91Ó°ÊÓ

The drawing shows a human figure 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 whole-body mass) has been ignored to simplify the drawing.

Step by step solution

01

Identify Masses and Coordinates

Let's identify the masses of each part of the figure and their respective coordinates. The torso, neck, and head have a mass of \(41 \, \text{kg}\) with a center of mass at \((0, 0.39)\). The upper legs have a mass of \(17 \, \text{kg}\) with a center of mass at \((0.17, 0)\). The lower legs and feet have a mass of \(9.9 \, \text{kg}\) with a center of mass at \((0.43, -0.26)\).

02

Calculate Total Mass

Add the masses of all parts to calculate the total mass: \(41 + 17 + 9.9 = 67.9 \, \text{kg}\).

03

Calculate Center of Mass - X Coordinate

Use the center of mass formula for the x-coordinate: \[ x_{cm} = \frac{m_1 \cdot x_1 + m_2 \cdot x_2 + m_3 \cdot x_3}{m_1 + m_2 + m_3} \] Substitute the given values: \[ x_{cm} = \frac{41 \cdot 0 + 17 \cdot 0.17 + 9.9 \cdot 0.43}{67.9} \] Calculate: \[ x_{cm} = \frac{0 + 2.89 + 4.257}{67.9} = \frac{7.147}{67.9} \approx 0.105 \text{m}\]

04

Calculate Center of Mass - Y Coordinate

Utilize the center of mass formula for the y-coordinate: \[ y_{cm} = \frac{m_1 \cdot y_1 + m_2 \cdot y_2 + m_3 \cdot y_3}{m_1 + m_2 + m_3} \] Insert the data: \[ y_{cm} = \frac{41 \cdot 0.39 + 17 \cdot 0 + 9.9 \cdot (-0.26)}{67.9} \] Compute: \[ y_{cm} = \frac{15.99 + 0 - 2.574}{67.9} = \frac{13.416}{67.9} \approx 0.197\text{m}\]

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Key Concepts

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

Physics Problem Solving

Physics problem solving involves breaking down a complex scenario into manageable parts to find a solution. Understanding the elements of the problem is crucial. Consider a human figure in a sitting position, where we want to find the center of mass. This involves identifying and analyzing each part of the figure to apply relevant physics formulas.

• First, you identify the masses and their positions in the figure.
• Subsequently, calculations based on these identified pieces of data are performed.
• Formulas for the center of mass are applied to find the result.

In this exercise, dividing the human body into three parts helps us calculate the center of mass coordinates. Students often find it easier to process each part separately before combining them. This problem-solving approach simplifies physics exercises by providing a clear direction and step-by-step guidance.

Mass and Weight Distribution

Mass and weight distribution play an essential role in understanding how objects balance and move. In physics, the center of mass is a point that acts as if all the mass of an object is concentrated there. It is crucial for determining how forces affect an object.

In the given exercise, we consider the torso, upper legs, and lower legs separately:

• The torso, neck, and head make up the heaviest part, positioning its mass along the y-axis.
• The upper legs have their mass on the x-axis, contributing horizontally.
• Finally, the lower legs and feet factor into both the x and y coordinates.

This distribution of mass impacts the total center of mass of the human figure. Understanding how mass is spread out helps predict how it will behave when subject to forces like gravity. This concept is essential in applications ranging from biomechanics to engineering.

Coordinate Systems in Physics

Coordinate systems are frameworks that allow us to describe an object's position in space using numbers. The most common system in physics is the Cartesian coordinate system, with x, y, and z axes typically denoting horizontal, vertical, and depth positions, respectively.

In our exercise, a 2D Cartesian coordinate system was used:

• The x-axis describes horizontal positioning, critical for calculating how parts of the figure are placed from side to side.
• The y-axis refers to vertical positioning, indicating elevation above or below a baseline, like the origin.
• Coordinates allow precise calculation of each part's center of mass in the problem given.

Understanding coordinate systems in physics is key to accurately solve problems related to positions and movements. This exercise shows how essential these systems are for simplifying complex real-world scenarios into solvable mathematical expressions.