Physics 1901 Advanced

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Physics 1901 (Advanced)

• Prof Geraint F. Lewis
• Rm 560, A29
• gfl_at_physics.usyd.edu.au
• www.physics.usyd.edu.au/gfl/Lecture

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Rotational Motion

• So far we have examined linear motion
• Newtons 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

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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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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 lt1.
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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 dont 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 x 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 Newtons
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?

http//www.physics.lsa.umich.edu/demolab/demo.asp?
id696
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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.