/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 89 BIO A Flea's Jump The resilin in... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 89

BIO A Flea's Jump The resilin in the upper leg (coxa) of a flea has a force constant of about \(26 \mathrm{N} / \mathrm{m}\), and when the flea cocks its jumping legs, the resilin in each leg is stretched by approximately \(0.10 \mathrm{mm}\). Given that the flea has a mass of \(0.50 \mathrm{mg}\) and that two legs are used in a jump, estimate the maximum height a flea can attain by using the energy stored in the resilin. (Assume the resilin to be an ideal spring.)

Short Answer

Expert verified
The maximum height a flea can jump is approximately 0.53 meters.

Step by step solution

01

Understand the Problem

The problem requires us to estimate the maximum height a flea can jump using the stored energy in its legs. We have a force constant, a stretch length, and the flea's mass. We'll consider the energy stored in the spring-like mechanism of the flea鈥檚 leg.
02

Calculate Energy Stored in One Leg

The energy stored in a spring is given by the formula \( E = \frac{1}{2} k x^2 \), where \( k \) is the force constant and \( x \) is the displacement. Here, \( k = 26 \text{ N/m} \) and the displacement \( x = 0.1 \text{ mm} = 0.0001 \text{ m} \). Thus, the energy stored in one leg is \( E = \frac{1}{2} \times 26 \times (0.0001)^2 = 1.3 \times 10^{-6} \text{ J} \).
03

Calculate Total Energy in Two Legs

Since the flea uses two legs for jumping, the total stored energy is double that of one leg. Therefore, total energy \( E_{total} = 2 \times 1.3 \times 10^{-6} = 2.6 \times 10^{-6} \text{ J} \).
04

Use Energy Conservation to Find Maximum Height

The energy stored in the flea's legs is converted entirely to gravitational potential energy at the maximum height. Set the stored energy equal to gravitational potential energy \( E_{potential} = mgh \). Here, \( m = 0.5 \times 10^{-6} \text{ kg} \), \( g = 9.8 \text{ m/s}^2 \), and \( h \) is the height. Solving \( 2.6 \times 10^{-6} = 0.5 \times 10^{-6} \times 9.8 \times h \), we find \( h \approx 0.53 \text{ m} \).

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Key Concepts

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

Spring Constant
The spring constant, often denoted by the symbol \( k \), is an essential part of understanding how a spring behaves. It relates to the stiffness of the spring, offering a direct measure of how much force is needed to stretch or compress it by a certain distance.
For example, in the case of the flea's leg, this constant is given as \( 26 \text{ N/m} \). This means that every meter the flea's leg is stretched requires a force of 26 Newtons.
  • A high spring constant means the spring is very stiff and requires more force to stretch.
  • A low spring constant means it's easier to stretch the spring with less force.
Spring constants are crucial in calculating the potential energy stored in the spring. In practical terms, a strong flea leg can store a lot of energy because of its higher spring constant, enabling the flea to jump high.
Energy Conservation
Energy conservation is a fundamental principle in physics that states energy cannot be created or destroyed, only transformed from one form to another. This principle is at work when a flea jumps. Initially, there is energy stored in the legs, and this gets converted to gravitational potential energy as the flea lifts off.
Here's how it works step by step:
  • First, the flea stores elastic potential energy in its legs. This happens because the resilin, which behaves like a spring, is stretched.
  • On jumping, this stored energy converts into kinetic energy as the flea moves upward.
  • Finally, when the flea reaches maximum height, all kinetic energy is converted into gravitational potential energy.
This energy transformation explains how fleas can leap impressively high despite their tiny size. The principle ensures that all energy used in stretching the leg goes into increasing the height of the jump.
Gravitational Potential Energy
Gravitational potential energy (GPE) is the energy an object possesses due to its position in a gravitational field, commonly defined as \( E = mgh \).
In the flea's jump, the formula helps us understand how high the flea can leap.
  • The mass \( m \) is the flea's mass, which is very small at \( 0.5 \text{ mg} \), converted into kilograms.
  • The gravitational constant \( g \) is standard Earth gravity, approximated as \( 9.8 \text{ m/s}^2 \).
  • The height \( h \) is what we aim to find, representing how far upward the flea ascends.
As the flea jumps, the energy stored in the stretched leg springs is converted into GPE at the peak of the jump. The greater the amount of stored energy, the higher the flea can jump. Understanding GPE is important for predicting how far the energy from a spring or other mechanism can push an object upwards.

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Most popular questions from this chapter

A \(23-\mathrm{kg}\) child swings back and forth on a swing suspended by \(2.5-\mathrm{m}-\) long ropes. Plot the gravitational potential energy of this system as a function of the angle the ropes make with the vertical, assuming the potential energy is zero when the ropes are vertical. Consider angles up to \(90^{\circ}\) on either side of the vertical.

Predict/Explain Ball 1 is thrown to the ground with an initial downward speed; ball 2 is dropped to the ground from rest. Assuming the balls have the same mass and are released from the same height, is the change in gravitational potential energy of ball 1 greater than, less than, or equal to the change in gravitational potential energy of ball \(2 ?\) (b) Choose the best explanation from among the following: I. Ball 1 has the greater total energy, and therefore more energy can go into gravitational potential energy. II. The gravitational potential energy depends only on the mass of the ball and the drop height. III. All of the initial energy of ball 2 is gravitational potential energy.

A 2.9-kg block slides with a speed of 1.6 \(\mathrm{m} / \mathrm{s}\) on a frictionless horizontal surface until it encounters a spring. (a) If the block compresses the spring \(4.8 \mathrm{cm}\) before coming to rest, what is the force constant of the spring? (b) What initial speed should the block have to compress the spring by \(1.2 \mathrm{cm} ?\)

At the local playground a child on a swing has a speed of \(2.02 \mathrm{m} / \mathrm{s}\) when the swing is at its lowest point. (a) To what maximum vertical height does the child rise, assuming he sits still and "coasts"? Ignore air resistance. (b) How do your results change if the initial speed of the child is halved?

A trapeze artist of mass \(m\) swings on a rope of length \(L\) Initially, the trapeze artist is at rest and the rope makes an angle \(\theta\) with the vertical. (a) Find the tension in the rope when it is vertical. (b) Explain why your result for part (a) depends on \(L\) in the way it does.
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