Problem 4 - Thomas Calculus

Question

You weld the ends of two steel rods into a right-angled frame. One rod is twice the length of the other. Where is the frame's center of mass? (Hint: Where is the center of mass of each rod?)

Step-by-Step Solution

Answer

The frame's center of mass is at \( \left( \frac{L}{6}, \frac{2L}{3} \right) \)."

Step 1: Understanding the Problem

We have two steel rods of different lengths forming an L shape. One rod is twice as long as the other. The task is to find the frame's center of mass.

Step 2: Locating the Center of Mass of Each Rod

For a uniform rod, the center of mass is at its midpoint. Let the shorter rod have length L, then the longer rod has length 2L.

Step 3: Finding Coordinates of the Center of Mass for Each Rod

Assume the corner of the L-frame is at the origin (0, 0). Let the shorter rod lie along the x-axis, and the longer along the y-axis. The center of mass of the shorter rod is at (L/2, 0); for the longer rod, it's at (0, L).

Step 4: Calculating the Frame's Center of Mass

The center of mass of the system of rods can be found using the formula for combined center of mass: \( \bar{x} = \frac{m_1x_1 + m_2x_2}{m_1 + m_2} \) and \( \bar{y} = \frac{m_1y_1 + m_2y_2}{m_1 + m_2} \),where \( m_1 \) and \( m_2 \) are the masses of the shorter and longer rod, respectively. Assuming uniform density, the mass of the longer rod is twice that of the shorter rod.

Step 5: Substituting Values into the Center of Mass Formula

Let the mass of the shorter rod be \( m \), making the longer rod \( 2m \). The x-coordinate of the center of mass is:\[ \bar{x} = \frac{m \cdot \frac{L}{2} + 2m \cdot 0}{m + 2m} = \frac{\frac{mL}{2}}{3m} = \frac{L}{6} \] The y-coordinate is:\[ \bar{y} = \frac{m \cdot 0 + 2m \cdot L}{m + 2m} = \frac{2mL}{3m} = \frac{2L}{3} \].

Step 6: Conclusion

The center of mass of the frame is located at \( \left( \frac{L}{6}, \frac{2L}{3} \right) \).

Key Concepts

Steel Rods

Steel rods are long, slender structures made of steel, which is an alloy known for its strength and durability. These rods are often used in construction and engineering due to their high tensile strength and resistance to deformation.
Steel rods can vary in length and thickness, and they have uniform density when the material is evenly distributed along the rod.
For this exercise, we consider two steel rods of different lengths. Importantly, the rods have consistent density throughout, which means each part of the rod has the same mass per unit length. This characteristic is crucial for calculating the center of mass, because it simplifies the mathematical calculations involved.

Right-Angled Frame

A right-angled frame, also known as an L-frame, is formed by joining two perpendicularly oriented components. In this scenario, these components are steel rods.
When welding the ends of two rods to create a right-angled frame, the rods form a 90-degree angle.
In our exercise, one rod is twice as long as the other, and each is oriented along an axis in a coordinate plane:

• The shorter rod is along the x-axis.
• The longer rod extends along the y-axis.

This arrangement plays an essential role in determining the coordinates of the center of mass as it provides a clear reference point, with the origin at the intersection of the rods.

Coordinates of Center of Mass

The center of mass of an object or system is the point at which the entire mass could be concentrated without affecting the object's motion. For this right-angled frame, we calculate the coordinates of the center of mass by focusing on each rod's midpoint.
To find this center:

• The shorter rod's center of mass is located at \( \left( \frac{L}{2}, 0 \right) \), halfway along its length on the x-axis.
• The longer rod's center is at \( \left( 0, L \right) \), halfway along its extension on the y-axis.

Once these points are established, the combined center of mass accounts for both rods' contribution, necessitating calculations based on their masses and positions. This ensures that the frame's balance point accurately reflects the combined distribution of mass.

Uniform Density

Uniform density refers to a situation where an object's material is distributed evenly throughout its volume or length, meaning the density remains constant at all points.
For steel rods in this exercise, having uniform density simplifies finding the center of mass because it indicates that the mass of each rod is proportional to its length.

• This proportionality enables easy calculation of the mass for each rod, knowing the density is consistent.
• In our example, if the shorter rod's mass is \( m \), then the longer rod, being twice as long, has a mass of \( 2m \).

This relationship is crucial for using the combined center of mass formula accurately, ensuring that the weighted average coordinates reflect the true balance point for the entire frame.