/*! 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 27 The elastic energy stored in you... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 27

The elastic energy stored in your tendons can contribute up to \(35 \%\) of your energy needs when running. Sports scientists have studied the change in length of the knee extensor tendon in sprinters and nonathletes. They find (on average) that the sprinters' tendons stretch \(41 \mathrm{mm},\) while nonathletes' stretch only 33 mm. The spring constant for the tendon is the same for both groups, \(33 \mathrm{N} / \mathrm{mm} .\) What is the difference in maximum stored energy between the sprinters and the nonathletes?

Short Answer

Expert verified
The difference in maximum stored energy between the sprinters and the nonathletes can be calculated by finding the potential energy stored in the tendons of both groups based on their stretch length and the given spring constant, then subtracting one from the other.

Step by step solution

01

Calculate the Energy Stored by the Sprinters

For the sprinters, the tendon stretches 41 mm. Using the formula for potential energy stored in a spring, \( U = \frac{1}{2}kx^2 \), where \( k \) is the spring constant and \( x \) is the change in length, calculate the potential energy stored: \( U_s = \frac{1}{2} \cdot 33 \, N/mm \cdot (41 \, mm)^2 \).
02

Calculate the Energy Stored by the Nonathletes

For the nonathletes, the tendon stretches 33 mm. Using the same formula, \( U = \frac{1}{2}kx^2 \), calculate the potential energy stored: \( U_n = \frac{1}{2} \cdot 33 \, N/mm \cdot (33 \, mm)^2 \).
03

Calculate the Difference in Energy

Subtract the potential energy stored in nonathletes from that stored in sprinters to find the difference in maximum stored energy: \( \Delta U = U_s - U_n \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Tendon Elasticity
Tendon elasticity plays a crucial role in human movement, particularly in activities like running and jumping. Tendons connect muscles to bones, and their ability to stretch and then return to their original shape is where elasticity comes in. When you run or jump, your tendons temporarily store energy as they stretch, which helps propel you forward or upward.
  • Elastic tendons absorb energy with every step, store it, and then release it efficiently.
  • This ability to store and release energy can significantly reduce the metabolic cost of running.
  • Tendon elasticity is beneficial in improving performance and reducing fatigue.
In sprinters, the tendons show greater stretching ability compared to non-athletes. This increased elasticity allows sprinters to harness more stored energy from their tendons, aiding in faster and more efficient movement. Thus, the elasticity of tendons is a natural advantage for athletes.
Spring Constant
The concept of the spring constant is vital in understanding how tendons function as springs. The spring constant (\( k )\) is a measure of a spring's stiffness. In the case of tendons, it determines how much force is required to stretch the tendon by a specific amount.
  • A higher spring constant means the spring (or tendon) is stiffer and requires more force to stretch.
  • A lower spring constant means it is more flexible and requires less force to stretch.
In our case, both the sprinters and non-athletes share the same spring constant of \( 33 \, \text{N/mm} .\)This means the stiffness and force response are the same for the tendons in both groups. However, the difference in the energy stored by these tendons arises due to both the spring constant and the amount by which they stretch. The more the tendon length changes, the more energy it can store.
Energy Calculation in Sports Science
In sports science, calculating the energy stored in tendons allows us to understand the efficiency and performance of athletes. The formula used to calculate the potential energy \( U \) in a tendon is: \[U = \frac{1}{2}kx^2\]where \( k \) is the spring constant and \( x \) is the deformation or change in length of the tendon.
  • For sprinters with a tendon stretch of 41 mm, this formula helps determine the energy stored is approximately \( U_s = \frac{1}{2} \cdot 33 \, \text{N/mm} \cdot (41 \, \text{mm})^2.\)
  • For non-athletes, a tendon stretch of 33 mm results in \( U_n = \frac{1}{2} \cdot 33 \, \text{N/mm} \cdot (33 \, \text{mm})^2.\)
  • The difference in these calculations gives the additional energy advantage sprinters have.
This difference illustrates how tendon elasticity can enhance performance. By better understanding these calculations, sports scientists can assess and improve athletic performance, aiding in training and injury prevention.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The maximum energy a bone can absorb without breaking is surprisingly small. For a healthy human of mass \(60 \mathrm{kg}\), experimental data show that the leg bones of both legs can absorb about \(200 \mathrm{J}\) a. From what maximum height could a person jump and land rigidly upright on both feet without breaking his legs? Assume that all the energy is absorbed in the leg bones in a rigid landing. b. People jump from much greater heights than this; explain how this is possible. Hint: Think about how people land when they jump from greater heights.

A crate slides down a ramp that makes a \(20^{\circ}\) angle with the ground. To keep the crate moving at a steady speed, Paige pushes back on it with a \(68 \mathrm{N}\) horizontal force. How much work does Paige do on the crate as it slides \(3.5 \mathrm{m}\) down the ramp?

When you ride a bicycle at constant speed, almost all of the energy you expend goes into the work you do against the drag force of the air. In this problem, assume that all of the energy expended goes into working against drag. As we saw in Section \(5.6,\) the drag force on an object is approximately proportional to the square of its speed with respect to the air. For this problem, assume that \(F \propto v^{2}\) exactly and that the air is motionless with respect to the ground unless noted Suppose a cyclist and her bicycle have a combined mass of \(60 \mathrm{kg}\) and she is cycling along at a speed of \(5 \mathrm{m} / \mathrm{s}\). If the drag force on the cyclist is \(10 \mathrm{N},\) how much energy does she use in cycling \(1 \mathrm{km} ?\) A. \(6 \mathrm{kJ}\) B. \(10 \mathrm{kJ}\) C. \(50 \mathrm{kJ}\) D. \(100 \mathrm{kJ}\)

The ball's kinetic energy just after the bounce is less than just before the bounce. In what form does this lost energy end up? A. Elastic potential energy B. Gravitational potential energy C. Thermal energy D. Rotational kinetic energy

The famous cliff divers of Acapulco leap from a perch \(35 \mathrm{m}\) above the ocean. How fast are they moving when they reach the water surface? What happens to their kinetic energy as they slow to a stop in the water?
See all solutions

Recommended explanations on Physics Textbooks

Scientific Method Physics

Read Explanation

Atoms and Radioactivity

Read Explanation

Electromagnetism

Read Explanation

Kinematics Physics

Read Explanation

Circular Motion and Gravitation

Read Explanation

Dynamics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.