   Chapter 5, Problem 28P

Chapter
Section
Textbook Problem

A flea is able to jump about 0.5 m. It has been said that if a flea were as big as a human, it would be able to jump over a 100-story building! When an animal jumps, it converts work done in contracting muscles into gravitational potential energy (with some steps in between). The maximum force exerted by a muscle is proportional to its cross-sectional area, and the work done by the muscle is this force times the length of contraction. If we magnified a flea by a factor of 1 000, the cross section of its muscle would increase by 1 0002 and the length of contraction would increase by 1 000. How high would this “superflea” be able to jump? (Don’t forget that the mass of the “superflea" increases as well.)

To determine
The height to which the supper flea is able to jump.

Explanation

Given Info:

The height that a flea is able to jump is 0.5m .

According to Work-energy theorem,

Wnc=(KEf+PEf)(KEi+KEi)

Consider the jump of the original flea. Since, the flea is at rest in the ground; the initial kinetic and potential energy of the flea is zero. At the top of the jump the final kinetic energy of the flea is also zero.

Thus,

The Work–Energy theorem for the jump of the original flea is,

Fmd=mghf

• Fm is the force exerted by the muscle
• hf is the maximum height that the flea jumps
• d is the length of contraction

On re-arranging,

hf=Fmdmg       (I)

When we scale the flea with a factor f, there will be an increase of f2 in muscle force

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