Question

The maximum energy that a bone can absorb without breaking depends on characteristics such as its cross-sectional area and elasticity. For healthy human leg bones of approximately 6.0 cm\(^2\) cross-sectional area, this energy has been experimentally measured to be about 200 J. (a) From approximately what maximum height could a 60-kg person jump and land rigidly upright on both feet without breaking his legs? (b) You are probably surprised at how small the answer to part (a) is. People obviously jump from much greater heights without breaking their legs. How can that be? What else absorbs the energy when they jump from greater heights? (\(Hint\): How did the person in part (a) land? How do people normally land when they jump from greater heights?) (c) Why might older people be much more prone than younger ones to bone fractures from simple falls (such as a fall in the shower)?

Short answer

(a) 0.34 meters; (b) Muscles and other tissues absorb more energy; (c) Older bones are less dense and elastic.

Step-by-step solution

Calculate Potential Energy

When a person jumps from a height, the gravitational potential energy at the height is given by \( PE = mgh \), where \( m \) is mass, \( g \) is the acceleration due to gravity, and \( h \) is the height. For a 60-kg person, \( PE = 60 \times 9.8 \times h \). We want this to equal 200 J, the energy absorbed by the bone: \( 588h = 200 \).

Solve for Height

Solve the equation \( 588h = 200 \) for \( h \). Divide both sides by 588 to find \( h = \frac{200}{588} \approx 0.34 \text{ meters} \). Thus, the maximum height is approximately 0.34 meters.

Discuss Energy Distribution

In part (b), consider the distribution of energy: People land by bending their knees, which allows muscles and other tissues to absorb some of the energy, reducing the impact on bones. This is how people can jump from higher heights without breaking bones.

Examine Risk for Older Adults

In part (c), older people are more prone to fractures due to decreased bone density and elasticity, which makes their bones more brittle and less able to absorb energy from impacts, such as falls.

Key concepts

Cross-Sectional Area of Bones

The cross-sectional area of a bone is essentially a measurement of its thickness or width. In the context of human leg bones, a typical cross-sectional area is about 6.0 cm². This area is crucial because it directly influences a bone's ability to withstand force without breaking.

A larger cross-sectional area generally indicates that the bone can absorb more energy without fracturing, as the force from impacts is spread over more material.

• Think of it like a sturdy pillar. The wider the pillar, the more it can support because the load is distributed over a larger surface.
• In bones, the cross-sectional area relates to both strength and stability.

This factor is why athletes often have stronger, thicker bones in specific areas relevant to their sports – an adaptation to continual impact forces.

Elasticity of Bones

Elasticity in bones refers to their ability to flex and return to their original shape after being subjected to force. It is a critical property that allows bones to absorb energy through flexing instead of breaking upon impact.

Unlike rigid materials that crack when stressed, elastic materials like healthy bone can bend slightly. This prevents damage by "giving way" a bit before returning to their original form. This elasticity helps in:

• Distributing impact forces smoothly.
• Preventing immediate fractures by flexing under stress.

Over time, particularly with age or due to certain medical conditions, bones can become less elastic. This reduced elasticity makes them more susceptible to fracturing under stress.

Bone Fractures in Older Adults

As people age, their bones naturally lose density and elasticity, making them more prone to fractures. This process, known as osteoporosis, results in bones becoming more brittle. A simple fall that might not harm a younger person can easily cause a fracture in an older adult.

This increased fragility is due to:

• A decrease in bone density, making them less robust.
• A reduction in elasticity, reducing their ability to absorb impacts.
• Changes in physiology that can affect balance and coordination, increasing fall risk.

Additionally, injuries can take longer to heal in older individuals, complicating recovery from such fractures. This emphasizes the importance of preventive measures like exercise and a proper diet rich in calcium and vitamin D.

Gravitational Potential Energy Calculation

Gravitational potential energy (GPE) is the energy an object possesses due to its position within a gravitational field. For example, when a person jumps from a height, their potential energy is calculated using the equation: \(PE = mgh\)where:

• \( m \) is the mass of the person in kilograms.
• \( g \) is the acceleration due to gravity, roughly 9.8 m/s² on Earth.
• \( h \) is the height in meters from which the person jumps.

In the exercise, calculating the maximum jump height that bones can withstand involves finding the height where the potential energy equals 200 J (the energy a bone can absorb). This understanding helps in designing safer physical activities and understanding the physical limits of the human body.