Worksheet: Center of Mass of Shapes in 3D

In this worksheet, we will practice finding the position of the center of mass of some uniform objects in three dimensions.

Q1:

A frustum of a solid right circular cone has circular faces of radii 2 cm and 4 cm and a height of 5 cm. A cylindrical hole of radius 1 cm was bored down the axis of symmetry of the frustum from one circular face to the other. Find the distance between the centre of mass of the solid formed and the centre of its larger face.

  • A 4 9 1 0 cm
  • B 3 7 2 0 cm
  • C 4 1 2 0 cm
  • D 1 9 1 0 cm
  • E 3 cm

Q2:

A uniform solid hemisphere of radius 4 cm and a uniform solid right circular cylinder of height 9 cm and radius 4 cm are joined by their bases to form a composite solid. The centres of the bases coincide at the point 𝑂 . Given that the hemisphere and the cylinder have the same density, determine the distance between 𝑂 and the centre of mass of the composite solid.

  • A 3 3 1 0 cm
  • B 9 7 0 cm
  • C 2 9 1 0 cm
  • D 2 1 9 7 0 cm
  • E 2 1 1 7 0 cm

Q3:

A uniform solid right circular cone of height 7 cm and base radius 3 cm and a uniform solid hemisphere of radius 3 cm are joined by their bases. The centres of their bases coincide at a point 𝑂 . Given that the hemisphere’s density is twice that of the cone, find the distance from 𝑂 to the centre of mass of the composite body.

  • A 1 2 cm
  • B 1 1 3 8 cm
  • C 1 6 8 cm
  • D 5 7 6 cm
  • E 3 1 3 8 cm

Q4:

A frustum of a right circular cone has radii 1 cm and 2 cm and height 8 cm. Find the distance between the frustum’s centre of mass and the centre of its cone’s base.

  • A 4 4 7 cm
  • B 2 7 7 cm
  • C 8 6 2 1 cm
  • D 2 2 7 cm
  • E 6 7 cm

Q5:

The base of a uniform hemisphere of radius 9 cm is joined to the top plane face of a uniform solid right circular cylinder of the same radius and height 6 cm. The centres of their adjoining circular faces coincide at the point 𝑂 . Given that the density of the hemisphere is five times that of the cylinder, determine the distance between the resulting solid’s centre of mass and 𝑂 .

  • A 7 3 2 cm
  • B 3 1 1 6 cm
  • C 1 1 6 cm
  • D 3 7 1 6 cm
  • E 1 3 4 cm

Q6:

The uniform solid frustum of a right circular cone is suspended by a string attached to a point on the rim of its smaller circular face. The two circular faces of the frustum have radii 3 cm and 5 cm, and the height of the frustum is 8 cm. Find, to the nearest degree, the angle between the axis of the frustum and the vertical when it hangs in equilibrium.

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