physics 1901 (advanced)

Semester 1 2009 http://www.physics.usyd.edu.au/~gfl/Lecture

Physics 1901 (Advanced)

Prof Geraint F. LewisRm 560, [email protected]/~gfl/Lecture


Rotational Motion

So far we have examined linear motion; 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.


Rotational VariablesRotation 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!


Angular VariablesAngular velocity is the change of angle with time

There is a simple relation between angular velocity and velocity


Angular Variables

Angular acceleration is the change of with time

Tangential acceleration is given by


Rotational KinematicsNotice 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+½t2


Net AccelerationRemember, 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?


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


Moment of Inertia

This quantity depends upon the distribution of the mass and the location of the axis of rotation.


Moment of InertiaLuckily, the moment of inertia is typically;

where c is a constant and is <1.

Object I

Solid sphere 2/5 M R2

Hollow sphere 2/3 M R2

Rod (centre) 1/12 M L2

Rod (end) 1/3 M L2


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.


Energy in RotationFor a mass at point P

Total kinetic energy


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?


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!


TorqueOpening 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.


Torque

Torque causes angular acceleration

Only the component of force tangential to the direction of motion has an effect

Torque is


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?


Vector Cross ProductThe magnitude of the resultant vector is

and is perpendicular to the plane containing vectors A and B.Right hand grip rule defines the

direction


Torque and AccelerationAt point P, the tangential force gives a tangential acceleration of

This becomes


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)


Rolling without Slipping

For a rolling wheel which does not slide, then the distance it travels is related to how much it turns.


Rolling without Slipping

The total kinetic energy is

and

Where C is the constant of the Moment of Inertia


Rolling without SlippingConservation of energy

Independent of mass & size

Any sphere beats any hoop!

What is the source of torque?


Rolling without SlippingTorque 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).


Rotational WorkIn 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.


Rotational Work

If the torque is constant

Hence, we now have a rotational work-kinetic energy theorem, except


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)


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.


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?


Angular MomentumI 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”.


Angular Momentum


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?id=696


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.


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.

physics 1901 (advanced)

Documents

axis of rotation

vector point

angular speed

angular displacement

vector quantity

particular axis

angular accelerationonly

tangential force