Transcript Document

Physics 1901 (Advanced)
A/Prof Geraint F. Lewis
Rm 557, A29
[email protected]
www.physics.usyd.edu.au/~gfl/Lecture
http://www.physics.usyd.edu.au/~gfl/Lecture
Rotational Motion
So far we have examined linear motion;
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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;
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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