Q64P - University Physics with Modern Physics

Question

While doing a chin-up, a man lifts his body 0.40 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 J of work per kilogram of muscle mass. If the man can just barely do a 0.40-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-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 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) The work done by the man per kilogram of body mass is 3.92J/kg.

(b) The percentage of his body’s mass do these muscles constitute is 5.6%.

(c) The percentage of the child's body’s mass do muscles constitute is  2.8%.

(d) children can commonly do chin-ups more easily than their fathers because maximal muscle force is lower in children than in fathers.

Step 1: Given data

The man lifts the body to a height of ${\mathrm{h}}_{\mathrm{m}}=0.40\hspace{0.33em}\mathrm{m}$  

The child lifts the body to a height of ${\mathrm{h}}_{\mathrm{c}}=0.20\hspace{0.33em}\mathrm{m}$  

Step 2: (a) Find the work done by the man per kilogram of body mass

Work done can be given as

 

$\mathrm{W}={\mathrm{m}}_{\mathrm{m}}{\mathrm{gh}}_{\mathrm{m}}$ ………………(1)

 

Here, $m_m$ is the mass of the man’s body, g is gravity, and ${\mathrm{h}}_{\mathrm{m}}$ is height lifted by man.

$$\begin{gathered} \mathrm{W}={\mathrm{m}}_{\mathrm{m}}\mathrm{gh} \\ \frac{\mathrm{W}}{{\mathrm{m}}_{\mathrm{m}}}=(9.8\hspace{0.33em}\mathrm{m}\cdot {\mathrm{s}}^{-2})(0.40\hspace{0.33em}\mathrm{m}) \\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}=3.92\hspace{0.33em}\frac{\mathrm{J}}{\mathrm{kg}} \end{gathered}$$ 

Hence, the work done by the man per kilogram of body mass is  3.92 J/kg.     

Step 3: (b) Find the percentage of his body’s mass do these muscles constitute

Work that the muscle can generate per kilogram of muscle mass is,

 

$\frac{W}{m_m}=70J/kg$ 

 

The percentage of body mass constituting the muscles is,

 

$$\begin{gathered} {\%}_{\mathrm{muscles}}=\frac{\left(3.92\mathrm{J}/\mathrm{kg}\right)}{70 \mathrm{J}/\mathrm{kg}}\times 100\% \\ =5.6\% \end{gathered}$$ 

 

Hence, the percentage of his body’s mass do these muscles constitute is 5.6% .

Step 4: (c) Find the percentage of the child's body mass do these muscles constitute

Work done can be given as,

 

 $\mathrm{W}\text{'}={\mathrm{m}}_{\mathrm{c}}{\mathrm{gh}}_{\mathrm{c}}...........................\left(2\right)$………………(2)

 

Here, ${\mathrm{m}}_{\mathrm{c}}$ is the mass of the child’s body,  is gravity, and ${\mathrm{h}}_{\mathrm{c}}$ is the height lifted by the child.

 

Substituting the given data in equation (2), we get,

 

$$\begin{gathered} \mathrm{W}\text{'}={\mathrm{m}}_{\mathrm{c}}{\mathrm{gh}}_{\mathrm{c}} \\ \frac{\mathrm{W}\text{'}}{{\mathrm{m}}_{\mathrm{c}}}=\left(9.8 \mathrm{m}.{\mathrm{s}}^{-2}\right)\left(0.20 \mathrm{m}\right) \\ =1.96 \mathrm{J}/\mathrm{kg} \end{gathered}$$ 

 

The percentage of body mass constituting the muscles is,

$$\begin{gathered} {\%}_{mucless}=\left(\frac{1.96{\displaystyle \frac{j}{kg}}}{70{\displaystyle \frac{j}{kg}}}\right)\times 100\% \\ =2.8\% \end{gathered}$$ 

 

 

Hence, the percentage of the child's body mass do muscles constitute is 2.8% .

Step 5: (d) Find the reason for the child’s easiness with chin-ups

If the man and child can do work at 70J/kg (the same rate) and if the child uses only 1.9J/kg, while the man uses 3.92 J/kg, the child can do more repetitions.

 

Maximal muscles force is lower in children than in fathers, even when 

size-normalized, for example, to body mass or to muscle cross-sectional area.

 

Children’s rate of force development also appears to be lower than that of adults.