AM1022 2014 Problem Sheet 3
L. Floyd
1
Centres of Mass
1. Compute the location of the centre of mass of a thin rod whose density is described by

1 + x, 0 ≤ x ≤ 1
ρ(x) = 3
x
 + , 1 ≤ x ≤ 2
2
2
2. Compute the location of the centre of mass of a thin rod whose density is described by

1 − x 2 , 0 ≤ x ≤ 1
ρ(x) = 1
x
 + , 1 ≤ x ≤ 2
2
2
3. Compute the location of the centre of mass of a thin rod whose density is described by
(
x3 , 0 ≤ x < 2
ρ(x) =
2, 2 ≤ x ≤ 3
4. A thin uniform rod is bent into the shape of a three-quarter circle of radius R. Determine the
position of the centre of gravity.
5. Compute the location of the centre of mass of the lamina bounded above by the curve f (x) = cos x,
and bounded below by the x-axis. Assume x ≥ 0.
6. Compute the location of the centre of mass of the lamina bounded above by the curve f (x) = sin x,
and bounded below by the x-axis. Assume x ≥ 0.
7. Compute the location of the centre of mass of the lamina bounded above by the curve f (x) = cos x,
and bounded below by the curve g(x) = sin x. Assume x ≥ 0.
x2
4a
+ √ and and bounded below by the x-axis on
a
5
the interval [−2a, 2a]. Compute the area bounded by the lamina. Find the location of the centre
of gravity of the lamina.
8. A lamina is bounded above by the curve y =
9. An object is constructed from two lamina. The first lamina is bounded above by the curve f1 (x) =
√
1
(x − 1) − (x − 1)3 and bounded below by the x-axis over the interval [1, 1 + 6]. The second
6
x 1
4 x
lamina is bounded above by the curve g1 (x) = − and bounded below by g2 (x) = − over
6 3
3 6
the interval [4, 8]. What is the centre of mass of the composite object.
1
10. Find the centre of mass of the object in the figure.
2b
2a
11. A lamina is bounded above by the curve
(x
g(x) =
, 0 ≤ x ≤ 4
2
√
8 − x, 4 ≤ x ≤ 8
and below by the x-axis. Compute the area bounded by the lamina. Compute the location of
the centre of mass of the lamina. Use Pappus’ theorem to find the volume of the solid formed by
rotating the lamina about the x-axis.
12. The diagram below shows a cone of height h and base radius R surmounted by a hemi-sphere of
radius R. Assuming the structure has uniform density compute the location of the √
centre of mass
of the combined solids, i.e. find ẑ in (0, 0, ẑ). What is the location of ẑ when h = 3 R?
R
h
2