W9.4 Another Balancing Act

The system in this challenge consists of a thin rod with a heavy brass collar that can be slid and fixed at a position along the length of the rod.

Graphic - Rod And Mass setup

The question is: I want to balance the rod vertically on the end of one finger. Where should the brass collar be fixed to enable me to do this most easily?

Should it be

  1. At the bottom
  2. In the middle
  3. At the top
  4. Or does it not matter?
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Hint

Think about what you have to give the rod as it begins to rotate about the balance point on your finger…
 

Solution

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Solution

Consider the following situation, as the rod is beginning to rotate clockwise about the balance point.

Graphic - Rod And Mass Solution setup

The weight (mg) of the collar produces a torque on the rod, causing an angular acceleration, about my finger. What we have to do is to provide an equal and opposite angular acceleration to keep the rod and collar exactly vertical.

This will be easiest if the angular acceleration is as small as possible, giving us time to react and provide an opposite angular acceleration before the mass is heading towards the floor and we cannot reverse its direction of travel.

So, the problem is really, where does the collar need to go to minimise the angular acceleration of the system?

The answer is at the top , as shown in the diagram.

The reasoning is as follows....

Recall the angular form of Newton’s 2nd Law (Key Point 5.18)

$$\tau = I \alpha \rightarrow \alpha = \frac{\tau}{I}$$

  • The torque from the mass of the collar (Key Point 5.14) is proportional to r
  • The moment of inertia (Key Point 5.12) of the collar is proportional to r2, thus
  • The angular acceleration is inversely proportional to r

Meaning…a large r results in the smallest angular acceleration to counteract, the opposite of what you might have thought…