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

Physics 1901 (Advanced)

A/Prof Geraint F. LewisRm 557, A29gfl@physics.usyd.edu.auwww.physics.usyd.edu.au/~gfl/Lecture

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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 SlippingConservation 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 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.

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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 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.

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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.