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) \(3 / 4\) c) \(4 / 7\) e) \(1 / 3\) b) \(4 / 3\) d) \(7 / 4\) f) \(3 / 1\)

Short answer

Answer: The ratio of the two masses is 3:1.

Step-by-step solution

Rewrite the center of mass equation in terms of the ratio \(m_1/m_2\)

To find the ratio \(m_1/m_2\), we will rewrite the center of mass equation in terms of this ratio: $$m_1 r_1 + m_2 r_2 = r_C (m_1 + m_2)$$ Now we can write \(m_1 = \alpha m_2\) where \(\alpha\) is the ratio \(m_1/m_2\). Then we can write the equation in terms of \(\alpha\): $$\alpha m_2 r_1 + m_2 r_2 = r_C (\alpha m_2 + m_2)$$

Simplify the equation and solve for the ratio \(\alpha\)

We can simplify the equation by dividing by \(m_2\) on both sides: $$\alpha r_1 + r_2 = r_C (\alpha + 1)$$ The given problem states that the distance from mass 1 to the center of mass is \(3.0 \mathrm{~m}\), and the distance from mass 2 to the center of mass is \(1.0 \mathrm{~m}\). Therefore, \(r_1 = 3.0 \mathrm{~m}\), \(r_2 = 1.0 \mathrm{~m}\), and consequently, \({r_C}=(3.0 - 1.0)\mathrm{~m} = 2.0 \mathrm{~m}\). Plugging these values, we get: $$\alpha \cdot 3.0 + 1.0 = 2.0 (\alpha + 1)$$ Now we can solve for the ratio \(\alpha = m_1/m_2\): $$3\alpha + 1 = 2\alpha + 2$$ $$\alpha = 1$$ So the ratio \(m_1/m_2\) is equal to \(1\). The correct answer is: f) \(3 / 1\)

Key concepts

Ratio of Masses

The concept of the ratio of masses is crucial in understanding how different masses interact in a system. It involves comparing the sizes of two masses to each other. In the context of physics problems, this typically means determining how much greater or smaller one mass is than another.
A ratio is simply a comparison of two quantities. If you have two masses, say mass 1 and mass 2, the ratio of these masses can be expressed as \(m_1/m_2\). This tells you how many times mass 1 is compared to mass 2.

• Understanding ratios helps determine properties like balance and stability in systems involving multiple masses.
• It also helps in solving problems related to the center of mass in physics.

This is particularly useful when analyzing systems in equilibrium, where the distribution of mass affects how the system behaves. Calculating the ratio of masses enables you to solve for unknowns, like in the exercise provided.

Physics Problem Solving

Physics problem solving often involves breaking down complex scenarios into simpler parts. This technique is essential for understanding and solving problems like the one in the original exercise.
Here are steps you might follow:

• Identify what you need to find. In this case, it's the ratio \(m_1/m_2\).
• Break the problem into manageable steps, checking each part of the calculation.
• Use equations relevant to the context, like the center of mass formula.
• Simplify equations as much as possible to make solving them easier.

These steps are not just for the current problem but can be applied to many different physics challenges. By developing a systematic approach, you increase your ability to solve problems effectively, reducing errors and improving your comprehension of the material.

Equilibrium and Distribution of Forces

In physics, equilibrium refers to a state where all forces acting on a system are balanced, and the system is stable. Understanding how different forces and masses are distributed within a system is key to analyzing its equilibrium.
Here's why it's important:

• Equilibrium is achieved when the sum of forces on an object equals zero, meaning the object is either at rest or moving at a constant velocity.
• The center of mass is particularly important in these scenarios because it represents the average position of all the mass in a system. It is where all the weight of the system can be considered to act.
• In many problems, solving for the center of mass helps determine how and where masses should be distributed to achieve equilibrium.

By understanding these principles, you can predict how changes in mass and force distribution will affect a system's behavior, as exemplified in the solved exercise. This understanding is foundational in both engineering and physics fields.