A child slides across a floor in a pair of rubber-soled...

A child slides across a floor in a pair of rubber-soled shoes. The friction force acting on each foot is $20 \mathrm{~N}$, the footprint area of each foot is $14 \mathrm{~cm}^{2}$, and the thickness of the soles is $5.0 \mathrm{~mm}$. Find the horizontal distance traveled by the sheared face of the sole. The shear modulus of the rubber is $3.0 \times 10^{6} \mathrm{~Pa}$.

Key Concepts

Shear Modulus

The shear modulus is a material property that quantifies its rigidity when subjected to a shear stress. It is defined as the ratio of shear stress to shear strain and indicates how much a material will deform under a given shear force. This constant is crucial for predicting material deformations in applications involving sliding or torsional forces.

Shear Stress

Shear stress is the force per unit area applied parallel to the surface of a material. When a force is applied in this manner, it causes layers within the material to slide relative to each other. Calculating shear stress is essential for determining the deformation behavior of materials under lateral forces.

Shear Strain

Shear strain is a dimensionless measure of deformation that describes the angular or lateral displacement between layers in a material under shear stress. It is defined as the ratio of the horizontal displacement to the original thickness of the material, providing insight into how a material's shape changes when a shear force is applied.

Hooke's Law for Shear

Hooke's Law for shear describes the linear relationship between shear stress and shear strain in elastic materials. This principle states that, within the elastic limit, the shear strain is directly proportional to the applied shear stress, with the shear modulus acting as the constant of proportionality. This law is used to predict how materials deform under shear forces.

Friction Force and Contact Area

Friction force arises from the interaction between two surfaces in contact and opposes relative motion. When this force is applied over a given contact area, it creates a shear stress on the material. Understanding the distribution of friction force over the contact area is essential for analyzing the resulting deformation, as the magnitude of the shear stress depends directly on both the applied friction force and the size of the contact area.

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