Environmental Engineering Reference
In-Depth Information
rotational and translational kinetic energies. In Section 3.3.1 we showed (at least
for a system of two particles
4
) that the total kinetic energy is equal to
1
2
MV
C
+
1
2
I
C
ω
2
,
K
=
(4.54)
where the kinetic energy of rotation is calculated about a fixed axis through the
centre of mass. Note that this expression only holds if the rotation axis passes
through the centre of mass. This is so since the second term, which represents the
kinetic energy of the body in the centre-of-mass frame, only reduces to
I
C
ω
2
/
2if
the centre-of-mass lies on a fixed axis of rotation.
Example 4.6.1
Determine an expression for the total kinetic energy of a solid
sphere of mass M and radius R that is rolling without slipping on a flat surface at
speed v.
Solution 4.6.1
The key to this problem is to recognise that the 'rolling without
slipping' aspect of the motion implies that ω and v are connected by v
Rω. This
means that the rotational and translational motions are no longer independent. Thus
=
2
I
C
v
2
1
2
Mv
2
1
K
=
+
R
2
.
2
5
MR
2
The moment of inertia of a uniform solid sphere about its centre is
so we
have
1
2
Mv
2
1
5
Mv
2
7
10
Mv
2
.
K
=
+
=
PROBLEMS 4
4.1 A turntable that rotates at a rate of 33
3
revolutions per minute has a mass
of 1.00 kg and a radius of 0.13 m. Assuming the turntable to be a uniform
disc, calculate the torque required if the operating speed is to be achieved in
a time of 2 seconds after it is switched on. The turntable then spins freely
and a lump of plasticine (mass 20 g) is dropped and sticks to the turntable
10 cm from the centre. What is the new angular frequency?
4.2 A stuntman stands on the roof of a bus, which is travelling at speed
v
around
a circular bend of radius
r
. The stuntman's feet are a distance 2
a
apart, he
has mass
m
, and his centre of mass is a height
h
above the roof of the bus.
Obtain expressions for the normal forces acting on each of his feet. Assume
that the roof of the bus remains horizontal and that his feet are equidistant
from the vertical axis through his centre of mass.
4.3 A thin circular ring of radius
R
and mass
m
lies in the horizontal plane on
a frictionless surface. It is free to rotate about a vertical axis fixed at some
point on the circumference. A bug (mass
m
b
) walks from the axis around the
ring at a constant speed
v
relative to the ring. Obtain an expression for the
angular speed of the ring when the bug is directly opposite the axis.
4
You may like to confirm that the result generalizes to any number of particles.
