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Chapter 9: Problem 4

Three uniform thin rods, cach of length \(L=22 \mathrm{~cm},\) form an inverted U. The vertical rods cach have a mass of \(14 \mathrm{~g}\); the horizontal rod has a mass of 42 g. What are (a) the \(x\) coordinate and (b) the \(y\) coordinate of the system's center of mass?

Short Answer

Expert verified
The center of mass is at \((11 \text{ cm}, 4.4 \text{ cm})\).

Step by step solution

01

Understand the System Configuration

We have three rods forming an inverted U-shape. The vertical rods each have a mass of \(m_1 = 14 \text{g}\) and the horizontal rod has a mass of \(m_2 = 42 \text{g}\). The length of each rod is \(L = 22 \text{cm}\). The horizontal rod forms the top bar of the inverted U, while the vertical rods form the legs.
02

Establish a Coordinate System

Let's place the origin of our coordinate system at the left end of the horizontal rod (the top of the inverted U) so that the horizontal rod runs along the x-axis from \(x = 0\) to \(x = 22\). The y-axis will run downward along the vertical rods from \(y = 0\) to \(y = 22\).
03

Determine Positions of Centers of Mass

- The center of mass of the left vertical rod is at \((0, 11)\).- The center of mass of the right vertical rod is at \((22, 11)\).- The center of mass of the horizontal rod is at \((11, 0)\).
04

Calculate x-coordinate of the Center of Mass

The x-coordinate of the center of mass of the system \(x_{cm}\) is calculated using the formula:\[x_{cm} = \frac{m_1x_1 + m_1x_2 + m_2x_3}{m_1 + m_1 + m_2}\]Substitute the values:\[x_{cm} = \frac{14\times0 + 14\times22 + 42\times11}{14 + 14 + 42} = \frac{0 + 308 + 462}{70} = \frac{770}{70} \approx 11 \text{ cm}\]Thus, the x-coordinate is 11 cm.
05

Calculate y-coordinate of the Center of Mass

The y-coordinate of the center of mass \(y_{cm}\) is given by:\[y_{cm} = \frac{m_1y_1 + m_1y_2 + m_2y_3}{m_1 + m_1 + m_2}\]Substitute the values:\[y_{cm} = \frac{14\times11 + 14\times11 + 42\times0}{70} = \frac{154 + 154 + 0}{70} = \frac{308}{70} \approx 4.4 \text{ cm}\]So, the y-coordinate is 4.4 cm.

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Key Concepts

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

Coordinate System
When dealing with problems involving the center of mass, establishing a coordinate system is a crucial first step. The coordinate system helps to simplify calculations by defining reference points and axes.
In this particular exercise, the origin of the coordinate system is placed at the left endpoint of the horizontal rod. This rod is part of the inverted U-shape.
The horizontal rod stretches across the x-axis from \(x = 0\) to \(x = 22\) cm. Meanwhile, the vertical rods move along the y-axis from \(y = 0\) to \(y = 22\) cm.
  • The x-axis represents the horizontal direction, covering the length of the horizontal rod.
  • The y-axis represents the vertical direction, marking the lengths of the vertical rods.
This setup ensures that the position of each rod within the coordinate system is clear and manageable for further calculations.
Mass Distribution
Understanding mass distribution is essential in calculating the center of mass. In this system, each of the three rods has a specific mass:
  • The two vertical rods each have a mass of 14 grams.
  • The horizontal rod has a mass of 42 grams.
The distribution of mass affects where the center of mass of the entire system lies. Since the horizontal rod is heavier, it has a more significant influence on the overall center of mass compared to the lighter vertical rods.
Each rod's mass contributes to the calculation of the center of mass in both the x and y directions.
By understanding mass distribution, we see that weighted averages are key to finding the system鈥檚 center of mass.
Inverted U-shape Rods
An inverted U-shape formation can be visualized by picturing three rods: two standing vertically, connected at the top by a horizontal rod. This shape is common in various engineering and physics applications.
In this exercise:
  • Each vertical rod stands perpendicularly aligned and descends from the common horizontal rod forming the legs of the U.
  • The horizontal rod rests on top, creating the "cap" of the U.
This particular arrangement requires us to consider how each rod's position and mass contribute to the overall structure of the inverted U-shape.
Each connection point where the rods meet is crucial for determining the center of mass, particularly given that the formation symmetrically encloses a space below.
Calculating x and y Coordinates
To locate the center of mass, we calculate the x and y coordinates, which tell us precisely where the balance point of the system is.
**X-coordinate Calculation:**
The x-coordinate \(x_{cm}\) is found using the formula:\[x_{cm} = \frac{m_1x_1 + m_1x_2 + m_2x_3}{m_1 + m_1 + m_2}\\]By substituting:
- \(m_1\) is the mass of each vertical rod,
- \(x_1\) and \(x_2\) are their respective x-positions,
- \(x_3\) is the x-position of the horizontal rod.
The formula gives us \(x_{cm} = 11\) cm as the x-coordinate.

**Y-coordinate Calculation:**
To find the y-coordinate \(y_{cm}\), use:\[y_{cm} = \frac{m_1y_1 + m_1y_2 + m_2y_3}{m_1 + m_1 + m_2}\\]Plugging values for y-positions and masses,
we get \(y_{cm} = 4.4\) cm for the y-coordinate.
  • These coordinates help determine the balance point, where the system can theoretically pivot in equilibrium.

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