Transcript Rotational Dynamics powerpoint

4. Rotational Dynamics
Rotational Motion and
Astrophysics
Advanced Higher
Moment of a Force
The moment of a force is the turning
effect it can produce – for example a long
handled screwdriver.
A force F is applied on the
screwdriver and it is also
applied perpendicularly to the
turning point.
The force is applied a
distance d from the turning
point.
Moment = F x d
Torque
When a force is applied and this causes a
rotation about an axis, this moment of force is
then known as the Torque.
The force is applied to the
object at a certain distance from
the pivot point.
T  Fr
T  Fr sin 
Perpendicular
Not Perpendicular
Unit (Nm)
Moment of Inertia
The moment of inertia, I, of an object is
described as its “resistance to change in its
angular motion”
The moment of inertia is dependant on the
mass of the object and the distribution of
mass about the axis of rotation.
It can also depend on what type of object it
is.
Moment of Inertia Formulae
Point Mass
I  mr
2
Rod about end
1 2
I  ml
3
Sphere about centre
2 2
I  mr
5
Rod about centre
1
2
I
ml
12
Disk about centre
1 2
I  mr
2
Example
1. A wheel has very light spokes. The
mass of the rim and tyre is 2 kg and
the radius of the wheel is 0.8 m.
Calculate the moment of inertia of the
wheel.
I  mr
I  2  0.8  0.8
2
I  1.28 kgm
-2
Unbalanced Torque
When an unbalanced torque is applied to
an object, this causes an angular
acceleration. The angular acceleration
produced depends on the unbalanced
torque and the moment of inertia.
T  I
Example
2. A cylindrical drum is free to rotate about an axis
AB as shown below.
The radius of the drum is 0.3 m and the moment of
inertia is 0.4 kgm-2. A rope of length 5 m is wound
round the drum and pulled with a constant force of
8 N.
(a) Calculate the torque on the drum
(b) Determine the angular acceleration
(c) Calculate the angular velocity of the drum just as
the rope leaves the drum. Assume is starts from
rest.
(a)
(c)
Solutions
T  Fr
T  8  0.3
T  2.4 Nm
(b)
T  I
T 2.4
-2
 
 6 rads
I 0.4
5  2

 16.6 rad
2r
  0  2
2
2
  0  2  6 16.6  14.1 rads -1
Angular Momentum
Similar to linear momentum, but the
object must be rotating.
The angular momentum L of a particle
about an axis is defined as the moment of
momentum
Linear Momentum p  mv
Moment of Momentum p  mvr
Angular Momentum L  mvr  mr  since v  r
2
Moment of Inertia I  mr
2
L  I
Units  kgm s
2 -1
Conservation of Angular
Momentum
Just like with linear momentum, memorise
this statement!
“The total angular momentum before and
impact will equal the total angular
momentum after an impact providing
there are no external torques acting!
Rotational Kinetic Energy
When an object rolls down a hill, it moves with
both translational (linear) kinetic energy and
rotational (angular) kinetic energy. For
conservation of energy, we must add these values
together to get the total kinetic energy
Ektot  Ektran  Ekrot
1 2 1 2
Ek  mv  I
2
2
Example
3. A shaft has a moment of inertia of 20
kgm2 about its central axis. The shaft is
rotating at 10 rpm. The shaft is locked
onto another shaft which is initially
stationery. The second shaft has a
moment of inertia of 30 kgm2
(a) Find the angular momentum of the
combination after the shafts are locked
together.
(b)What is the angular momentum after the
shafts are locked together.
(a)Angular Momentum Before  Angular Momentum After
L  I
10  2
L  20 
60
L  20.9 kgm 2s -1
(b) Angular Momentum Before  Angular Momentum After
I101  I 20 2  I1  I 2 
20 1.04  0  50
  0.41 rads -1