Worksheet: Center of Mass

In this worksheet, we will practice calculating the location of the center of mass of a system of objects with various masses and positions relative to each other.

Q1:

Three point masses are placed at the corners of a triangle, as shown in the accompanying figure. The origin of the system is defined to be at the position of the 150 g mass, with displacement to the right of the origin corresponding to positive 𝑥-values and displacement above the origin corresponding to positive 𝑦-values.

Find the 𝑥-coordinate of the center of mass of the system.

Find the 𝑦-coordinate of the center of mass of the system.

Q2:

If half of the population of Earth were to somehow be transferred to the Moon, the position of the center of mass of the Earth–Moon system would be slightly changed. Assume that the average mass of a human is 60.0 kg and the human population is 7.00×10 people. Use 5.97×10 kg for the mass of Earth, 7.34×10 kg for the mass of the Moon, and 3.84×10 m as the radius of the Moon’s orbit. Assume that the human population is evenly distributed over either Earth’s surface or the Moon’s surface. What is the magnitude of the change of the center of mass of the Earth–Moon system?

Q3:

The structure shown has a uniform thickness of 20 cm and a uniform density of 1.0 g/cm3. Assume an origin at the floor and at the structure’s centerline.

What is the horizontal distance from the origin of the center of mass of the object?

What is the vertical distance from the origin of the center of mass of the object?

Q4:

Two particles of masses 145 g and 210 g respectively are separated by a horizontal distance of 36 cm. How far from the 145 g mass particle is the center of mass of the particles?

Q5:

A 0.75 m long rod of iron with a density of 8.0 g/cm3 is joined end to end with a 0.75 m long rod of copper with a density of 2.7 g/cm3. If the rods have an equal cross-sectional area to each other, how far from the unjoined end of the iron rod is the center of mass of the object?

Q6:

Two particles of masses 2.0 kg and 4.0 kg move in uniform circles with radii of 5.0 cm and 𝑅 cm respectively. The 𝑥-coordinate of the particle moving in the 5.0 cm radius circle is given by 𝑥(𝑡)=5.0(2𝑡)cos and the 𝑦-coordinate is given by 𝑦(𝑡)=5.0(2𝑡)sin. The 𝑥-coordinate of the center of mass of the particles is given by 𝑥(𝑡)=6.0(2𝑡)cmcos and the 𝑦-coordinate of the center of mass of the particles is given by 𝑦(𝑡)=6.0(2𝑡)cmsin. Find 𝑅.

Q7:

A cubic volume of side length 𝑎 = 1.0 m is cut out of a solid cube of side length 𝑏 = 3.0 m, as shown in the diagram. What are the 𝑥- and 𝑦-coordinates of the center of mass of the cube? Assume that the solid cube is of uniform density.

  • A(1.5,1.5) m
  • B(3.0,1.0) m
  • C(1.6,1.6) m
  • D(3.1,1.5) m
  • E(3.1,3.1) m

Q8:

A system comprised of a sphere and a cylinder can be arranged in different ways, as shown in the diagram. The cylinder has a length 𝑙=15 cm and a radius 𝑟=3.5 cm. The sphere has a radius 𝑟=4.0 cm. The cylinder and the sphere have the same density. In arrangement (a), the axis of the cylinder along its length passes through the center of the sphere. In arrangement (b), the axis of the cylinder along the vertically directed radius of its circular face, horizontally half-way along the cylinder’s length, passes through the center of the sphere.

What are the 𝑥- and 𝑦-coordinates of the center of mass of the system in arrangement (a)?

  • A(0, 15) cm
  • B(0, 11) cm
  • C(15, 15) cm
  • D(0, 0) cm
  • E(11, 11) cm

What are the 𝑥- and 𝑦-coordinates of the center of mass of the system in arrangement (b)?

  • A(0, 0) cm
  • B(5.9, 5.9) cm
  • C(7.0, 7.0) cm
  • D(0, 7.0) cm
  • E(0, 5.9) cm

Q9:

Earth and the Moon form the Earth-Moon system, which has a center of mass at a point somewhere between the centers of Earth and the Moon. In modeling the center of mass of the Earth-Moon system, use a value of 5.97×10 kg for Earth’s mass, 7.36×10 kg for the Moon’s mass, and 3.82×10 km for the length of the line 𝐿 that intersects the centers of Earth and the Moon.

What is the magnitude of the displacement of the center of Earth from the center of mass of the Earth-Moon system?

  • A7.21×10 m
  • B4.65×10 m
  • C2.15×10 m
  • D1.91×10 m
  • E1.38×10 m

Determine how far from the surface of Earth 𝐿 intersects the center of mass of the Earth-Moon system. Use a value of 6.37×10 m for Earth’s radius and take displacement from Earth’s surface toward the Moon as corresponding to positive values.

  • A4.79×10 m
  • B1.58×10 m
  • C1.72×10 m
  • D1.58×10 m
  • E1.38×10 m

Q10:

Ions of sodium and chlorine form a cubic crystal lattice in the substance NaCl. The smallest possible cube, known as the unit cell, of NaCl consists of four sodium ions and four chlorine ions, as shown in the diagram. The length of one edge of the unit cell of NaCl is 2.36×10 m. Taking the left, front, and bottom corners of the unit cell as the origin of a coordinate system where displacement to the right corresponds to positive 𝑥-values, displacement backward corresponds to positive 𝑦-values, and displacement upward corresponds to positive 𝑧-values, state the position vector of the center of mass of the unit cell. Use a value of 35.45 𝑢 for the atomic mass of chloride ions and 22.99 𝑢 for the atomic mass of sodium ions.

  • A1.18×10+1.54×10+2.36×10ijk m
  • B1.18×10+1.18×10+1.18×10ijk m
  • C1.54×10+1.54×10+1.54×10ijk m
  • D1.18×101.18×101.18×10ijk m
  • E1.18×10+0.590×10+1.76×10ijk m

Q11:

A passenger car has a wheelbase of 2.5 m. When on a horizontal surface, the weight of the car is distributed with 52% of the weight supported by its front wheels and 48% supported by its rear wheels, as shown in the diagram. How far from the car’s rear axle is the car’s center of mass?

Q12:

The Sun, Earth, and the Moon form the Sun-Earth-Moon system, which has a center of mass at a point somewhere between the center of the Sun and the center of mass of the Earth-Moon system. In modeling the center of mass of the Sun-Earth-Moon system, use a value of 1.989×10 kg for the Sun’s mass, a value of 5.97×10 kg for Earth’s mass, a value of 7.36×10 kg for the Moon’s mass, and a value of 1.49×10 m for the separation from the center of the Sun to the center of Earth, and approximate the distance between Earth and the Moon as negligible compared to the Sun-Earth separation. What is the distance from the center of the Sun to the center of mass of the Sun-Earth-Moon system? Give your answer to two significant figures.

  • A4.4×10 m
  • B4.5×10 m
  • C9.1×10 m
  • D2.1×10 m
  • E1.8×10 m

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