Question

c5 - center of mass and moments: problem 1
(4 points)
the following masses ( m_i ) are located at the given points ( p_i: m_1 = 6, p_1(1,5)m_2 = 5, p_2(3, - 2)m_3 = 10, p_3(-2, - 1) ) find the moments ( m_x,m_y ), and the ( x ) and ( y ) coordinates of the center of mass of the system.
( m_x=)
( m_y=)
( x ) - coordinate of the center of mass ( overline{x}=)
( y ) - coordinate of the center of mass ( overline{y}=)
note: you can earn partial credit on this problem.

Explanation:

Step1: Calculate the total mass \(M\)

The total mass \(M=m_1 + m_2+m_3\). Given \(m_1 = 6\), \(m_2=5\), \(m_3 = 10\), so \(M=6 + 5+10=21\).

Step2: Calculate the moment \(M_x\)

The formula for \(M_x\) is \(M_x=m_1y_1 + m_2y_2+m_3y_3\). For \(P_1(1,6)\), \(y_1 = 6\); for \(P_2(3,-2)\), \(y_2=-2\); for \(P_3(-2,-1)\), \(y_3=-1\).

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Step3: Calculate the moment \(M_y\)

The formula for \(M_y\) is \(M_y=m_1x_1 + m_2x_2+m_3x_3\). For \(P_1(1,6)\), \(x_1 = 1\); for \(P_2(3,-2)\), \(x_2=3\); for \(P_3(-2,-1)\), \(x_3=-2\).

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Step4: Calculate the \(x\) - coordinate of the center of mass \(\overline{x}\)

The formula for \(\overline{x}=\frac{M_y}{M}\). Since \(M_y = 1\) and \(M = 21\), \(\overline{x}=\frac{1}{21}\).

Step5: Calculate the \(y\) - coordinate of the center of mass \(\overline{y}\)

The formula for \(\overline{y}=\frac{M_x}{M}\). Since \(M_x = 16\) and \(M = 21\), \(\overline{y}=\frac{16}{21}\).

Answer:

\(M_x = 16\), \(M_y=1\), \(x\) - coordinate of the center of mass \(\overline{x}=\frac{1}{21}\), \(y\) - coordinate of the center of mass \(\overline{y}=\frac{16}{21}\)