Question 11.1: Working Off Breakfast BIO GOAL Relate caloric energy to mech...
Working Off Breakfast BIO
GOAL Relate caloric energy to mechanical energy.
PROBLEM A student eats a breakfast consisting of a bowl of cereal and milk, containing a total of 3.20 × 10² Calories of energy. He wishes to do an equivalent amount of work in the gymnasium by performing curls with a 25.0-kg barbell (Fig. 11.1). How many times must he raise the weight to expend that much energy? Assume he raises it through a vertical displacement of 0.400 m each time, the distance from his lap to his upper chest.
STRATEGY Convert the energy in Calories to joules, then equate that energy to the work necessary to do n repetitions of the barbell exercise. The work he does lifting the barbell can be found from the work-energy theorem and the change in potential energy of the barbell. He does negative work on the barbell going down, to keep it from speeding up. The net work on the barbell during one repetition is zero, but his muscles expend the same energy both in raising and lowering.
Convert his breakfast Calories, E, to joules:
E=(3.20\times10^{2}\cancel{Cal})\left({\frac{1.00~\times~10^{3}{\cancel{cal}}}{1.00\ \cancel{Cal}}}\right)\left({\frac{4.186{~\mathrm{g}}}{{\cancel{cal}}}}\right)
=1.34\times10^{6}~J
Use the work–energy theorem to find the work necessary to lift the barbell up to its maximum height:
W=\Delta K E+\Delta P E=(0-0)+(m g h-0)=m g h
The student must expend the same amount of energy lowering the barbell, making 2mgh per repetition.
Multiply this amount by n repetitions and set it equal to the food energy E:
n(2mgh) = E
Solve for n, substituting the food energy for E:
n={\frac{E}{2m g h}}={\frac{1.34~\times~10^{6}\mathrm{J}}{2(25.0\mathrm{~kg})(9.80\mathrm{~m}/s^{2})(0.400\mathrm{~m})}}
= 6.84 3 10³ times
REMARKS If the student does one repetition every 5 seconds, it will take him 9.5 hours to work off his breakfast! In exercising, a large fraction of energy is lost through heat, however, due to the inefficiency of the body in doing work. This transfer of energy dramatically reduces the exercise requirement by at least three-quarters, a little over two hours. Further,
some small fraction of the energy content of the cereal may not actually be absorbed. All the same, it might be best to forgo a second bowl of cereal!