Worksheet: Center of Gravity of Particles in a Plane

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

Q2:

A rhombus 𝐴 𝐵 𝐶 𝐷 in which 𝐴 𝐶 = 2 𝐵 𝐷 = 8 c m 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 , 4 0 1 9
  • B 2 0 7 , 4 0 2 3
  • C 6 0 2 9 , − 4 1 3
  • D 6 0 2 3 , − 8 2 3

Q3:

The equilateral triangle 𝐴 𝐵 𝐶 in the figure has a side length is 36 cm. Point 𝐷 is the intersection of its medians (its centroid) and 𝐸 is the midpoint of 𝐵 𝐶 . 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 centre of gravity of the system.

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

Q4:

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

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

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 , 6 9 1 7
  • B 2 4 1 7 , 4 5 1 7
  • C 6 9 1 7 , 4 5 1 7
  • D 4 5 1 7 , 2 4 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 centre of gravity of the system.

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

Q7:

An equilateral triangle 𝐴 𝐵 𝐶 of side length 4 cm. The points 𝐷 , 𝐸 , and 𝐹 are the midpoints of 𝐵 𝐶 , 𝐶 𝐴 , and 𝐴 𝐵 , respectively. Weights of 5 kg, 3 kg, 10 kg, 7 kg, 2 kg, and 3 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 7 3 3 0 , 5 √ 3 7
  • B 5 3 1 5 , 2 √ 3 3
  • C 4 6 1 5 , 7 √ 3 8
  • D 7 3 3 0 , √ 3 2

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 centre 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 5 𝑙 , √ 3 1 0 𝑙
  • B √ 3 2 0 𝑙 , 1 1 0 𝑙
  • C √ 3 1 0 𝑙 , 1 5 𝑙
  • D 1 1 0 𝑙 , √ 3 2 0 𝑙

Q9:

A rectangle, 𝐴 𝐵 𝐶 𝐷 , in which 𝐴 𝐵 = 2 2 c m and 𝐵 𝐶 = 2 6 c m . Four masses of 6, 7, 5, and 9 g placed at the vertices 𝐴 , 𝐷 , 𝐵 , and 𝐶 respectively. Another mass of magnitude 8 grams is attached to the midpoint of 𝐴 𝐷 . Determine the coordinates of the centre of mass of the system.

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

Q10:

Four masses of 630 g are placed at the vertices of a square 𝐴 𝐵 𝐶 𝐷 of side length 𝐿 . Determine the coordinates of the centre of gravity of the system relative to the axes 𝐴 𝐵 and 𝐴 𝐷 .

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

Q11:

A square 𝐴 𝐵 𝐶 𝐷 has a sidelength of 70 cm. When four equal masses are placed at the vertices of the square, the centre of mass of the system is 𝐺 . When the mass at vertex 𝐴 is removed, the centre of mass of the system is 𝐺 ′ . Find the coordinates of the centre 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 𝐴 𝐵 𝐶 𝐷 . The mass of each point is detailed in the table. Find the coordinates of the centre of gravity of the system.

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

Q13:

A square 𝐴 𝐵 𝐶 𝐷 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 𝐴 𝐵 . Determine the coordinates of the centre of mass of the system.

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

Q14:

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

  • A 5 2 √ 2 3 cm
  • B 2 6 √ 3 3 cm
  • C 5 2 √ 3 3 cm
  • D 2 6 √ 2 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 centre of mass of the four particles?

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

Q16:

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

  • A ( 5 . 7 , 0 )
  • B ( 0 , 3 . 8 )
  • C ( 5 . 9 , 4 )
  • D ( 5 . 7 , 3 . 8 )
  • 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 centre of mass of all four masses to be at the origin.

  • A ( 5 , 0 )
  • B ( 0 , 2 )
  • C ( 2 , 0 )
  • D ( 0 , 5 )
  • E ( 2 4 , 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 centre of gravity of the system.

Position 𝐴 𝐵 𝐶 𝐷
Mass 75 kg 29 kg 71 kg 85 kg
  • A 1 2 5 , 4 8 1 3
  • B 3 0 1 3 , 1 8 5
  • C 4 8 1 3 , 1 2 5
  • 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 centre 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 centre of the hexagon and the centre of gravity of the system.

  • A 4 2 1 1 cm
  • B √ 1 0 cm
  • C 2 √ 3 1 cm
  • D 2 √ 3 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 1 1 √ 3 , 1 8
  • C 2 4 √ 3 , 1 4 4 5
  • D 1 8 , 1 1 √ 3

Q24:

A triangle 𝐴 𝐵 𝐶 , where 𝐴 𝐵 = 3 3 c m , 𝐵 𝐶 = 4 4 c m , 𝐶 𝐴 = 5 5 c m , and 𝐷 and 𝐸 are the midpoints of 𝐴 𝐵 and 𝐴 𝐶 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 2 2 3 , 3 3 2
  • B 4 4 3 , 1 1
  • C 1 1 , 2 2 3
  • D 2 2 3 , 1 1

Q25:

Find the coordinates of the centre of gravity of the following system: 𝑚 = 3 1 k g at ( 6 , 5 ) , 𝑚 = 7 2 k g at ( 1 , 2 ) , and 𝑚 = 6 3 k g at ( − 8 , 3 ) .

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

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