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Worksheet: Center of Mass

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 π‘₯ values and displacement above the origin corresponding to positive 𝑦 values.

Find the π‘₯ coordinate of the centre of mass system.

  • A βˆ’ 1 . 3 cm
  • B – 1 . 4 cm
  • C βˆ’ 1 . 1 cm
  • D βˆ’ 1 . 2 cm
  • E βˆ’ 1 . 0 cm

Find the 𝑦 coordinate of the centre of mass system.

  • A 0.69 cm
  • B 0.91 cm
  • C 0.26 cm
  • D βˆ’ 0 . 2 6 cm
  • E 1.1 cm

Q2:

If half of the population of Earth were to somehow be transferred to the Moon, the position of the centre 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 . 0 0 Γ— 1 0 9 people. Use 5 . 9 7 Γ— 1 0 2 4 kg for the mass of Earth, 7 . 3 4 Γ— 1 0 2 2 kg for the mass of the Moon, and 3 . 8 4 Γ— 1 0 5 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 centre of mass of the Earth-Moon system?

  • A 29.5 nm
  • B 23.7 nm
  • C 42.0 nm
  • D 13.3 nm
  • E 55.2 nm

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 centreline.

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

  • A 2.0 cm
  • B 1.1 cm
  • C βˆ’ 1 . 3 cm
  • D 0 cm
  • E βˆ’ 1 . 9 cm

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

  • A 86 cm
  • B 80 cm
  • C 76 cm
  • D 68 cm
  • E 97 cm

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 centre of mass of the particles?

  • A 18 cm
  • B 15 cm
  • C 27 cm
  • D 21 cm
  • E 32 cm

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 π‘₯ -coordinate of the particle moving in the 5.0 cm radius circle is given by π‘₯ ( 𝑑 ) = 5 . 0 ( 2 𝑑 ) c o s and the 𝑦 -coordinate is given by 𝑦 ( 𝑑 ) = 5 . 0 ( 2 𝑑 ) s i n . The π‘₯ -coordinate of the centre of mass of the particles is given by π‘₯ ( 𝑑 ) = 6 . 0 ( 2 𝑑 ) c m c o s and the 𝑦 -coordinate of the centre of mass of the particles is given by 𝑦 ( 𝑑 ) = 6 . 0 ( 2 𝑑 ) c m s i n . Find 𝑅 .

  • A 5.0 cm
  • B 2.5 cm
  • C 7.5 cm
  • D 6.5 cm
  • E 12 cm

Q7:

A cubic volume of side length π‘Ž = 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 π‘₯ - and 𝑦 -coordinates of the center of mass of the cube? Assume that the solid cube is of uniform density.

  • A ( 1 . 6 , 1 . 6 ) m
  • B ( 3 . 1 , 3 . 1 ) m
  • C ( 3 . 1 , 1 . 5 ) m
  • D ( 1 . 5 , 1 . 5 ) m
  • E ( 3 . 0 , 1 . 0 ) 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 = 1 5 cm and a radius π‘Ÿ = 3 . 5 1 cm. The sphere has a radius π‘Ÿ = 4 . 0 2 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 𝐡 , 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.