Question

Two point masses are located in the same plane. The distance from mass 1 to the center of mass is \(3.0 \mathrm{~m} .\) The distance from mass 2 to the center of mass is \(1.0 \mathrm{~m}\). What is \(m_{1} / m_{2}\), the ratio of mass 1 to mass 2 ? a) \(3 / 4\) b) \(4 / 3\) c) \(4 / 7\) d) \(7 / 4\) c) \(4 / 7\) d) \(7 / 4\)

Short answer

Answer: The ratio of mass 1 to mass 2 is 1/3.

Step-by-step solution

Write the given information

We have the following information: - Distance from mass 1 to center of mass: \(x_1=3.0 \mathrm{~m}\) - Distance from mass 2 to center of mass: \(x_2=1.0 \mathrm{~m}\) We have to find the ratio \(m_1/m_2\).

Apply the definition of center of mass

For a system of two point masses, the center of mass is given by the equation \(x_1 m_1 = x_2 m_2\). We can rearrange the equation to solve for the mass ratio: \(\frac{m_1}{m_2} = \frac{x_2}{x_1}\)

Substitute the values

We can now substitute the given distances \(x_1\) and \(x_2\) into the equation: \(\frac{m_1}{m_2} = \frac{1.0 \mathrm{~m}}{3.0 \mathrm{~m}}\)

Calculate the mass ratio

Divide the given distances to find the mass ratio \(m_1/m_2\): \(\frac{m_1}{m_2} = \frac{1}{3}\) The correct answer is that the ratio of mass 1 to mass 2 is \(\frac{1}{3}\). However, this answer is not among the given options. The problem seems to have an error in the provided choices. Please verify the available options or consult the teacher for clarification.

Key concepts

Physics Education

Physics education aims to construct a deep understanding of natural laws and principles, starting from foundational concepts and building up to more complex theories. One key aspect of such education is developing the ability to model real-world situations with simplified representations.

In our example, the concept of the center of mass is fundamental to understanding motion and balance. It's important to include in physics education because it ties closely with both theoretical aspects of physics and practical applications. Educators should emphasize the significance of the center of mass in various contexts—ranging from the orbits of planets to the stability of a standing ladder.

The exercise discussed involves calculations related to the center of mass and provides a concrete chance to apply these theoretical concepts. Ensuring that the problem-solving process is clear allows students to see how the abstract formulas relate to actual physical scenarios, hence reinforcing their understanding and promoting engagement with the material.

Point Masses

In physics, the term 'point mass' refers to an object whose size is negligible in comparison to the distances involved in the problem. Point masses allow us to focus solely on the mass of the object and its location in space without worrying about the object's shape or volume.

In our textbook exercise, the two objects are treated as point masses, making the calculation of the center of mass straightforward. This is because treating objects as point masses simplifies calculations—it allows us to use linear equations and directly apply Newton's laws of motion. Visual aids, like diagrams showing the point masses and their distances from the center of mass, can be very helpful in understanding these types of problems.

Mass Ratio

The mass ratio, in the context of this exercise, is a comparison of the quantities of matter in two objects. It is a dimensionless number that tells us how much more massive one object is compared to another. In our textbook problem, the ratio of mass 1 to mass 2, denoted as \(m_1/m_2\), is crucial for determining the location of the center of mass.

Understanding the concept of mass ratios can be significantly improved by using proportions and visualizing how changing one mass affects the position of the center of mass. It's also useful to consider real-world examples where mass ratios play a role, such as in the balancing of seesaws or the design of stable structures. Developing an intuition for mass ratios is helpful not only in physics but in various engineering applications as well.