THIS POST WAS ORIGINALLY WRITTEN FOR MY OWN IB CLASS. THERE ARE HANDOUTS AND PROBLEMS HERE THAT WE DID IN CLASS BUT NEWCOMERS SHOULD FIND THEM HELPFUL. GO AHEAD AND TRY.
It’s useful to bear in mind that if you can do SUVAT problems, you should have no trouble with their circular equivalents.
Four handouts in total to download; please make sure you work through them carefully. Any difficulty, get in touch.
The main arguments here are the idea of rotational motion and torque as force x distance from pivot x sin(angle between them)
Moments of Inertia need not be calculated for this course – if necessary, you’ll be given them. However, here’s a little problem to think about. You have 2 balls of identical diameter and weight. One is solid, one is hollow. You can’t tell which is which just by knocking on them. Devise a simple way of finding out which one is the solid one (hint: think about the balls rolling down an inclined plane from the same height. Now, compare the moments of inertia of the two balls. The rest is conservation of energy so quite easy.) If you can’t write out the solution, message me for help.
This little animation is quite fun to glance at – the angular displacement is, however, in degrees, not radians, so be careful. Here’s a screenshot, showing displacement – time graphs for the ant and the ladybug – the constant time period implies constant angular velocity. Notice too that the ladybug leads the ant by 90 degrees
Notice, angular velocity is constant, but the linear speed of the ant and the ladybug are not the same. The ladybug, being closer to the axis of rotation has a smaller linear speed, because:(Notice the vertical displacements of the bugs execute SHM – AHLs will study this later)
- A review of circular-motion-1
- Key Ideas, torque and couple
- Basic concepts about moment-of-inertia
- From the Specimen Paper specimen-question-for-option-b
Moment of inertia I is defined as the ratio of the angular momentum L of a system to its angular velocity ω around a principal axis. Just as inertial mass is the ratio of linear momentum to speed – its resistance to acceleration, in other words: Angular momentum in a closed system is, just like its linear counterpart, conserved. The ice skater rotates faster when the arms drop to the sides because moment of inertia is reduced and thus angular velocity increases.
Look at the Wolfram demonstration
You will need to download the Wolfram CDF player in order to run the demo.
Practically, we find I by imagining a flat sheet of any shape like this having an infinite number of mass elements m at their respective distances r from the pivot, each contributing torque about the axis of rotation.
We have to add all the torques up, normally requiring integration. But, to keep it simple, we can write:
which is, as the handout shows, the basis for finding I for lots of other shapes and axes of rotation. Remember, you’ll be given I for a particular shape as required.
Bear in mind that when we do problems, the total energy of the system is the sum of the rotational and linear parts – important when we think about an object rolling (instead of sliding) down a hill, for example. Take a look at this solid-cylinder-rolling-down-an-inclined-plane. which runs through a few basic ideas plus some possible lab work.
Finally, for now, a use for all that stored energy.
The great flywheel on Richard Trevithick’s 1802 locomotive, used to level out the power supplied by a single cylinder. Rotational inertia kept the wheel turning.