Problem 71 - College Physics

Question

Shoes for the sports of bouldering and rock climbing are designed to provide a great deal of friction between the foot and the surface of the ground. On smooth rock these shoes might have a coefficient of static friction of 1.2 and a coefficient of kinetic friction of 0.90 . For a person wearing these shoes, what's the maximum angle (with respect to the horizontal) of a smooth rock that can be walked on without slipping? A. \(42^{\circ}\) B. \(50^{\circ}\) C. \(64^{\circ}\) D. Greater than \(90^{\circ}\)

Step-by-Step Solution

Answer

B. \(50^{\circ}\)

Step 1: Understand the Problem

We need to find the maximum angle at which a person can stand on a smooth rock using the coefficient of static friction. This maximum angle is known as the angle of repose.

Step 2: Identify the Relevant Formula

The angle of repose \( \theta \) can be calculated using the formula: \[ \tan(\theta) = \mu_s \]where \( \mu_s \) is the coefficient of static friction.

Step 3: Apply the Formula

Given that the coefficient of static friction \( \mu_s \) is 1.2, we use:\[ \tan(\theta) = 1.2 \]

Step 4: Solve for the Angle

We find \( \theta \) by taking the inverse tangent (arctan) of 1.2:\[ \theta = \tan^{-1}(1.2) \]Using a calculator, this evaluates to approximately \( \theta = 50^{\circ} \).

Step 5: Evaluate the Options

The calculated angle \( 50^{\circ} \) matches the option B.

Key Concepts

Angle of Repose

In physics, the angle of repose is the steepest angle at which a sloping surface can hold material without the material sliding down. This concept is extremely important when discussing problems related to static friction. It helps us understand how objects remain stationary on inclined surfaces.
For example, when someone is standing on a slope, the angle of repose determines whether they will slide down or remain standing. The angle depends heavily on the friction between their shoes and the ground. An interesting fact about the angle of repose is that it acts as a direct measure of the static frictional forces at play. If the angle of repose for a surface is equal to a given angle, then the surface can maintain an object at rest at that angle without slippage. Key Points to Remember about the Angle of Repose:

• An increased angle of repose indicates higher static friction.
• It is determined using the formula: \( \tan(\theta) = \mu_s \), where \( \theta \) is the angle of repose and \( \mu_s \) is the coefficient of static friction.
• It directly determines the maximum tilt of a surface without causing an object to slide.

Coefficient of Static Friction

The coefficient of static friction is a crucial part of understanding how objects interact with surfaces without moving. This dimensionless number represents the ratio of the maximum static frictional force between surfaces before motion begins to the normal force acting upon the surfaces.
An important detail about this coefficient is that it varies between different pairs of materials. For instance, rubber-soled shoes on a rock would yield a different coefficient than steel on steel.Crucial Aspects of Coefficient of Static Friction:

• The symbol for the coefficient of static friction is \( \mu_s \).
• A higher \( \mu_s \) indicates more friction, meaning it is harder for objects to start moving.
• In the given exercise, a \( \mu_s \) of 1.2 means a relatively high gripping ability, allowing a person to stand on a rocky slope without slipping easily.

Physics Problem Solving

Solving physics problems doesn't always have to be complicated. By breaking down problems into manageable steps, you can make even the most complex physics challenges more approachable. Let's look at the general process using the previous example of static friction on a slope.
Basic Steps for Effective Physics Problem Solving:

• Step 1: Understand the given problem. Identify what is known and what needs to be found. In this example, it was finding the angle of repose.
• Step 2: Apply the relevant laws and formulas. Here, using \( \tan(\theta) = \mu_s \) to find the angle of repose was key.
• Step 3: Solve using mathematical operations, like inverse functions, to attain solutions to variables, such as using \( \tan^{-1}(1.2) \) to find \( \theta \).
• Step 4: Confirm the solution aligns with the problem options presented, ensuring logical reasoning throughout the entire process.

Breaking problems into these steps not only simplifies solving them but also deepens understanding of the underlying physics.