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/m2 In 3D, a measure of mass per unit volume, units kg/m3

Terminology and Modelling

x

y

x

y

aR

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

bxy2dx where V = π∫a

by2dx V y = π∫c

dyx2dy where V = π∫c

dx2dy

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 BodyVolume/Curved

Surface Area

Height of COM above

base

Solid sphere, radius r

43 πr3 r

Solid hemisphere, radius r

23 πr3

38 r

Hollow hemisphere,

radius r2πr2

12 r

Solid cone or pyramid, height h,

radius r

13 πr2h

14 h

Hollow cone or pyramid, height h,

radius r

πrl(l is sloping

height)

13 h

Moments about vertical through vertex

3πr3ρ × h = πr3ρ × 94 r + 2πr3ρ ×

278 r

h = 3r

4rr

Shape Mass Distance from Vertex

23 πr3 × 3ρ(= 2πr3ρ)

3r + 3r8

(= 278 r)

13 πr2 × 3r × ρ

(= πr3ρ)

34 (3r) =

94 r

4rr πr3ρ + 2πr3 ρ

(= 3πr3ρ)h

h is the distance of the centre of mass vertically above the vertex of the cone.

3D Problems

A level Further Maths6.3 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

3aA D

C

Shape Area/mass Distance from AB

Distancefrom AD

12

πa2ρ (–)4a3π a

3a2ρ a2a3

πa2ρ 0 aB

a

3aA D

C a2ρ (32 π + 3) x y

Moments about AB Moments about AD

a2ρ (32 π + 3) x = (12 πa2ρ) (– 4a3π )

+ (3a2ρ)(a)

x = 7

3(32 π+ 3) a

a2ρ (32 π + 3) y = (12 πa2ρ) (a) + (3a2ρ) (2a3 )

+ (πa2ρ)(a)

y = (32 π + 2)(32 π + 3)

a

2D Problems Centres of Mass of Uniform Bodies

Triangular lamina: 23

along median from vertex

Semi circle: 4r3π

from straight edge

along axis of symmetry

Quarter circle: x = 4r3π

y = 4r3π

from vertex

Suspension from a fixed point

3a

a

a

xy

(x, y)a – y

2a – xθ a – y

xθ

α

y

xθ

α

y

3a – xθ

tan θ = a – y2a – x

tan θ = y

3a – x

tan θ = a – y

x

tan θ = yx

tan a = x

a – y

tan a = xy