Stress on the Shin Bone. Compressive strength of our bones...

Stress on the Shin Bone. Compressive strength of our bones is important in everyday life. Young's modulus for bone is about $1.4 \times 10^{10}$ Pa. Bone can take only about a 1.0$\%$ change in its length before fracturing. (a) What is the maximum force that can be applied to a bone whose minimum cross-sectional area is 3.0 $\mathrm{cm}^{2} ?$ (This is approximately the cross-sectional area of a tibia, or shin bone, at its narrowest point) (b) Estimate the maximum height from which a $70-k g$ man could jump and not fracture the tibia. Take the time between when he first touches the floor and when he has stopped to be 0.030 $\mathrm{s}$ , and assume that the stress is distributed equally between his legs.

Stress on the Shin Bone. Compressive strength of our bones is important in everyday life. Young's modulus for bone is about $1.4 \times 10^{10}$ Pa. Bone can take only about a 1.0$\%$ change in
its length before fracturing. (a) What is the maximum force that can be applied to a bone whose minimum cross-sectional area is 3.0 $\mathrm{cm}^{2} ?$ (This is approximately the cross-sectional area of a tibia, or shin bone, at its narrowest point) (b) Estimate the maximum height from which a $70-k g$ man could jump and not fracture the tibia. Take the time between when he first touches the floor and
when he has stopped to be 0.030 $\mathrm{s}$ , and assume that the stress is distributed equally between his legs.

Key Concepts

Hooke's Law

Hooke's law relates the stress in an elastic material to the strain it experiences, stating that, within the elastic limit, the strain is directly proportional to the applied stress. This law underpins the calculation of maximum stresses that materials can sustain before they begin to deform permanently or fracture.

Impulse-Momentum Theorem

The impulse-momentum theorem connects the force applied over a time interval with the change in momentum of an object. It is an essential concept in dynamics, particularly when estimating the forces involved during impacts or deceleration, such as when landing from a jump.

Energy Conservation

Energy conservation in mechanics involves the transformation of energy from one form to another, such as from gravitational potential energy to kinetic energy. This principle is used to relate the height from which an object is dropped or jumps to the impact forces experienced upon deceleration, providing insight into material limits during dynamic events.

Young's Modulus

Young's modulus is a material property that defines the relationship between stress and strain in the elastic region of a material's deformation. It quantifies the stiffness of a material, with higher values indicating that the material deforms less under a given load, making it essential for evaluating material performance under stress.

Stress

Stress is the force applied per unit area on an object and it quantifies the internal forces that neighboring particles of a material exert on each other. It is critical for determining whether a material can withstand applied loads without failure and is usually measured in Pascals (Pa).

Strain

Strain is the measure of deformation representing the relative change in length of a material under stress. It is a dimensionless quantity that indicates how much a material stretches or compresses, and it is used alongside stress to assess the material’s elastic behavior.

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