Worksheet: Center of Mass of Particles

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In this worksheet, we will practice finding the position of the center of gravity (or center of mass) of a set of particles arranged in a two-dimensional plane.

Q1:

The figure shows three weights arranged in an equilateral triangle of side length 12 cm. Find the coordinates of the centre of gravity of the system.

  • A 1 3 √ 3 5 , 3 1 5
  • B 1 1 √ 3 , 2 9 5
  • C 3 1 5 , 1 3 √ 3 5
  • D 2 9 5 , 1 1 √ 3

Q2:

A rhombus 𝐴𝐵𝐶𝐷 in which 𝐴𝐶=2𝐵𝐷=8cm such that point 𝐴 is located in the first quadrant of a Cartesian plane, 𝐵 is at the origin, and point 𝐷 is on the 𝑥-axis. Masses of 4 g, 3 g, 6 g, and 10 g are attached at vertices 𝐴, 𝐵, 𝐶 and 𝐷 respectively. Find the coordinates of the centre of gravity of the system.

  • A 6 0 2 3 , − 8 2 3
  • B 6 0 2 3 , 4 0 1 9
  • C 2 0 7 , 4 0 2 3
  • D 6 0 2 9 , − 4 1 3

Q3:

The equilateral triangle 𝐴𝐵𝐶 in the figure has a side length of 36 cm. Point 𝐷 is the intersection of its medians (its centroid) and 𝐸 is the midpoint of 𝐵𝐶. Masses of magnitudes 15 g, 27 g, 40 g, 12 g, and 50 g are fixed at the points 𝐴, 𝐵, 𝐶, 𝐷, and 𝐸 respectively. Determine the coordinates of the center of gravity of the system.

  • A 1 3 1 8 , 1 9 √ 3 8
  • B 1 1 9 8 , √ 3 2
  • C √ 3 2 , 1 1 9 8
  • D 1 9 √ 3 8 , 1 3 1 8

Q4:

A square 𝐴𝐵𝐶𝐷 has side length 𝐿. Three masses of 610 g are placed at 𝐴, 𝐵, and 𝐷. Find the coordinates of the center of mass of the system.

  • A 𝐿 3 , 𝐿 3
  • B 𝐿 3 , 𝐿
  • C 𝐿 2 , 𝐿 2
  • D ( 𝐿 , 𝐿 )

Q5:

The figure shows a system of point masses placed at the vertices of a triangle. The mass placed at each point is detailed in the table. Determine the coordinates of the center of gravity of the system.

Position 𝐴 𝐵 𝐶
Mass 13 kg 6 kg 15 kg
  • A 6 9 1 7 , 4 5 1 7
  • B 4 5 1 7 , 2 4 1 7
  • C 6 9 1 7 , 6 9 1 7
  • D 2 4 1 7 , 4 5 1 7

Q6:

The figure shows a system of point masses. The mass placed at each point is detailed in the table. Determine the coordinates of the center of gravity of the system.

Position 𝐴 𝐶 𝐸 𝐹
Mass 9 kg 5 kg 4 kg 3 kg
  • A 6 , 9 0 7
  • B 3 2 7 , 9
  • C 9 0 7 , 6
  • D 9 , 3 2 7

Q7:

An equilateral triangle 𝐴𝐵𝐶 of side length 10 cm. The points 𝐷, 𝐸, and 𝐹 are the midpoints of 𝐵𝐶, 𝐶𝐴, and 𝐴𝐵, respectively. Weights of 7 kg, 9 kg, 5 kg, 2 kg, 3 kg, and 4 kg are placed at the points 𝐴, 𝐵, 𝐶, 𝐷, 𝐸, and 𝐹, respectively. Find the coordinates of the centre of mass of the system.

  • A 1 7 4 , 7 √ 3 4
  • B 2 7 4 , 7 √ 3 3
  • C 1 3 2 , 2 9 √ 3 1 0
  • D 2 3 4 , 2 1 √ 3 1 0

Q8:

The figure shows a system of point masses placed at the vertices of a hexagon of side length 𝑙. The mass placed at each point is detailed in the table. Determine the coordinates of the center of gravity of the system.

Position at 𝐴 at 𝐹 at 𝐷 at 𝐶
Mass 18 g 26 g 6 g 30 g
  • A 1 1 0 𝑙 , √ 3 2 0 𝑙
  • B √ 3 1 0 𝑙 , 1 5 𝑙
  • C √ 3 2 0 𝑙 , 1 1 0 𝑙
  • D 1 5 𝑙 , √ 3 1 0 𝑙

Q9:

In a rectangle 𝐴𝐵𝐶𝐷, 𝐴𝐵=22cm and 𝐵𝐶=26cm. Four masses of 6, 7, 5, and 9 g are placed at the vertices 𝐴, 𝐷, 𝐵, and 𝐶 respectively. Another mass of magnitude 8 grams is attached to the midpoint of 𝐴𝐷. Determine the coordinates of the center of mass of the system.

  • A 5 2 5 , 6 6 5
  • B 7 8 7 , 6 6 5
  • C 9 1 5 , 5 2 8 3 5
  • D 7 8 7 , 5 5 7

Q10:

Four masses of 630 g are placed at the vertices of a square 𝐴𝐵𝐶𝐷 of side length 𝐿. Determine the coordinates of the center of gravity of the system relative to the axes 𝐴𝐵 and 𝐴𝐷.

  • A 𝐿 , 𝐿 2
  • B ( 𝐿 , 𝐿 )
  • C 𝐿 2 , 𝐿 2
  • D 𝐿 4 , 𝐿

Q11:

A square 𝐴𝐵𝐶𝐷 has a side length of 70 cm. When four equal masses are placed at the vertices of the square, the center of mass of the system is 𝐺. When the mass at vertex 𝐴 is removed, the center of mass of the system is 𝐺′. Find the coordinates of the center of mass of the two systems 𝐺 and 𝐺′.

  • A 𝐺 ( 3 5 , 3 5 ) , 𝐺 ′ 3 5 , 7 0 3
  • B 𝐺 ( 3 5 , 7 0 ) , 𝐺 ′ 7 0 3 , 7 0 3
  • C 𝐺 ( 3 5 , 7 0 ) , 𝐺 ′ 3 5 , 7 0 3
  • D 𝐺 ( 3 5 , 3 5 ) , 𝐺 ′ 7 0 3 , 7 0 3

Q12:

The figure shows a system of point masses 𝐴𝐵𝐶𝐷. The mass of each point is detailed in the table. Find the coordinates of the center of gravity of the system.

Position at 𝐴 at 𝐵 at 𝐶 at 𝐷
Mass 𝑚 kg 𝑚 kg 𝑚 kg 𝑚 kg
  • A 9 2 , 7 2
  • B 6 , 1 4 3
  • C 1 4 3 , 6
  • D 7 2 , 9 2

Q13:

𝐴 𝐵 𝐶 𝐷 is a square of side length 3 cm. Four masses of 2, 6, 3, and 2 grams are placed at 𝐴, 𝐵, 𝐶, and 𝐷 respectively. Another mass of 8 g is placed at the midpoint of 𝐴𝐵. Using a scale of 1 centimeter for 1 unit on both the 𝑥-axis and 𝑦-axis, determine the coordinates of the center of mass of the system.

  • A 5 1 1 9 , 2 7 1 3
  • B 1 3 5 , 1 3 7
  • C 1 6 7 , 8 7
  • D 2 3 1 0 , 1 2 7

Q14:

Equal masses are suspended from six of the vertices of a regular octagon 𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻. The masses are placed at 𝐴, 𝐵, 𝐶, 𝐷, 𝐸, and 𝐺. Given that the distance from any vertex to the center of the octagon 𝑀 is 52 cm, find the distance between 𝑀 and the center of gravity of the system of the six masses.

  • A 5 2 √ 2 3 cm
  • B 2 6 √ 2 3 cm
  • C 5 2 √ 3 3 cm
  • D 2 6 √ 3 3 cm

Q15:

Four particles of masses 9 kg, 10 kg, 4 kg, and 7 kg are placed on the 𝑥-axis at the points (4,0), (3,0), (8,0), and (1,0) respectively. What is the position of the center of mass of the four particles?

  • A ( 1 6 , 0 )
  • B ( 3 . 5 , 3 0 )
  • C ( 3 . 5 , 0 )
  • D ( 2 6 . 2 , 3 0 )
  • E ( 1 6 , 3 0 )

Q16:

Four particles are positioned at the points (0,𝑎), (0,5), (0,1), and (0,3). The center of mass of the four particles is the point 𝐺(0,2). Given that the masses of the four particles are 10𝑚, 5𝑚, 4𝑚, and 3𝑚 respectively, find the value of 𝑎.

Q17:

Two particles of weights 8 N and 18 N are separated by a distance of 39 m. Find the distance between the particle of weight 8 N and the center of gravity of the system.

Q18:

Three particles are placed on a line. Particle 𝐴 of mass 4 kg is located at the origin, particle 𝐵 of mass 6 kg at (9,6), and particle 𝐶 of mass 10 kg at (6,4). Determine the coordinates of the center of mass of the three particles.

  • A ( 5 . 7 , 0 )
  • B ( 0 , 3 . 8 )
  • C ( 5 . 7 , 3 . 8 )
  • D ( 5 . 9 , 4 )
  • E ( 3 , 2 )

Q19:

Three masses of 5 kg, 3 kg, and 9 kg are on the 𝑦-axis at the coordinates (0,2), (0,−3), and (0,−4) respectively. Determine the position of a fourth particle whose mass is 7 kg that needs to be added to the system for the center of mass of all four masses to be at the origin.

  • A ( 2 4 , 0 )
  • B ( 0 , 5 )
  • C ( 2 , 0 )
  • D ( 0 , 2 )
  • E ( 5 , 0 )

Q20:

The figure shows a system of point masses placed at the vertices of a square of side length 6 units. The mass placed at each point is detailed in the table. Determine the coordinates of the center of gravity of the system.

Position 𝐴 𝐵 𝐶 𝐷
Mass 75 kg 29 kg 71 kg 85 kg
  • A 3 0 1 3 , 1 8 5
  • B 4 8 1 3 , 1 2 5
  • C 1 2 5 , 4 8 1 3
  • D 1 8 5 , 3 0 1 3

Q21:

The points (0,6), (0,9), and (0,4) on the 𝑦-axis are occupied by three solids of masses 9 kg, 6 kg, and 𝑚 kg respectively. Determine the value of 𝑚 given the center of mass of the system is at the point (0,7).

Q22:

Six masses of 70, 30, 70, 50, 70, and 10 kilograms are placed at the vertices 𝐴, 𝐵, 𝐶, 𝐷, 𝐸, and 𝐹 of a uniform hexagon of side length 30 cm. Find the distance between the center of the hexagon and the center of gravity of the system.

  • A √ 1 0 cm
  • B 2 √ 3 1 cm
  • C 2 √ 3 cm
  • D 4 2 1 1 cm

Q23:

The figure shows a system of point masses. The mass placed at each point is detailed in the table. Given that 𝐴 and 𝐷 are on the same horizontal line, find the coordinates of the centre of gravity.

Position at 𝐴 at 𝐵 at 𝐶 at 𝐷
Mass 7 𝑚 kg 9 𝑚 kg 4 𝑚 kg 4 𝑚 kg
  • A 1 4 4 5 , 2 4 √ 3
  • B 2 4 √ 3 , 1 4 4 5
  • C 1 1 √ 3 , 1 8
  • D 1 8 , 1 1 √ 3

Q24:

A triangle 𝐴𝐵𝐶, where 𝐴𝐵=33cm, 𝐵𝐶=44cm, 𝐶𝐴=55cm, and 𝐷 and 𝐸 are the midpoints of 𝐴𝐵 and 𝐴𝐶 respectively, is located in the first quadrant of a Cartesian plane such that 𝐵 is at the origin, and the point 𝐶 is on 𝑥-axis. Three equal masses are placed at points 𝐵, 𝐷, and 𝐸. Determine the coordinates of the centre of gravity of the system.

  • A 1 1 , 2 2 3
  • B 4 4 3 , 1 1
  • C 2 2 3 , 3 3 2
  • D 2 2 3 , 1 1

Q25:

Find the coordinates of the centre of gravity of the following system: 𝑚=3kg at (−8,9), 𝑚=1kg at (6,−7), and 𝑚=4kg at (0,5).

  • A 5 , − 9 4
  • B − 1 2 , 5 4
  • C − 9 4 , 5
  • D 5 4 , − 1 2

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