Problem 105 - Algebra and Trigonometry Real Mathematics, Real People

Question

The forces acting on an object weighing \(W\) units on an inclined plane positioned at an angle of \(\theta\) with the horizontal are modeled by $$\mu W \cos \theta=W \sin \theta$$ where \(\mu\) is the coefficient of friction (see figure). Solve the equation for \(\mu\) and simplify the result.

Step-by-Step Solution

Answer

The coefficient of friction \(mu\) in terms of \(theta\) is \(mu = tan theta\).

Step 1: Understand The Given Equation

The given equation is \(mu W cos theta = W sin theta\). This equation contains the product of \(W\) (the object's weight) and \(mu\) (the coefficient of friction), which is balanced against the product of \(W\) and the sin of \(theta\) (the angle of the surface).

Step 2: Isolate The Coefficient Of Friction

The goal is to isolate \(mu\) in the equation. To do this, first divide both sides of the equation by \(W cos theta\). This results in \(mu = sin theta / cos theta\).

Step 3: Simplify The Result

Simplifying the right side of the equation, we know that \(sin theta / cos theta\) is equal to \(tan theta\). Therefore, \(mu = tan theta\).

Key Concepts

Coefficient of Friction

The coefficient of friction, often denoted as \(\mu\), is a number that represents the amount of friction between two surfaces. In physics, it's crucial for understanding how objects move or stay at rest when in contact with one another. Think of it as a score that tells us how 'grippy' a surface is. A higher value means there is more resistance to motion (think of rubber on asphalt), while a lower value indicates less resistance (like ice on metal).

In the context of solving trigonometric equations, particularly for objects on an inclined plane, \(\mu\) enters the equation as the proportionality constant in the force of friction opposing the down-slope component of the weight of the object. Mathematically, the force due to friction is expressed as \(\mu \times N\), where \(N\) represents the normal force—essentially the force pressing the object to the surface, which is equal to \(W \cos \theta\) here. This coefficient is dimensionless and typically falls between 0 and 1 for most common materials.

Inclined Plane Physics

When studying objects on an inclined plane, we deal with the classic physics problem of decomposing forces into components parallel and perpendicular to the surface of the plane. This is essential to understand how the object will behave when placed on a slope, whether it will slide down or remain stationary.

The inclined plane simplifies complex problems into manageable calculations by converting a two-dimensional problem into two, one-dimensional problems. Weight, represented by \(W\), is broken down into a force acting downwards along the plane (\(W \sin \theta\)) and a force acting normal to the plane (\(W \cos \theta\)). The normal force is particularly crucial as it determines the frictional force, which counteracts the motion along the plane. These are fundamental concepts that intertwine with trigonometry to provide a complete understanding of the system's mechanics.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables. They are powerful tools for simplifying and solving equations in both pure mathematics and applied physics contexts, such as those involving inclined planes.

A critical identity used in this problem is the quotient identity, which relates the sine and cosine of an angle: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). This identity simplifies the process of isolating and solving for \(\mu\) in our equation. By recognizing that \(\sin \theta / \cos \theta\) is the definition of the tangent of \(\theta\), we arrive at a neater expression for our problem, which is easy to interpret. Practicing and knowing these identities can greatly expedite solving trigonometric equations, as they help to condense and clarify the steps needed to find a solution.