Question

four very thin rods, each 5.6 m long, are joined to form a square, as part (a) of the drawing shows. the center of mass of the square is located at the coordinate origin. the rod on the right is then removed, as shown in part (b) of the drawing. what are the x - and y - coordinates of the center of mass of the remaining three - rod system? x coordinate of the center of mass = number units y coordinate of the center of mass = number units

Explanation:

Step1: Determine the mass and center - of - mass coordinates of each rod

Let the mass of each rod be \(m\).
For a thin rod of length \(L = 5.6\ m\) forming a square:

• The left - hand rod: \(m_1=m\), \(x_1=-\frac{L}{2}=- 2.8\ m\), \(y_1 = 0\)
• The top - hand rod: \(m_2=m\), \(x_2 = 0\), \(y_2=\frac{L}{2}=2.8\ m\)
• The bottom - hand rod: \(m_3=m\), \(x_3 = 0\), \(y_3=-\frac{L}{2}=-2.8\ m\)
• The right - hand rod (which is removed later): \(m_4=m\), \(x_4=\frac{L}{2}=2.8\ m\), \(y_4 = 0\)

Step2: Use the formula for the \(x\) - coordinate of the center of mass \(x_{cm}=\frac{\sum_{i = 1}^{n}m_ix_i}{\sum_{i = 1}^{n}m_i}\)

After removing the right - hand rod (\(n = 3\), \(m_1=m\), \(m_2=m\), \(m_3=m\)):

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Step3: Use the formula for the \(y\) - coordinate of the center of mass \(y_{cm}=\frac{\sum_{i = 1}^{n}m_iy_i}{\sum_{i = 1}^{n}m_i}\)

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Answer:

The \(x\) - coordinate of the center of mass is \(-0.93\ m\) and the \(y\) - coordinate of the center of mass is \(0\ m\)