

Understanding physics is essential when it comes to explaining real-world phenomena. One classic example is determining the force necessary to keep an object static on an inclined plane. This force is known as the static frictional force. When an object is stationary on a slope, it's the physics behind statics and friction that stops it from sliding down.
To calculate the coefficient of static friction in a specific scenario, such as a shoe on a plank, physics principles guide us through analyzing forces, decomposing them, and setting up equilibrium conditions. This calculation is crucial for various applications, including safety in vehicle design and sports gear, where traction is key. For the student of physics, visualizing forces, understanding their components and how they interact with the surfaces at play, opens the door to practical applications of academic knowledge.

Frictional Force

Frictional force, a concept every student encounters, is the resistance that one surface or object encounters when moving over another. It has two main types: static and kinetic. Static friction acts on objects that are not in motion. It prevents surfaces from sliding past each other and must be overcome by an external force to start motion.
The coefficient of static friction, usually denoted by the Greek letter \(\mu\), is a dimensionless value that represents the ratio of the maximum static frictional force to the normal force exerted by the surface. A higher coefficient means that more force is required to initiate movement. Understanding how this coefficient works not only deepens physics knowledge but also has everyday implications, like choosing the right basketball shoes for better grip on the court.

Statics

Statics is the branch of mechanics that deals with bodies at rest and forces in equilibrium. In statics, we explore how objects are affected by the various forces acting on them and how these forces ensure the objects remain stationary.
In the context of a shoe resting on an inclined plane, the principles of statics allow us to set up equations based on the forces acting perpendicularly and parallel to the surface. Through statics, we can determine the necessary conditions for equilibrium, which in turn leads to the calculation of the coefficient of static friction. The subject lays the groundwork for more complex studies in engineering and physics, emphasizing the importance of balance and distribution of forces in designing stable structures and systems.

Trigonometry

Trigonometry, an essential branch of mathematics, plays a vital role in calculating forces within physics. By using trigonometric ratios such as sine and cosine, we can decompose forces into perpendicular components to better understand their effects on objects.
In the exercise of finding a shoe's coefficient of static friction, trigonometry allows us to relate the angle of the plank to the frictional force required to keep the shoe from sliding. By finding that the coefficient of static friction \(\mu\) is equal to the tangent of the angle \(\theta\), we use trigonometry to bridge the gap between theoretical concepts and practical measurements. This illustrates how trigonometry is not just about solving triangles but is also instrumental in solving real-world problems across various disciplines.