Centre of mass – The point through which the mass of an object is concentrated. Moment – A measure of the turning effect of a force on a rigid body. Coplanar – Acting in the same plane. Lamina – A 2D object whose thickness can be ignored. Uniform – Mass is distributed evenly. Density – Often denoted by ρ. In 2D, a measure of mass per unit area, units kg/m 2 In 3D, a measure of mass per unit volume, units kg/m 3 Terminology and Modelling x y x y a R b a x b y = f (x) f (x) } x y y x y y d d R c c x = u(y) u(y) } V x = π∫ a b xy 2 dx where V = π∫ a b y 2 dx V y = π∫ c d yx 2 dy where V = π∫ c d x 2 dy Volume of revolution 3D Composite body A composite solid body consists of a uniform cone capped by a uniform hemisphere. The total height of the solid is 4r cm, where r represents the common radius. The ratio of the density of the hemisphere to that of the cone is 3:1. The following results are not provided: Diagram Body Volume/ Curved Surface Area Height of COM above base Solid sphere, radius r 4 3 πr 3 r Solid hemisphere, radius r 2 3 πr 3 3 8 r Hollow hemisphere, radius r 2πr 2 1 2 r Solid cone or pyramid, height h, radius r 1 3 πr 2 h 1 4 h Hollow cone or pyramid, height h, radius r πrl (l is sloping height) 1 3 h Moments about vertical through vertex 3πr 3 ρ × h = πr 3 ρ × 9 4 r + 2πr 3 ρ × 27 8 r h = 3r 4r r Shape Mass Distance from Vertex 2 3 πr 3 × 3ρ (= 2πr 3 ρ) 3r + 3r 8 ( = 27 8 r ) 1 3 πr 2 × 3r × ρ (= πr 3 ρ) 3 4 (3r) = 9 4 r 4r r πr 3 ρ + 2πr 3 ρ (= 3πr 3 ρ) h h is the distance of the centre of mass vertically above the vertex of the cone. 3D Problems A level Further Maths 6.3 Moments and Centre of Mass From page 8 of the formula booklet. ACB is the diameter of a semi-circular lamina with centre C. A right-angled triangular lamina ADB is added as well as a particle at C to form a uniform lamina, as shown below. The particle at C has a mass equal to twice that of the semi-circle. B a 3a A D C Shape Area/mass Distance from AB Distance from AD 1 2 πa 2 ρ (–) 4a 3π a 3a 2 ρ a 2a 3 πa 2 ρ 0 a B a 3a A D C a 2 ρ ( 3 2 π + 3 ) x y Moments about AB Moments about AD a 2 ρ ( 3 2 π + 3 ) x = ( 1 2 πa 2 ρ ) ( – 4a 3π ) + (3a 2 ρ)(a) x = 7 3 ( 3 2 π+ 3 ) a a 2 ρ ( 3 2 π + 3 ) y = ( 1 2 πa 2 ρ ) (a) + (3a 2 ρ) ( 2a 3 ) + (πa 2 ρ)(a) y = ( 3 2 π + 2 ) ( 3 2 π + 3 ) a 2D Problems Centres of Mass of Uniform Bodies Triangular lamina: 2 3 along median from vertex Semi circle: 4r 3π from straight edge along axis of symmetry Quarter circle: x = 4r 3π y = 4r 3π from vertex Suspension from a fixed point 3a a a x y (x, y) a – y 2a – x θ a – y x θ α y x θ α y 3a – x θ tan θ = a – y 2a – x tan θ = y 3a – x tan θ = a – y x tan θ = y x tan a = x a – y tan a = x y