Class Notes Physics I, 11/11/98

Stable and Unstable Equilibrium; Rotational KE

If, however, the object is tilted, the action of the weight vector does exert a rotational influence about the point on the base of the object which is in contact with the level service.

• In the second figure below, it should be clear that the weight of the object will tend to rotate it in a clockwise direction around its point of contact with the service.
• The object will therefore tend to rotate in the clockwise direction whenever the line of action of the weight vector falls outside the point of contact.
• The result of this torque will be to cause the object to rotate further and further from its equilibrium position, and hence the fall.
• If the line of action falls inside the point of contact, the rotation will be in the counterclockwise direction, and the object will tend to return to its equilibrium position.

If we compare this situation with the object shown below, we see that compared to the previous object it takes a much smaller rotation to result in the clockwise torque necessary to make the object fall.  That is, we don't have to rotate the object very far it all in order for the line of action of its weight to file outside the point contact.

• The axis of symmetry of the beam is marked with a purple dotted line, which is aligned with a central axis of the marker.
• The center of mass will be at the central point of the beam, assuming a uniform distribution of mass throughout the beam, and is indicated by the + at this point.

In the second figure below, the beam has been tilted and the line of action of its weight force has moved past the point contact.

• The beam will therefore tend to rotate in the clockwise direction and thus to topple.

In the second figure below, we depict the same situation, but with a pack of washers attached below the beam.

• As the beam rotates to the right, we can see that its center of mass moves to the right and in the picture moves slightly outside of the point contact.
• The torque of the beam will then tend to rotate the system clockwise.
• The washers, however, move in the opposite direction, with their center of mass moving to the left and tending to rotate the system counterclockwise about the point of contact.
• The center of mass of the beam itself (not including the washers) will therefore have to move significantly past the point of contact before the beam can topple, in contrast to the situation before the washers were attached, when any displacement of the center of mass beyond the point contact would cause disaster.

A closer look at this system shows how the two forces provide opposing rotational influences.

• From this change in altitude we can calculate the change in the potential energy of the washers.
• By making time and angular position measurements on the beam, we were able to determine the angular velocity attained by the beam.

Using your measurements of the mass of the beam, and the angular velocity attained by the beam, assume that the beam consists of 8 equal pieces and determine the approximate resulting kinetic energy of each piece. Compare with the potential energy loss of the washers.

http://youtu.be/vGSO5623Znc