Question

c5 - center of mass and moments: problem 1
(4 points)
results for this submission
2 of the answers are not correct.
the following masses ( m _ { i } ) are located at the given points ( p _ { i } : m _ { 1 } = 6,quad p _ { 1 } ( 1,5 ) m _ { 2 } = 5,quad p _ { 2 } ( 3, - 2 ) m _ { 3 } = 10,quad 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: ( x = )
( y ) - coordinate of the center of mass: ( y = )
note: you can earn partial credit on this problem.
preview my answers submit answers
your score was recorded.
your score was successfully sent to canvas.
you have attempted this problem 1 time.
you received a score of ( 50 % ) for this attempt.
your overall recorded score is ( 50 % ).
you have 14 attempts remaining.

Explanation:

Step1: Calculate the total mass \(m\)

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

Step2: Calculate the moment \(M_y\)

The formula for \(M_y\) is \(M_y=m_1x_1+m_2x_2+m_3x_3\). Here \(x_1 = 1\), \(x_2=3\), \(x_3=-2\). So \(M_y=6\times1 + 5\times3+10\times(-2)=6 + 15-20=1\)

Step3: Calculate the moment \(M_x\)

The formula for \(M_x\) is \(M_x=m_1y_1+m_2y_2+m_3y_3\). Here \(y_1 = 5\), \(y_2=-2\), \(y_3=-1\). So \(M_x=6\times5+5\times(-2)+10\times(-1)=30-10 - 10=10\)

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

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

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

The formula for \(\overline{y}\) is \(\overline{y}=\frac{M_x}{m}\). Since \(M_x = 3\) and \(m = 21\), then \(\overline{y}=\frac{3}{21}=\frac{1}{7}\)

Answer:

\(M_x = 16\), \(M_y = 1\), \(x=\frac{1}{21}\), \(y = \frac{1}{7}\)