A-LevelMathematicsMechanics9709/51 · Oct/Nov 2017 · Q69 Marks

A solid object consists of a uniform hemisphere of radius 0.4m attached to a uniform cylinder of radius 0.4 m so that the circumferences of their circular faces coincide. The hemisphere and cylinder each have weight 20 N. The centre of mass of the object lies at the centre O of their common circular face.

(i)Show that the height of the cylinder is 0.3 m.[2]
(ii)Find the distance of the centre of mass of the new object from O.[4]
(iii)The new object is placed with its hemispherical part on a rough horizontal surface. The new object is held in equilibrium by a force of magnitude P N acting along its axis of symmetry, which is inclined at 30° to the horizontal. Find P.[3]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2017). The mark or mark type is shown in brackets.

[M1] 20 × 3 × 0.4/8 = 20 × h/2 Takes moments about the common surface.
[A1] h = 0.3 m AG
[B1] Cylinder moment = 10 × 0.15/2
[M1A1] 20 × 3 × 0.4/8 – 10 × 0.15/2 = 30x Takes moments about the base of the cylinder.
[A1] x = 0.075 m
[M1A1] 30 × 0.075sin60 = P × 0.4sin60 Takes moments about point of contact of the cylinder with the surface.
[A1] P = 5.625

On Oracle Prep you can write or photograph your own answer and have it marked by AI against this mark scheme, with feedback on what you missed.