Understanding Center of Mass

Oct 13, 2025 by miodeya.com 135 views

Introduction

In this problem, we are tasked with finding the center of mass of three beads placed on the vertices of an equilateral triangle. The first bead has a mass of 145 g and is placed on the top vertex, while the other two beads have masses of 120 g and 150 g, respectively. The side length of the equilateral triangle is given as 3.9 cm. To solve this problem, we will use the concept of center of mass and the formula for calculating it.

Understanding Center of Mass

The center of mass of a system of particles is the point where the entire mass of the system can be considered to be concentrated. It is a point that represents the average position of the mass of the system. The center of mass is a useful concept in physics, as it allows us to simplify complex problems and make predictions about the motion of objects.

Calculating Center of Mass

To calculate the center of mass of the three beads, we need to use the formula:

\vec{r}_{\text{CM}} = \frac{m_1 \vec{r}_1 + m_2 \vec{r}_2 + m_3 \vec{r}_3}{m_1 + m_2 + m_3}

where $\vec{r}_{\text{CM}}$ is the position vector of the center of mass, $m_1$ , $m_2$ , and $m_3$ are the masses of the three beads, and $\vec{r}_1$ , $\vec{r}_2$ , and $\vec{r}_3$ are the position vectors of the three beads.

Given Information

The side length of the equilateral triangle is $d = 3.9 \, \text{cm}$ .
The mass of the first bead is $m_1 = 145 \, \text{g}$ .
The mass of the second bead is $m_2 = 120 \, \text{g}$ .
The mass of the third bead is $m_3 = 150 \, \text{g}$ .

Finding the Position Vectors

To find the position vectors of the three beads, we need to use the coordinates of the vertices of the equilateral triangle. Let's assume that the vertices of the triangle are at $(0, 0)$ , $(d, 0)$ , and $(d/2, \sqrt{3}d/2)$ .

The position vector of the first bead is $\vec{r}_1 = (0, 0)$ .

The position vector of the second bead is $\vec{r}_2 = (d, 0)$ .

The position vector of the third bead is $\vec{r}_3 = (d/2, \sqrt{3}d/2)$ .

Substituting Values into the Formula

Now that we have the position vectors of the three beads, we can substitute the values into the formula for calculating the center of mass:

\vec{r}_{\text{CM}} = \frac{m_1 \vec{r}_1 + m_2 \vec{r}_2 + m_3 \vec{r}_3}{m_1 + m_2 + m_3}

\vec{r}_{\text{CM}} = \frac{(145 \, \text{g})(0, 0) + (120 \, \text{g})(3.9 \, \text{cm}, 0) + (150 \, \text{g})(1.95 \, \text{cm}, 3.29 \, \text{cm})}{145 \, \text{g} + 120 \, \text{g} + 150 \, \text{g}}

\vec{r}_{\text{CM}} = \frac{(0, 0) + (464.4 \, \text{g} \cdot \text{cm}, 0) + (491.5 \, \text{g} \cdot \text{cm}, 493.35 \, \text{g} \cdot \text{cm})}{415 \, \text{g}}

\vec{r}_{\text{CM}} = \frac{(0, 0) + (0, 0) + (0, 0)}{415 \, \text{g}}

\vec{r}_{\text{CM}} = (0, 0)

Conclusion

In this problem, we calculated the center of mass of three beads placed on the vertices of an equilateral triangle. We used the formula for calculating the center of mass and substituted the values of the masses and position vectors of the three beads. The result showed that the center of mass of the three beads is at the origin, which is the center of the equilateral triangle.

Final Answer

Q: What is the center of mass of a system of particles?

A: The center of mass of a system of particles is the point where the entire mass of the system can be considered to be concentrated. It is a point that represents the average position of the mass of the system.

Q: How do you calculate the center of mass of a system of particles?

A: To calculate the center of mass of a system of particles, you need to use the formula:

\vec{r}_{\text{CM}} = \frac{m_1 \vec{r}_1 + m_2 \vec{r}_2 + m_3 \vec{r}_3}{m_1 + m_2 + m_3}

where $\vec{r}_{\text{CM}}$ is the position vector of the center of mass, $m_1$ , $m_2$ , and $m_3$ are the masses of the three particles, and $\vec{r}_1$ , $\vec{r}_2$ , and $\vec{r}_3$ are the position vectors of the three particles.

Q: What is the significance of the center of mass in physics?

A: The center of mass is a useful concept in physics, as it allows us to simplify complex problems and make predictions about the motion of objects. It is also a point that represents the average position of the mass of a system, which can be useful in understanding the behavior of complex systems.

Q: Can the center of mass be located outside of the system of particles?

A: Yes, the center of mass can be located outside of the system of particles. This occurs when the system of particles is not symmetrical, and the center of mass is not located at the center of the system.

Q: How do you find the position vectors of the particles in a system?

A: To find the position vectors of the particles in a system, you need to use the coordinates of the particles. For example, if the particles are located at $(x_1, y_1)$ , $(x_2, y_2)$ , and $(x_3, y_3)$ , then the position vectors of the particles are $\vec{r}_1 = (x_1, y_1)$ , $\vec{r}_2 = (x_2, y_2)$ , and $\vec{r}_3 = (x_3, y_3)$ .

Q: Can the center of mass be used to predict the motion of a system of particles?

A: Yes, the center of mass can be used to predict the motion of a system of particles. By knowing the position and velocity of the center of mass, you can predict the motion of the system of particles.

Q: What are some real-world applications of the center of mass?

A: The center of mass has many real-world applications, including:

Designing buildings and bridges: By knowing the center of mass of a building or bridge, engineers can design it to withstand external forces and ensure its stability.
Predicting the motion of celestial bodies: By knowing the center of mass of a celestial body, astronomers can predict its motion and behavior.
Designing vehicles: By knowing the center of mass of a vehicle, engineers can design it to be stable and maneuverable.

Conclusion

In this Q&A article, we have discussed the concept of the center of mass and its significance in physics. We have also answered some common questions about the center of mass and its applications. By understanding the center of mass, you can gain a deeper understanding of the behavior of complex systems and make predictions about their motion.

Final Answer

The final answer is: The center of mass is a useful concept in physics that allows us to simplify complex problems and make predictions about the motion of objects.