1.3 Solutions
41
1.51 If the rod is to move with pure translation without rotation, then it should be
struck at C, the centre of mass of the loaded rod. Let C be located at distance
x from A so that
GC =
1
2 L − x, Fig. 1.30. Let M be the mass of the rod and 2M be attached
at A. Take torques about C
Fig. 1.30
2M x = M
L
2
− x
∴ x =
L
6
Thus the rod should be struck at a distance
L
6 from the loaded end.
1.52 Volume of the cone, V =
1
3 π R 2 h where R is the radius of the base and h
is its height, Fig. 1.31. The volume element at a depth z below the apex is
dV = πr 2 dz, the mass element dm = ρdV = πr 2 dz f
Fig. 1.31
dm = ρdv = ρπr
2 dz
z
r
=
h
R
∴ dz =
h
R
dr
For reasons of symmetry, the centre of mass must lie on the axis of the cone.
Take the origin at O, the apex of the cone:
Z CM =
Z dm
dm
=
R
0
hr
R
ρπr 2 h
r dr
1
3 π R 2 hρ
=
3h
4