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Chin-Ups. While doing a chin-up, a man lifts his body 0.40 \(\mathrm{m}\) . (a) How much work must the man do per kilogram of body mass? (b) The muscles involved in doing a chin-up can generate about 70 \(\mathrm{J}\) of work per kilogram of muscle mass. If the man can just barely do a \(0.40-\mathrm{m}\) chin-up, what percentage of his body's mass do these muscles constitute? (For comparison, the total percentage of muscle in a typical \(70-k g\) man with 14\(\%\) body fat is about 43\(\%\) . (c) Repeat part (b) for the man's young son, who has arms half as long as his father's but whose muscles can also generate 70 \(\mathrm{J}\) of work per kilogram of muscle mass. (d) Adults and children have about the same percentage of muscle in their bodies. Explain why children can commonly do chin-ups more easily than their fathers.

Step by step solution

01

Understanding the Problem

We need to calculate the work done by the man in lifting his body 0.40 meters during a chin-up, and relate it to the muscle capability to assess what percentage of body mass is constituted by muscle.

02

Calculating Work Done Per Kilogram

For part (a), the work done per kilogram of the man's body is given by the equation \( W = mgh \), where \( h = 0.40 \, \text{m} \), \( g = 9.8 \, \text{m/s}^2 \) is the acceleration due to gravity, and \( m = 1 \, \text{kg} \). Thus, \( W = 1 \, \text{kg} \times 9.8 \, \text{m/s}^2 \times 0.40 \, \text{m} = 3.92 \, \text{J/kg} \).

03

Calculating Muscle Mass Percentage

For part (b), if the man just barely manages to lift himself, then his muscles are providing exactly the amount of work needed. Each kilogram of muscle mass can generate 70 J. Thus, the percentage of muscle mass is calculated by \( \frac{3.92 \, \text{J/kg}}{70 \, \text{J/kg}} \times 100\% \approx 5.6\% \).

04

Repeating for the Son

For part (c), since the son's arms are half as long, the height is \( h = 0.20 \, \text{m} \). Calculating work per kilogram: \( 9.8 \times 0.20 = 1.96 \, \text{J/kg} \). Muscle mass percentage: \( \frac{1.96 \, \text{J/kg}}{70 \, \text{J/kg}} \times 100\% \approx 2.8\% \).

05

Explaining the Child's Advantage

For part (d), though muscles make up a similar body percentage for adults and children, the child's smaller size and shorter arms result in less work against gravity per chin-up, making the task easier relative to their body scale.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Work and Energy

When we talk about work and energy in physics, we're discussing how force causes movement. Work is done when a force causes an object to move over a distance. The formula for work is expressed as:

• Work, \( W = F \times d \times \cos(\theta) \)

where:

• \( F \) is the force applied,
• \( d \) is the displacement, and
• \( \theta \) is the angle between the force and the direction of motion.

In the example of a chin-up, the entire weight of the body is lifted a certain distance, which is 0.40 meters in the problem. The force exerted by the muscles equals the gravitational force to lift the body upward. The energy expended translates directly to the mechanical work done, demonstrating energy conversion from muscular energy to gravitational potential energy.

Muscle Mass Percentage

Understanding muscle mass percentage is crucial in evaluating a person's physical capabilities. It compares the muscle mass against the total body mass. In this problem, knowing the percentage helps us figure out how much of the man's body can actually generate the needed work to lift his body during a chin-up.

• Muscle mass percentage helps determine efficiency in energy use during physical activities.
• The problem simplifies finding which part of mass contributes most to energy production.

By assessing how much work the man can do per kilogram of body weight, in conjunction with potential energy generated per kilogram of muscle, we can derive that roughly 5.6% of his body is muscle responsible for the chin-up.

Biomechanics

Biomechanics is the study of structure, function, and motion of the body’s mechanical systems. It provides insights into how humans move and interact with their environment. In exercises like chin-ups, biomechanics helps understand how the body's levers (bones) and engines (muscles) function.

• Analyzing body movement helps optimize energy use and reduces injury risk.
• Understanding the role of muscle groups can enhance performance in exercises such as pull-ups and chin-ups.

Through biomechanics, we see how muscles work together to distribute force and motion efficiently. Additionally, it can elucidate why children often have an easier time performing chin-ups due to their different body mechanics, such as shorter arms needing less work to lift against gravity.

Gravitational Force

Gravitational force is a force of attraction between two masses. On Earth, it gives weight to physical objects and causes them to fall to the ground when dropped. It is calculated by the formula:

• Force of gravity, \( F = mg \)

where:

• \( m \) is the mass of the object, and
• \( g \approx 9.8 \, \text{m/s}^2 \) is the acceleration due to Earth's gravity.

In the chin-up problem, gravitational force must be overcome to lift the body upward. This requires the muscles to exert an equal and opposite force to counteract the gravitational pull. Less gravitational force is needed by children because they typically have less mass and shorter limbs, decreasing the distance over which the force must act.