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Moment of Inertia

A sewer pipe rolls more slowly down an incline than a bowling ball with the same mass. Understand this and more by learning about Moment of Inertia, a measure of how compact objects are.

Center of mass of a collection of points

         

Three particles of respective masses \(m_1=12.0\text{ kg},\) \(m_2=25.0\text{ kg}\) and \(m_3=38.0\text{ kg}\) form an equilateral triangle of side length \(a=140\text{ cm}.\) If we locate \(m_1\) at the origin on the \(xy\)-plane, and put \(m_2\) to the right of \(m_1\) on the \(x\)-axis, as shown in the above figure, what are the approximate coordinates of the center of mass of this system?

The value of \(\sqrt{3}\) is \(1.732.\)

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Three particles are on the \(xy\)-plane, as shown in the above figure. The masses of the three particles are \(m_1=2.0\text{ kg},\) \(m_2=5.0\text{ kg}\) and \(m_3=9.0\text{ kg}.\) If the scales on the axes are set by \(x_s=4.0\text{ m}\) and \(y_s=6.0\text{ m},\) what are the \(xy\)-coordinates of the system's center of mass?

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Consider a uniform bar of length \(3L\) and mass \(m_3=5m.\) Two balls are hanging on strings with negligible mass from the two ends of the bar, and their masses are \(m_1=m\) and \(m_2=3m.\) The lengths of the string on which the balls are hanging are \(L\) and \(2L,\) respectively, as shown in the above figure. What is the center of mass of this system relative to the midpoint of the bar?

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Three particles \(A,\) \(B\) and \(C\) are on the \(xy\)-plane. Their masses are \(m_A=2.00\text{ kg},\) \(m_B=4.00\text{ kg}\) and \(m_C=3.00\text{ kg},\) and the coordinates of \(A\) and \(B\) are \((-1.40\text{ m},0.48\text{ m})\) and \((0.70\text{ m},-0.72\text{ m}),\) respectively. If the coordinates of the center of mass of the three-particle system is \((-0.50\text{ m},-0.70\text{ m}),\) what is the coordinates of particle \(C ?\)

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Three particles of respective masses \(m_1=13.0\text{ kg},\) \(m_2=29.0\text{ kg}\) and \(m_3=37.0\text{ kg}\) form an equilateral triangle of side length \(a=140\text{ cm}.\) If we locate \(m_1\) at the origin on the \(xy\)-plane, and put \(m_2\) to the right of \(m_1\) on the \(x\)-axis, as shown in the above figure, what are the approximate coordinates of the center of mass of this system?

The value of \(\sqrt{3}\) is \(1.732.\)

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by Brilliant Staff
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