Problem 64 - University Physics with Modern Physics

Question

BIO 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 \(70-\mathrm{kg}\) 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

Answer

(a) 3.92 J/kg. (b) 5.6%. (c) 2.8%. Children have shorter arms, so they do less work.

Step 1: Understand the Problem

In part (a), we're asked to calculate the work done per kilogram as a man lifts his body 0.40 m during a chin-up. In part (b), we need to determine what percentage of the man's total body mass is muscle, given the information about work per kilogram of muscle. In part (c), we will do the same calculation for a boy whose arms are half the length. Part (d) requires an explanation of why children do chin-ups more easily than adults.

Step 2: Calculate Work Done per Kilogram (Part a)

Using the gravitational potential energy formula: \[ W = mgh \]where \( m \) is the body mass, \( g = 9.8 \ \mathrm{m/s^2} \), and \( h = 0.40 \ \mathrm{m} \). Per kilogram of body mass, the work \( W \) is calculated as:\[ W = 1 \cdot 9.8 \cdot 0.40 = 3.92 \ \mathrm{J} \].

Step 3: Determine Muscle Mass Percentage (Part b)

We know each kilogram of muscle can generate 70 J of work. If the man lifts his entire body, which requires 3.92 J per kg of body mass, the percentage of his body's mass that is muscle is found by:\[ \text{Muscle Mass Percentage} = \frac{3.92}{70} \times 100\% \approx 5.6\% \].

Step 4: Solve for the Boy's Muscle Mass Percentage (Part c)

The boy's arms are half as long, meaning he only lifts his body half the adult height: \( h = 0.20 \ \mathrm{m} \). The work done per kilogram is now:\[ W = 1 \cdot 9.8 \cdot 0.20 = 1.96 \ \mathrm{J} \].His muscle mass percentage is:\[ \text{Muscle Mass Percentage} = \frac{1.96}{70} \times 100\% \approx 2.8\% \].

Step 5: Explain Chin-Up Ease in Children (Part d)

Children often perform chin-ups more easily than adults because they typically have proportionally shorter arms, which minimizes the distance they need to lift their bodies. This reduced distance requires less work.

Key Concepts

Gravitational Potential Energy

Gravitational potential energy is the energy an object possesses due to its position in a gravitational field. The core aim is to determine the energy needed to lift an object against gravity, which is vital in understanding movements like chin-ups.
When you lift your body during a chin-up, you are working against gravitational forces. The work done is calculated using the formula:\[ W = mgh \]This formula states that work, \( W \), is the product of mass \( m \), gravitational force \( g \), often approximated as \( 9.8 \ \text{m/s}^2 \), and height \( h \) the object is lifted. In the context of a chin-up, it allows us to quantify the energy expended to lift one's body a certain height.
The beauty of gravitational potential energy lies in its simplicity—it helps us understand the energy transformations happening when performing a physical activity like a chin-up.

Muscle Mass Percentage

Muscle mass percentage is a measure of how much muscle makes up one's body weight. It’s a precise way to assess physical strength and the ability to perform tasks requiring muscle exertion, such as chin-ups. Understanding the percentage of your body that is muscle gives insight into how efficient your body might be at performing work.
For someone doing a chin-up, muscles work to generate energy. If a muscle can create about 70 J of energy per kilogram, knowing your muscle mass percentage can predict how easily and efficiently you can perform a chin-up. If a person has a higher muscle mass percentage, it means more muscle is available to perform work, potentially making tasks like chin-ups easier.
By calculating this percentage, we obtain valuable information about one's strength capacities.

Ergonomics in Physics

Ergonomics is the science of designing tasks and environments for comfort, efficiency, and productivity. In physics, it plays a role in optimizing how people interact with their environments during physical tasks.
Consider ergonomics when performing chin-ups. Chin-ups involve the lifting of one's body, where the body’s physical dimensions, like arm and torso length, affect the energy needed for the action. For example, children with shorter arms can find chin-ups easier because they have less distance to lift, thus minimizing the work done against gravity.
Understanding ergonomics in physics helps us design exercise routines that maximize benefits while reducing strain and energy expenditure, improving overall physical performance.