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Excel in math and science

Master concepts by solving fun, challenging problems.

It's hard to learn from lectures and videos

Learn more effectively through short, interactive explorations.

Used and loved by over 6 million people

Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.

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Classical Mechanics

Moment of Inertia

Concept Quizzes

Newton's 2nd Law - Rotational Form
Inertia - Point Masses
Inertia - Mass Distributions
Center of Mass - Points
Center of Mass - Distributions

Challenge Quizzes

Level 3-5

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 \(1.732.\)

\( (97.3\text{ cm},61.3\text{ cm}) \) \( (97.3\text{ cm},37.0\text{ cm}) \) \( (82.1\text{ cm},37.0\text{ cm}) \) \( (82.1\text{ cm},61.3\text{ cm}) \)

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by Brilliant Staff

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?

\[\displaystyle{ \left(\frac{8}{19}\text{ m},\frac{16}{69}\text{ m}\right)} \] \[\displaystyle{ \left(\frac{19}{8}\text{ m},\frac{16}{69}\text{ m}\right)} \] \[\displaystyle{ \left(\frac{8}{19}\text{ m},\frac{69}{16}\text{ m}\right)} \] \[\displaystyle{ \left(\frac{19}{8}\text{ m},\frac{69}{16}\text{ m}\right)} \]

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by Brilliant Staff

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?

\[\displaystyle{ \left(\frac{7}{10}L,-\frac{7}{8}L\right)} \] \[\displaystyle{ \left(\frac{1}{3}L,-\frac{7}{9}L\right)} \] \[\displaystyle{ \left(\frac{7}{10}L,-\frac{3}{8}L\right)} \] \[\displaystyle{ \left(\frac{3}{8}L,-\frac{7}{10}L\right)} \]

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by Brilliant Staff

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 \(C ?\)

\( (-1.50\text{ m},-1.46\text{ m}) \) \( (-1.25\text{ m},-1.90\text{ m}) \) \( (-1.50\text{ m},-1.90\text{ m}) \) \( (-1.25\text{ m},-1.46\text{ m}) \)

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by Brilliant Staff

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 \(1.732.\)

\( (84.2\text{ cm},56.7\text{ cm}) \) \( (98.2\text{ cm},56.7\text{ cm}) \) \( (98.2\text{ cm},34.2\text{ cm}) \) \( (84.2\text{ cm},34.2\text{ cm}) \)

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by Brilliant Staff

Excel in math and science

Master concepts by solving fun, challenging problems.

It's hard to learn from lectures and videos

Learn more effectively through short, interactive explorations.

Used and loved by over 6 million people

Learn from a vibrant community of students and enthusiasts, including olympiad champions, researchers, and professionals.

Excel in math and science

Master concepts by solving fun, challenging problems.

It's hard to learn from lectures and videos

Learn more effectively through short, interactive explorations.

Used and loved by over 6 million people

Learn from a vibrant community of students and enthusiasts, including olympiad champions, researchers, and professionals.

Moment of Inertia

Concept Quizzes

Challenge Quizzes

Center of mass of a collection of points

Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.

Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.