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Chapter 9: Problem 15

Two particles of equal mass \(m\) are at the vertices of the base of an equilateral triangle. The triangle's center of mass is midway between the base and the third vertex. What's the mass at the third vertex?

Short Answer

Expert verified
The mass at the third vertex, given the conditions, is \(3m\). This is the result when following through with the calculations based on the center of mass concept and the provided conditions.

Step by step solution

01

Understand the concept of Center of Mass

The center of mass for a system of particles is given by \[r_{cm} = \frac{1}{M} \sum m_i r_i\] where \(r_{cm}\) is the position of the center of mass, \(M\) is the total mass of the system, \(m_i\) are the individual masses and \(r_i\) are their respective positions.
02

Apply given conditions

The center of mass of the triangle is given as midway between the third vertex and the base where there are two point masses, each of mass \(m\). In terms of distances, if we assume the side length of the triangle to be \(a\), the position of the center of mass from the base would be \(\frac{a}{2}\), and so \[r_{cm} = \frac{a}{2}\]
03

Calculate the unknown mass

Replace the position of the center of mass in the formula in Step 1. Compute the total mass and the mass of vertex as \[m_{3} = M - 2m = \frac{2m(a/2)}{a/3} - 2m\] Solve the expression to find mass at the third vertex.

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