a level further maths - resource.download.wjec.co.uk
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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
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