Problem 73 - 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 . A person wearing these shoes stands on a smooth horizontal rock. She pushes against the ground to begin running. What is the maximum horizontal acceleration she can have without slipping? A. \(0.20 g\) B. \(0.75 g\) C. \(0.90 g\) D. \(1.2 g\)

Step-by-Step Solution

Answer

The correct answer is D: \(1.2g\).

Step 1: Understand the Question

To find the maximum horizontal acceleration of the person without slipping, we need to consider the coefficient of static friction. The question involves the interaction between the friction force and the acceleration caused by this force.

Step 2: Identify the Given Values

The coefficient of static friction \( \mu_s \) is given as 1.2. The acceleration due to gravity \( g \) is typically approximated as \( 9.8 \text{ m/s}^2 \).

Step 3: Calculate Maximum Frictional Force

The maximum static frictional force \( f_{\text{max}} \) is given by \( f_{\text{max}} = \mu_s \cdot N \), where \( N \) is the normal force, equal to the gravitational force \( mg \). Therefore, \( f_{\text{max}} = \mu_s \cdot mg = 1.2 \cdot mg \).

Step 4: Relate Frictional Force to Maximum Acceleration

Using Newton's second law, \( f = ma \), where \( f \) is the maximum frictional force and \( a \) is the acceleration, we have: \( 1.2 \cdot mg = ma \). Solving for \( a \), we get \( a = 1.2g \).

Step 5: Select the Correct Answer Based on Calculation

From the calculation in step 4, the maximum horizontal acceleration without slipping is \( 1.2g \). This matches with option D.

Key Concepts

kinetic friction

Kinetic friction, often considered lesser-known than its static counterpart, plays a vital role after an object has already started moving. It refers to the resistive force opposing the motion of two surfaces sliding past each other. In our context, once the climber begins moving, her shoes interact with the rock through kinetic friction. Unlike static friction which prevents slipping in the first place, kinetic friction comes into play during motion and is usually lower than static friction. For instance, the coefficient of kinetic friction for the bouldering shoes on smooth rock is 0.90, indicating less resistance when in motion as compared to an idle state.

acceleration

To understand acceleration in the context of climbing shoes on smooth rock, we need to look at how it affects movement. Acceleration is the rate of change of velocity of an object. When the climber pushes against the rock to start running, she accelerates, needing a strong initial push without slipping. In this scenario, the climber’s acceleration is bound by friction, as it determines how fast she can speed up without losing grip. By maximizing the frictional force, she can increase her acceleration to the threshold of slipping, measured in terms of gravitational acceleration, "g."

Newton's second law

Newton's second law is fundamental in calculating the maximum acceleration the climber can achieve without slipping. This law states that the force acting on an object is equal to the mass of the object multiplied by its acceleration, represented by the equation \( f = ma \). In our example, the force is the static frictional force, which must counterbalance the force required for acceleration. The equation becomes \( 1.2mg = ma \), where \( 1.2 \) is the coefficient of static friction. By solving for acceleration, we find the maximum possible rate of acceleration without slipping is \( 1.2g \), enforcing Newton's rule in a tangible way.

coefficient of friction

The coefficient of friction is a crucial parameter that influences the interaction between surfaces. It's a dimensionless number representing how easily one object will slide over another. There are two types of coefficients to consider: static and kinetic. The static coefficient, usually higher, determines the initiation of movement, acting as the threshold to overcome to begin slipping. For the shoes on smooth rock, this is 1.2, marking a high resistance to initial slip. Meanwhile, the kinetic coefficient is 0.90, reflecting diminished resistance during motion. Recognizing these values helps to understand the limits of adhesion and movement, crucial for safely maximizing acceleration.