Problem 84 - University Physics with Modern Physics

Question

If the coefficient of static friction between a table and a uniform massive rope is \(\mu_{s},\) what fraction of the rope can hang over the edge of the table without the rope sliding?

Step-by-Step Solution

Answer

The fraction of the rope that can hang over is \( \frac{\mu_s}{1 + \mu_s} \).

Step 1: Understand the Problem

We need to find the maximum length of the rope that can hang over the edge of a table without causing it to slide, given the static friction coefficient, \( \mu_s \).

Step 2: Set Up the Forces

When the rope is on the verge of sliding, the static friction force \( f_s \) at the edge of the table equals the weight of the hanging part. If \( L \) is the total length of the rope and \( x \) is the length that hangs, the weight of the hanging part is \( x \cdot \lambda \cdot g \), where \( \lambda \) is the linear mass density and \( g \) is the acceleration due to gravity.

Step 3: Calculate Static Frictional Force

The maximum static friction force is \( f_s = \mu_s \cdot \lambda \cdot (L-x) \cdot g \). This acts on the part of the rope still on the table, \( (L-x) \cdot \lambda \).

Step 4: Set Forces Equal

Set the weight of the hanging part equal to the maximum static friction force: \( \mu_s \cdot \lambda \cdot (L-x) \cdot g = x \cdot \lambda \cdot g \).

Step 5: Simplify the Equation

Cancel \( \lambda \cdot g \) from both sides to get \( \mu_s (L-x) = x \).

Step 6: Solve for x

Solve \( \mu_s (L-x) = x \) for \( x \):1. \( \mu_s L - \mu_s x = x \)2. \( \mu_s L = x + \mu_s x \)3. \( x (1 + \mu_s) = \mu_s L \)4. \( x = \frac{\mu_s}{1 + \mu_s} L \).

Step 7: Find the Fraction of the Rope

The fraction of the rope that can hang over is \( \frac{x}{L} = \frac{\mu_s}{1 + \mu_s} \).

Key Concepts

Coefficient of Static Friction

The coefficient of static friction, denoted as \( \mu_s \), is a measure of how much force can be applied to an object before it starts sliding over a surface. This value is usually less than 1 and reflects the interaction between the two surfaces in contact. The higher the coefficient, the greater the resistance to sliding, which makes it fantastic for preventing unwanted movement. In the context of our rope problem, \( \mu_s \) determines how much weight or length of the rope can hang over the edge of a table before the force of gravity overcomes the frictional grip holding the rope in place.

Uniform Rope Problem

A common physics problem, the Uniform Rope Problem, deals with a rope that has a consistent mass distribution. This uniformity means each section of the rope weighs the same, making calculations simpler. In this scenario, we are asked to determine the length of the rope that can hang off a table without falling due to gravity. The total length of the rope is \( L \), and a length \( x \) is hanging off the table. The solution involves balancing the forces to find out how much weight the friction can hold, considering the rope's uniformity.

Physics Problem Solving

Solving physics problems effectively requires a clear understanding and application of various concepts and mathematical skills. Here's a basic approach you can use:

• Analyze the Problem: Understand what is being asked. Here, it’s about finding the length of rope hanging over the edge.


• Set Up the Forces: Figure out the forces at play; in our problem, static friction and the gravitational pull on the rope are key.


• Use Relevant Equations: Apply physics equations appropriately, like equating frictional force to gravitational force in our problem.


• Simplify and Solve: Simplify the equations and solve for the unknowns, such as the length \( x \) of the rope hanging over.


• Interpret the Results: Consider whether your solution makes sense in practical terms.

Frictional Forces

Frictional forces are essential in preventing sliding between two surfaces. Static friction happens when there is no relative movement. The friction attempts to resist motion until a certain threshold, defined by \( \mu_s \), is reached. It is crucial to understand that static friction changes depending on the angle or length of the surface in question. In the rope problem, friction acts up to the edge of the table to hold the rope, but can no longer resist once the weight of the hanging rope surpasses the maximum static frictional force. This force can be modeled as: \( f_s = \mu_s \cdot \lambda \cdot (L-x) \cdot g \).

Length of Rope on Table

The length of rope remaining on the table provides the balance required to prevent the hanging part from pulling the whole rope off. When the rope reaches the maximum extension possible without slipping, we balance the static friction force and the force exerted by gravity on the hanging rope. The length \( (L-x) \) remains on the table, contributing to the static frictional force that counteracts the gravitational pull on \( x \). To find the fraction of the hanging rope, the calculation reduces to \( \frac{x}{L} = \frac{\mu_s}{1 + \mu_s} \), showing how much of the rope can hang without sliding.