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Lesson Worksheet: Center of Gravity of Particles Mathematics

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

Q1:

In the given figure, three weights of magnitudes 2 N, 5 N and 3 N are placed on the vertices of an equilateral triangle of side length 8 cm.

Find the center of gravity of the system.

  • A65,265 cm
  • B265,435 cm
  • C265,635 cm
  • D635,265 cm
  • E435,265 cm

Q2:

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)

Q3:

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(16,0)
  • B(3.5,30)
  • C(3.5,0)
  • D(26.2,30)
  • E(16,30)

Q4:

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).

Q5:

Suppose three masses of 1 kg, 4 kg, and 6 kg are located at points whose position vectors are (6)ij, (29)ij, and (7+8)ij. Determine the position vector of the center of gravity for this system of masses.

  • A(4+)ij
  • B(4+)ij
  • C(+)ij
  • D(44)ij
  • E(+4)ij

Q6:

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.

Position𝐴𝐵𝐶𝐷
Mass75 kg29 kg71 kg85 kg
  • A3013,185
  • B4813,125
  • C125,4813
  • D185,3013

Q7:

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.

  • A𝐿3,𝐿3
  • B𝐿3,𝐿
  • C𝐿2,𝐿2
  • D(𝐿,𝐿)

Q8:

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 given that C is placed at the origin, and the scales of the axes are defined such that each unit represents 1 cm of distance.

  • A525,665
  • B787,665
  • C915,52835
  • D787,557

Q9:

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 center of gravity of the system.

  • A11,223
  • B443,11
  • C223,332
  • D223,11

Q10:

Suppose three masses of 6 kg, 9 kg, and 𝑚 kg are at points (5,9), (0,6), and (4,3) respectively. If the center of mass of the three masses is (1,𝑦), what are the values of 𝑚 and 𝑦?

  • A𝑚=5, 𝑦=8.2
  • B𝑚=3.75, 𝑦=6.36
  • C𝑚=3, 𝑦=7.8
  • D𝑚=5, 𝑦=5.85
  • E𝑚=3, 𝑦=6.5

This lesson includes 50 additional questions and 283 additional question variations for subscribers.

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