Classical Mechanics

Moment of Inertia

Center of mass of a collection of points

         

Three particles of respective masses m1=12.0 kg,m_1=12.0\text{ kg}, m2=25.0 kgm_2=25.0\text{ kg} and m3=38.0 kgm_3=38.0\text{ kg} form an equilateral triangle of side length a=140 cm.a=140\text{ cm}. If we locate m1m_1 at the origin on the xyxy-plane, and put m2m_2 to the right of m1m_1 on the xx-axis, as shown in the above figure, what are the approximate coordinates of the center of mass of this system?

The value of 3\sqrt{3} is 1.732.1.732.

Three particles are on the xyxy-plane, as shown in the above figure. The masses of the three particles are m1=2.0 kg,m_1=2.0\text{ kg}, m2=5.0 kgm_2=5.0\text{ kg} and m3=9.0 kg.m_3=9.0\text{ kg}. If the scales on the axes are set by xs=4.0 mx_s=4.0\text{ m} and ys=6.0 m,y_s=6.0\text{ m}, what are the xyxy-coordinates of the system's center of mass?

Consider a uniform bar of length 3L3L and mass m3=5m.m_3=5m. Two balls are hanging on strings with negligible mass from the two ends of the bar, and their masses are m1=mm_1=m and m2=3m.m_2=3m. The lengths of the string on which the balls are hanging are LL and 2L,2L, respectively, as shown in the above figure. What is the center of mass of this system relative to the midpoint of the bar?

Three particles A,A, BB and CC are on the xyxy-plane. Their masses are mA=2.00 kg,m_A=2.00\text{ kg}, mB=4.00 kgm_B=4.00\text{ kg} and mC=3.00 kg,m_C=3.00\text{ kg}, and the coordinates of AA and BB are (1.40 m,0.48 m)(-1.40\text{ m},0.48\text{ m}) and (0.70 m,0.72 m),(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 m,0.70 m),(-0.50\text{ m},-0.70\text{ m}), what is the coordinates of particle C?C ?

Three particles of respective masses m1=13.0 kg,m_1=13.0\text{ kg}, m2=29.0 kgm_2=29.0\text{ kg} and m3=37.0 kgm_3=37.0\text{ kg} form an equilateral triangle of side length a=140 cm.a=140\text{ cm}. If we locate m1m_1 at the origin on the xyxy-plane, and put m2m_2 to the right of m1m_1 on the xx-axis, as shown in the above figure, what are the approximate coordinates of the center of mass of this system?

The value of 3\sqrt{3} is 1.732.1.732.

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