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 20.0 $\mathrm{N}$ . The foot-print area of each shoe sole is $14.0 \mathrm{cm}^{2},$ and the thickness of cach sole is 5.00 $\mathrm{mm}$ . Find the horizontal distance by which the upper and lower surfaces of each sole arfset. The shear modulus of the rubber is 3.00 $\mathrm{MN} / \mathrm{m}^{2}$ .

Key Concepts

Force Distribution and Frictional Force

The frictional force provides the lateral force that acts over the contact area of the material. Understanding how this force is distributed over the area is key, as it results in a certain shear stress on the material. When this stress is known, it can be related to the material’s properties (such as its shear modulus) to compute the resulting shear strain, and subsequently, the displacement or offset in the material.

Shear Modulus

Shear modulus, also known as the modulus of rigidity, quantifies a material’s resistance to shear deformation. It is defined as the ratio of shear stress to shear strain and is crucial in predicting how materials deform when subjected to lateral forces. In problems like these, the shear modulus helps determine how much a material, such as rubber, will shear (or slide layer over layer) under a given force.

Shear Stress and Shear Strain

Shear stress is the force per unit area acting parallel to a surface, while shear strain is the relative displacement between layers divided by the distance over which that displacement occurs. The relationship between these two, given by the equation shear stress = shear modulus × shear strain, is essential in determining the deformation experienced by materials under applied forces.

Dimensional Analysis in Material Deformation

In problems involving material deformation, the dimensions of the material (like thickness) play a critical role in relating shear strain to actual displacement. The thickness of an object is used to convert the dimensionless shear strain (ratio of displacement to thickness) into a measurable offset. This principle is widely applicable in solid mechanics and helps bridge the gap between theoretical modulus and observable deformation.

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