Transcript ClassicalMechanics_4..

Physics 1901 (Advanced)
A/Prof Geraint F. Lewis
Rm 560, A29
[email protected]
www.physics.usyd.edu.au/~gfl/Lecture
Semester 1 2008
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Rotational Motion
So far we have examined linear motion;
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
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Newton’s laws
Energy conservation
Momentum
Rotational motion seems quite different, but is
actually familiar.
Remember: We are looking at rotation in fixed
coordinates, not rotating coordinate systems.
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Rotational Variables
Rotation is naturally described in polar
coordinates, where we can talk about
an angular displacement with respect to
a particular axis.
For a circle of radius r, an angular
displacement of  corresponds to an arc
length of
Remember: use radians!
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Angular Variables
Angular velocity is the
change of angle with time
There is a simple relation
between angular velocity
and velocity
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Angular Variables
Angular acceleration is
the change of  with
time
Tangential acceleration
is given by
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Rotational Kinematics
Notice that the form of rotational relations is the same as
the linear variables. Hence, we can derive identical
kinematic equations:
Linear
Rotational
a=constant
=constant
v=u+at
=o+ t
s=so+ut+½at2
=o+ot+½t2
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Net Acceleration
Remember, for circular
motion, there is always
centripetal acceleration
The total acceleration is
the vector sum of arad
and atan.
What is the source of arad?
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Rotational Dynamics
As with rotational kinematics, we will see that the
framework is familiar, but we need some new
concepts;
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Linear
Rotational
Mass
Moment of Inertia
Force
Torque
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Moment of Inertia
This quantity depends
upon the distribution of
the mass and the
location of the axis of
rotation.
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Moment of Inertia
Luckily, the moment of inertia is typically;
where c is a constant and is <1.
Object
Solid sphere
Hollow sphere
Rod (centre)
Rod (end)
I
2/5 M R2
2/3 M R2
1/12 M L2
1/3 M L2
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Energy in Rotation
To get something moving, you do work on it, the
result being kinetic energy.
To get objects spinning also takes work, but what
is the rotational equivalent of kinetic energy?
Problem: in a rotating object, each bit of mass
has the same angular speed , but different linear
speed v.
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Energy in Rotation
For a mass at point P
Total kinetic energy
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Parallel Axis Theorem
The moment of inertia depends upon the mass
distribution of an object and the axis of rotation.
For an object, there are an infinite number of
moments of inertia!
Surely you don’t have to do an infinite number of
integrations when dealing with objects?
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Parallel Axis Theorem
If we know the moment of
inertia through the centre of
mass, the moment of inertia
along a parallel axis d is;
The axis does not have to
be through the body!
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Torque
Opening a door requires not only an application of
a force, but also how the force is applied;
 It is ‘easier’ pushing a door further away
from the hinge.
 Pulling or pushing away from the hinge
does not work!
From this we get the concept of torque.
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Torque
Torque causes angular
acceleration
Only the component of
force tangential to the
direction of motion has
an effect
Torque is
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Torque
Like force, torque is a vector quantity (in fact, the
other angular quantities are also vectors). The
formal definition of torque is
where the £ is the vector cross product.
In which direction does this vector point?
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Vector Cross Product
The magnitude of the
resultant vector is
and is perpendicular to
the plane containing
vectors A and B.
Right hand grip rule defines the direction
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Torque and Acceleration
At point P, the tangential
force gives a tangential
acceleration of
This becomes
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Torque and Acceleration
For an arbitrarily shaped object
We have the rotational equivalent of Newton’s
second law!
Torque produces an angular acceleration.
Notice the vector quantities. All rotational
variables point along the axis of rotation.
(Read torques & equilibrium 11.0-11.3 in textbook)
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Rolling without Slipping
For a rolling wheel which does not slide, then the distance it
travels is related to how much it turns.
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Rolling without Slipping
The total kinetic energy is
and
Where C is the constant of the Moment of Inertia
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Rolling without Slipping
Conservation of energy
 Independent of mass & size
 Any sphere beats any hoop!
What is the source of torque?
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Rolling without Slipping
Torque is provided by
friction acting at the
surface (otherwise the
ball would just slide).
Note that the normal
force does not
produce a torque
(although it can with
deformable surfaces
and rolling friction).
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Rotational Work
In linear mechanics, the
work-kinetic energy
theorem can be used to
solve problems.
In rotational mechanics,
we note that a force,
Ftan, applied to a point
on a wheel always
points along the
direction of motion.
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Rotational Work
If the torque is constant
Hence, we now have a rotational work-kinetic
energy theorem, except
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Angular Momentum
In linear dynamics, complex interaction
(collisions) can be examined using the
conservation of momentum.
In rotational dynamics, the concept of angular
momentum similarly eases complex interactions.
(Derivation similar to all other rotational quantities)
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Angular Momentum
In linear dynamics:
In rotational dynamics:
Hence, the net torque is equal to the rate of
change of angular momentum. Hence, if there is
no net torque, angular momentum is conserved.
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Angular Momentum
We can change the
angular velocity by
modifying the moment
of inertia.
Angular momentum is
conserved, but where
has the extra energy
come from?
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Angular Momentum
I have to apply a
force on the
mass to change
its linear velocity.
Through NIII, the
mass applies a
force on me.
For every torque there is an equal and opposite retorque.
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Angular Momentum
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Angular Momentum
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Angular Momentum
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Angular Momentum
Consider a lecturer on a
rotating stool holding a
spinning wheel, with
the axis of the wheel
pointing towards the
ceiling.
What happens when the
wheel is turned over?
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Angular Momentum
As with linear momentum, we
can use conservation of
angular momentum without
having to worry about the
various (internal) torques in
action.
External torques will change
the value of the total angular
momentum.
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Linear & Angular Momentum
What is the angular momentum
of an object moving along a
straight line?
Objects moving linearly have constant angular momentum.
Rotational mechanics is linear mechanics in a different
coordinate system.
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