91Ó°ÊÓ

In physics, equilibrium means that all forces acting on an object are balanced, resulting in no movement. For the washcloth, our goal is for it to stay in place and not slide off the table. This means the forces pulling the washcloth towards the floor, primarily its overhanging part's weight, need to be countered by the frictional force on the table. These forces must reach a balance, which we refer to as the equilibrium condition.

• The gravitational force pulling the overhanging part downwards is calculated as the mass times gravity, i.e., \( m_{\text{oft}} \cdot g \).
• The frictional force that prevents it from sliding is determined by \( \mu_s \cdot m_{\text{ca}} \cdot g \), where \( \mu_s \) is the static friction coefficient.

To maintain equilibrium, the frictional force must equal or surpass the gravitational force of the overhang, leading us to our main inequality \( \mu_s \cdot m_{\text{ca}} \geq m_{\text{oft}} \). This defines the maximum portion that can hang without slipping.

Gravitational force is the pull that draws objects towards the center of the Earth. This force is a crucial factor in determining whether the washcloth can stay on the table or will fall completely off. Gravitational force can be calculated using the formula \( F_g = m \cdot g \), where \( m \) is mass and \( g \) is the acceleration due to gravity, approximately \( 9.8 \text{ m/s}^2 \) on Earth.
For our washcloth problem, two masses are involved:

• \( m_{\text{ca}} \), the portion's mass on the table.
• \( m_{\text{oft}} \), the portion's mass overhanging the edge.

The gravitational force acting on the overhanging part attempts to pull it down, threatening equilibrium. Hence, understanding and calculating gravitational force is essential to analyze how the cloth behaves under its influence.

Mass distribution in an object influences how it interacts with forces such as gravity and friction. With our washcloth, part is resting on the table while another hangs over, each part contributing differently to the system. It’s key to see how the mass is 'distributed' or spread out, as it affects equilibrium and the forces at play.

In our equation, we separate two different parts:

• Mass on the table, \( m_{\text{ca}} \).
• Mass over the edge, \( m_{\text{oft}} \).

This distinction is significant since a wide mass distribution can change the stability of our setup. If most of the mass is overhanging, friction may not suffice to prevent sliding. Therefore, understanding mass distribution is vital in calculating the maximum safe overhang.

The friction coefficient, denoted as \( \mu_s \), quantifies how much friction acts between two surfaces. It varies between different material pairs. In our problem, it is given to be 0.40, which indicates a moderate level of friction.
The frictional force which stops the washcloth from sliding is calculated as \( \mu_s \cdot m_{\text{ca}} \cdot g \). This shows the role of \( \mu_s \) in providing the necessary force to resist the downward pull of gravity on the overhanging portion.

• A higher coefficient implies stronger grip, supporting more mass without slipping.
• A lower coefficient suggests weaker friction, allowing less mass to hang without sliding.

This coefficient allows us to define an inequality showing that the part’s weight should not exceed the frictional resistance — crucial for maintaining equilibrium and stability.