# Worksheet: Center of Mass of Particles

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
• B
• C
• D

Q2:

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

Q3:

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

Q4:

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

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 Mass 𝐴 𝐵 𝐶 13 kg 6 kg 15 kg
• A
• B
• C
• D

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 Mass 𝐴 𝐶 𝐸 𝐹 9 kg 5 kg 4 kg 3 kg
• A
• B
• C
• D

Q7:

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

Q8:

The figure shows a system of point masses placed at the vertices of a hexagon of side length . The mass placed at each point is detailed in the table. Determine the coordinates of the center of gravity of the system. Position Mass at 𝐴 at 𝐹 at 𝐷 at 𝐶 18 g 26 g 6 g 30 g
• A
• B
• C
• D

Q9:

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

Q10:

Four masses of 630 g are placed at the vertices of a square of side length . Determine the ordered pairs of the center of gravity of the system relative to the axes and .

• A
• B
• C
• D

Q11:

A square 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 . When the mass at vertex is removed, the center of mass of the system is . Find the coordinates of the center of mass of the two systems and . • A,
• B,
• C,
• D,

Q12:

The figure shows a system of point masses . The mass of each point is detailed in the table. Find the coordinates of the center of gravity of the system. Position Mass at 𝐴 at 𝐵 at 𝐶 at 𝐷 𝑚 kg 𝑚 kg 𝑚 kg 𝑚 kg
• A
• B
• C
• D

Q13:

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

Q14:

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

• A cm
• B cm
• C cm
• D cm

Q15:

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

• A
• B
• C
• D
• E

Q16:

Four particles are positioned at the points , , , and . The center of mass of the four particles is the point . Given that the masses of the four particles are , , , and respectively, find the value of .

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 of mass 4 kg is located at the origin, particle of mass 6 kg at , and particle of mass 10 kg at . Determine the coordinates of the center of mass of the three particles.

• A
• B
• C
• D
• E

Q19:

Three masses of 5 kg, 3 kg, and 9 kg are on the -axis at the coordinates , , and 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
• B
• C
• D
• E

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 Mass 𝐴 𝐵 𝐶 𝐷 75 kg 29 kg 71 kg 85 kg
• A
• B
• C
• D

Q21:

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

Q22:

Six masses of 70, 30, 70, 50, 70, and 10 kilograms are placed at the vertices , , , , , and 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 cm
• B cm
• C cm
• D cm

Q23:

The figure shows a system of point masses. The mass placed at each point is detailed in the table. Given that and are on the same horizontal line, find the coordinates of the centre of gravity. Position Mass at 𝐴 at 𝐵 at 𝐶 at 𝐷 7𝑚 kg 9𝑚 kg 4𝑚 kg 4𝑚 kg
• A
• B
• C
• D

Q24:

A triangle , where , , , and and are the midpoints of and respectively, is located in the first quadrant of a Cartesian plane such that is at the origin, and the point is on -axis. Three equal masses are placed at points , , and . Determine the coordinates of the centre of gravity of the system. • A
• B
• C
• D

Q25:

Find the coordinates of the centre of gravity of the following system: at , at , and at .

• A
• B
• C
• D