A-LevelMathematicsMechanics9709/51 · Oct/Nov 2018 · Q26 Marks
A uniform object is made by attaching the base of a solid hemisphere to the base of a solid cone so that the object has an axis of symmetry. The base of the cone has radius 0.3 m, and the hemisphere has radius 0.2 m. The object is placed on a horizontal plane with a point A on the curved surface of the hemisphere and a point B on the circumference of the cone in contact with the plane [Figure 2].
- (i)Given that the object is on the point of toppling about B, find the distance of the centre of mass of the object from the base of the cone.[3]
- (ii)Given instead that the object is on the point of toppling about A, calculate the height of the cone. [The volume of a cone is ar²h. The volume of a hemisphere is r³.][3]
This question uses a diagram from the original paper, which isn't shown here. Open the full paper from the past papers archive to see it.
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2018). The mark or mark type is shown in brackets.
- [B1] cos0 = 0.2/0.3 (Axis makes an angle 0 with the horizontal)
- [M1] tan0 = x/0.3
- [A1] x = 0.335(41..)
- [M1] (π0.3²h/3)×(h/4) = (2π0.2³/3)(3×0.2/8) (Attempt to take moments about A)
- [A1] h = 0.231
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