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 and Label Masses and Coordinates

There are three parts to the human figure: (1) The torso, neck, and head with a mass of 41 kg positioned at (0, 0.39); (2) The upper legs with a mass of 17 kg positioned at (0.17, 0); (3) The lower legs and feet with a mass of 9.9 kg positioned at (0.43, -0.26). These coordinates are based on their respective center of mass locations.

02

Calculate Total Mass

Compute the total mass of the human figure by adding the masses of all three parts. This is done using the formula:\[M_{total} = m_1 + m_2 + m_3 = 41 + 17 + 9.9 = 67.9 \text{ kg}\]

03

Calculate x-coordinate of the Center of Mass

To find the x-coordinate, use the formula:\[\bar{x} = \frac{m_1x_1 + m_2x_2 + m_3x_3}{M_{total}} = \frac{41 \times 0 + 17 \times 0.17 + 9.9 \times 0.43}{67.9}\]Calculate the total, then divide by the total mass to find \(\bar{x}\). After calculation:\[\bar{x} \approx 0.11 \text{ m}\]

04

Calculate y-coordinate of the Center of Mass

To find the y-coordinate, use the formula:\[\bar{y} = \frac{m_1y_1 + m_2y_2 + m_3y_3}{M_{total}} = \frac{41 \times 0.39 + 17 \times 0 + 9.9 \times (-0.26)}{67.9}\]Calculate the total, then divide by the total mass to find \(\bar{y}\). After calculation:\[\bar{y} \approx 0.20 \text{ m}\]

05

Conclude the Center of Mass Coordinates

The center of mass for the human figure is found at: \(\bar{x} \approx 0.11 \text{ m}, \bar{y} \approx 0.20 \text{ m}\). This takes into account the distribution of mass across the given body parts.

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Key Concepts

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

The System of Particles

Understanding the concept of a system of particles is fundamental when analyzing objects made up of multiple components. In physics, a system of particles refers to a collection of numerous small particles, where each part can be considered to have its own mass and position. When dealing with a complex structure like a human body, treating it as a system of particles allows you to simplify calculations by breaking it down into manageable pieces. Each part, such as the torso or legs, acts as a particle with its own center of mass and specific positioning in space. This approach simplifies the complex problem of calculating the overall center of mass by addressing each particle individually and then summing their effects.

Coordinate System

A coordinate system is crucial in precisely determining the location of different components within a system. It provides a framework consisting of an origin point and axes to define the position of points in space. In two-dimensional analyses, objects can be plotted on an x-y plane where the position of each part is denoted by coordinates \(x, y\). For instance, the torso in the exercise is situated at (0, 0.39), which means it's aligned along the y-axis 0.39 meters from the origin. Choosing a coordinate system that simplifies calculations makes an analysis more efficient and helps visualize the spatial arrangement of parts within the system of particles. When dealing with systems like a human figure, selecting the right coordinate origin ensures that all subsequent calculations are both accurate and easier to perform.

Mass Distribution

Understanding mass distribution is key when calculating the center of mass in a system of particles. Mass distribution refers to how mass is spread out in space, affecting the object's balance and stability. In the context of the exercise, different body parts have different masses that are distributed at various coordinates. The torso has a mass of 41 kg, acting at (0,0.39), while the upper legs have a mass of 17 kg at (0.17,0), and the lower legs and feet weigh 9.9 kg positioned at (0.43,-0.26). These distributions influence the overall center of mass position because each component's mass and distance from the origin contribute to it proportionally. Calculating the center of mass involves weighing the position of each mass based on its distribution to establish the balance point of the whole system.

Two-Dimensional Motion

Two-dimensional motion involves analyzing movements and positions on a flat plane. It's critical to understand that this type of motion involves changes along both the x and y axes. When determining the center of mass in two-dimensional systems, each component's motion must be considered for both dimensions. In the given exercise, the separate body parts of the human figure each have coordinates in an x-y plane, requiring calculations that take into account their two-dimensional positions. By using the nature of two-dimensional motion, focus on how masses are placed along both axes. This helps ensure accuracy in calculating where the center of mass resides in such systems, given its dependency on both x-dimension and y-dimension placements.