Environmental Engineering Reference
In-Depth Information
4.4 A uniform circular disc of mass
m
rolls without slipping such that the linear
velocity of the centre of mass is
v
. Show that the kinetic energy is given by
K
3
mv
2
/
4
.
4.5 A uniform thin beam of length
l
and mass
m
is pivoted at one end and
supported at the other. The beam is initially horizontal before the support is
removed and the beam rotates (under gravity) in a vertical plane on the pivot.
=
(a) Obtain an expression for the force acting on the beam due to the pivot
before the support is removed.
(b) Show that the instantaneous force acting at the pivot immediately after
the support is removed is
F
=
mg/
4.
(c) By considering the mechanical energy of the system show that the angular
speed of rotation about the pivot when the beam makes an angle
θ
to the
horizontal is given by
3
g
sin
θ
l
ω
=
.
4.6 A uniform cylindrical drum of radius
b
and mass
m
rolls without slipping
down a plane inclined at an angle
θ
to the horizontal. Find the acceleration
of the centre of mass of the drum.
Use the principle of conservation of energy to determine the speed of the
centre of mass after the drum has rolled a distance
x
down the slope. Show
that this result is consistent with the expression for the linear acceleration that
you just determined.
4.7 Show that the moment of inertia of a solid sphere of mass
m
and radius
R
is
2
5
mR
2
. A spin bowler is able to impart a frictional force of 10 N on the seam
of a cricket ball (mass 0.15 kg and radius 4.0 cm) for 0.1 seconds. Estimate
the rotational speed of the ball.
4.8 The height of the cushion on a snooker table is chosen to be
5
R
,where
R
is
the radius of the snooker ball. This unique choice of height enables the ball
to roll without slipping when it rebounds. Prove this result. You will need to
use the result that the moment of inertia of the snooker ball about an axis
through the centre of mass is
2
5
MR
2
,
where
M
is the mass of the snooker
ball.
