/*! 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 44 The drawing shows a human figure... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 44

The drawing shows a human figure approximately in a sitting position. For purposes of this problem, there are three parts to the figure, and the center of mass of each one is shown in the drawing. These parts are: (1) the torso, neck, and head (total mass \(=41 \mathrm{~kg}\) ) with a center of mass located on the \(y\) axis at a point \(0.39 \mathrm{~m}\) above the origin, (2) the upper legs (mass \(=17 \mathrm{~kg}\) ) with a center of mass located on the \(x\) axis at a point \(0.17 \mathrm{~m}\) to the right of the origin, and (3) the lower legs and feet (total mass \(=9.9 \mathrm{~kg}\) ) with a center of mass located \(0.43 \mathrm{~m}\) to the right of and \(0.26 \mathrm{~m}\) below the origin. Find the \(x\) and \(y\) coordinates of the center of mass of the human figure. Note that the mass of the arms and hands (approximately \(12 \%\) of the wholebody mass) has been ignored to simplify the drawing.

Short Answer

Expert verified
The center of mass is at approximately (0.105, 0.198) m.

Step by step solution

01

Identify the components and their properties

We have three parts of the human figure: 1. Torso, neck, and head with a mass of 41 kg and center of mass at (0, 0.39) m. 2. Upper legs with a mass of 17 kg and center of mass at (0.17, 0) m. 3. Lower legs and feet with a mass of 9.9 kg and center of mass at (0.43, -0.26) m.
02

Calculate total mass

Sum up the masses of all parts: \[ M = 41 + 17 + 9.9 = 67.9 \, \text{kg} \]
03

Calculate x-coordinate of the center of mass

Use the formula for the x-coordinate of the center of mass: \[ x_{\text{cm}} = \frac{m_1 x_1 + m_2 x_2 + m_3 x_3}{M} \]Substitute the known values: \[ x_{\text{cm}} = \frac{41 \times 0 + 17 \times 0.17 + 9.9 \times 0.43}{67.9} \]Calculate: \[ x_{\text{cm}} = \frac{0 + 2.89 + 4.257}{67.9} = \frac{7.147}{67.9} \approx 0.1052 \, \text{m} \]
04

Calculate y-coordinate of the center of mass

Use the formula for the y-coordinate of the center of mass: \[ y_{\text{cm}} = \frac{m_1 y_1 + m_2 y_2 + m_3 y_3}{M} \]Substitute the known values: \[ y_{\text{cm}} = \frac{41 \times 0.39 + 17 \times 0 + 9.9 \times (-0.26)}{67.9} \]Calculate: \[ y_{\text{cm}} = \frac{15.99 + 0 - 2.574}{67.9} = \frac{13.416}{67.9} \approx 0.1976 \, \text{m} \]

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.

Understanding Equilibrium
Equilibrium is a fundamental concept in physics that describes a state where all forces or moments are balanced. In the context of this exercise, equilibrium helps us understand how the different parts of the human body are balanced when in a sitting position, especially focusing on the center of mass.

The center of mass is the point where the total mass of a system can be considered to be concentrated. When an object is in equilibrium, such as a sitting human figure, its center of mass is aligned in such a way that all individual parts work together to maintain a stable position.

To determine equilibrium, it's crucial to ensure that the calculated center of mass keeps the object balanced. If the center of mass shifts outside the base support, the object鈥攐r person鈥攎ay tip over. Therefore, knowing how to find the center of mass allows you to predict whether or not a system will remain stable when subjected to external forces.
Using a Coordinate System
Coordinate systems are an essential tool in physics to locate points in space, helping us apply mathematical techniques to real-world problems. Here, the coordinate system is used to describe the location of the different parts of the human figure with respect to an origin point.

In this exercise, we used a Cartesian coordinate system where each part of the human figure is given positions (x, y) relative to an origin. This provides a clear and organized way to calculate the center of mass by breaking down the problem into manageable pieces.

The origin, where the axes intersect, serves as a reference point. By plotting each mass in relation to this point, you can easily apply mathematical formulas to find the overall center of mass. This systematic approach simplifies complex problems and is widely used in various fields, from engineering to computer graphics.
Exploring Mass Distribution
Mass distribution refers to how mass is spread out in an object or system. For the human figure in this exercise, mass distribution affects the position of the center of mass. Each body part鈥攖orso with neck and head, upper legs, and lower legs and feet鈥攈as a different mass and location in the coordinate system.

Understanding mass distribution is key to determining the center of mass. The formula used to find both the x and y coordinates of the center of mass takes into account the mass of each part and its position.

  • For the x-coordinate: sum of the product of each mass and its x position, divided by the total mass.
  • For the y-coordinate: sum of the product of each mass and its y position, divided by the total mass.
This accounting of mass helps in visualizing how different parts contribute to the overall balance and position of the center of mass. It's a crucial concept when analyzing stability and posture in biomechanics, architecture, and even space engineering.

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

(a) John has a larger mass than Barbara has. He is standing on the \(x\) axis at \(x_{\mathrm{J}}=+9.0 \mathrm{~m}\), while she is standing on the \(x\) axis at \(x_{\mathrm{B}}=+2.0 \mathrm{~m}\). Is their centerof-mass point closer to the \(9.0\) -m point or the \(2.0\) -m point? (b) They switch positions. Is their center-of-mass point now closer to the \(9.0\) -m point or the \(2.0\) -m point? (c) In which direction, toward or away from the origin, does their center of mass move as a result of the switch?

Object \(\mathrm{A}\) is moving due east, while object \(\mathrm{B}\) is moving due north. They collide and stick together in a completely inelastic collision. Momentum is conserved. (a) Is it possible that the two-object system has a final total momentum of zero after the collision? (b) Roughly, what is the direction of the final total momentum of the two-object system after the collision?

ssm A golf ball bounces down a flight of steel stairs, striking several steps on the way down, but never hitting the edge of a step. The ball starts at the top step with a vertical velocity component of zero. If all the collisions with the stairs are elastic, and if the vertical height of the staircase is \(3.00 \mathrm{~m}\), determine the bounce height when the ball reaches the bottom of the stairs. Neglect air resistance.

During July 1994 the comet Shoemaker-Levy 9 smashed into Jupiter in a spectacular fashion. The comet actually consisted of 21 distinct pieces, the largest of which had a mass of approximately \(4.0 \times 10^{12} \mathrm{~kg}\) and a speed of \(6.0 \times 10^{4} \mathrm{~m} / \mathrm{s}\). Jupiter, the largest planet in the solar system, has a mass of \(1.9 \times 10^{27} \mathrm{~kg}\) and an orbital speed of \(1.3 \times 10^{4} \mathrm{~m} / \mathrm{s} .\) If this piece of the comet had hit Jupiter head-on, what would have been the change (magnitude only) in Jupiter's orbital speed (not its final speed)?

A cannon of mass \(5.80 \times 10^{3} \mathrm{~kg}\) is rigidly bolted to the earth so it can recoil only by a negligible amount. The cannon fires an \(85.0-\mathrm{kg}\) shell horizontally with an initial velocity of \(+551 \mathrm{~m} / \mathrm{s}\). Suppose the cannon is then unbolted from the earth, and no external force hinders its recoil. What would be the velocity of a shell fired by this loose cannon? (Hint: In both cases assume that the burning gunpowder imparts the same kinetic energy to the system.)
See all solutions

Recommended explanations on Physics Textbooks

Fields in Physics

Read Explanation

Wave Optics

Read Explanation

Dynamics

Read Explanation

Classical Mechanics

Read Explanation

Kinematics Physics

Read Explanation

Physical Quantities And Units

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.