Question

The centre of gravity of a right circular cone of diameter (d) and height (h) lies at a distance of ____________ from the base measured along the vertical radius.

• A. h/2
• B. h/3
• C. h/4
• D. h/6

Hint:

Remember that a solid cone's centroid is closer to the base than the midpoint, because the cone has more volume near its base.

Answer 1

The Correct Option is C

Step 1: For a solid right circular cone of height h, the centre of gravity lies on the central axis, and its position is a standard result from integrating the volume of thin circular slices from the base to the apex.

Step 2: Setting up the integral, if x is measured from the apex, the centroid works out to \(\bar{x} = \dfrac{3h}{4}\) from the apex, which means it is \(h - \dfrac{3h}{4} = \dfrac{h}{4}\) from the base.

Step 3: So the centre of gravity of a solid cone lies at h/4 from the base (equivalently 3h/4 from the apex), measured along the vertical axis.

Step 4: The diameter d does not affect this position at all, since the centroid along the height depends only on how the cross sectional area varies with height, not on the actual size of the base.

Step 5: h/2, h/3 and h/6 are values people often confuse with the centroid of a triangle (h/3 from the base) or a hemisphere, but for a solid cone the correct distance from the base is h/4.