A gymnast does a one-arm handstand. The humerus, which is...

A gymnast does a one-arm handstand. The humerus, which is the upper arm bone (between the elbow and the shoulder joint), may be approximated as a 0.30 -m-long cylinder with an outer radius of $1.00 \times 10^{-2} \mathrm{m}$ and a hollow inner core with a radius of $4.0 \times 10^{-3} \mathrm{m} .$ Excluding the arm, the mass of the gymnast is 63 $\mathrm{kg} . \quad$ (a) What is the compressional strain of the humerus? (b) By how much is the humerus compressed?

A gymnast does a one-arm handstand. The humerus, which is the upper arm bone (between the elbow and the shoulder joint), may be approximated as a 0.30 -m-long cylinder with an outer radius of
$1.00 \times 10^{-2} \mathrm{m}$ and a hollow inner core with a radius of $4.0 \times 10^{-3} \mathrm{m} .$ Excluding the arm, the mass of the gymnast is 63 $\mathrm{kg} . \quad$ (a) What is the
compressional strain of the humerus? (b) By how much is the humerus
compressed?

Key Concepts

Hollow Cylinder Geometry

The geometry of a hollow cylinder is important for calculating properties such as the cross-sectional area, which directly affects the stress experienced under a load. The cross-sectional area in a hollow cylinder is determined by subtracting the area of the inner circle from the area of the outer circle. This calculation is necessary for determining the effective area that carries the load, which in turn is used to compute the compressive stress when the object is under axial compression.

Strain

Strain is a measure of deformation representing the ratio of the change in length to the original length of an object under load. In the context of compressional strain, it quantifies how much a material shortens under a compressive force. Strain is a dimensionless quantity and is central to the analysis of material deformation and the evaluation of a structure’s response when subject to various forces.

Compressive Stress

Compressive stress is the internal force per unit area that develops within a material when it is subjected to an axial load (a force applied along its length). It is calculated by dividing the total applied force by the cross?sectional area over which the force is distributed. This concept is fundamental in understanding how materials like bones or structural elements respond when they are compressed, and it is crucial in determining whether a material can bear a given load without failing.

Young's Modulus

Young’s modulus, also known as the modulus of elasticity, is a material property that relates stress and strain in the elastic region of a material's behavior. It is defined as the ratio of stress to strain and provides an indication of a material's stiffness. A higher Young’s modulus means that the material will deform less under a given load, and this concept is essential when calculating how much a structure such as a bone or metal rod compresses under a specific compressive stress.

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