Mechanics · Moments
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Non-uniform rods and tilting
Beams whose weight does not act at the centre — finding the weight or the position of the centre of mass by taking moments — and "on the point of tilting" problems, where the reaction at the support about to be left is zero.
Mechanics · Moments
Non-uniform rods and tilting
Beams whose weight does not act at the centre — finding the weight or the position of the centre of mass by taking moments — and "on the point of tilting" problems, where the reaction at the support about to be left is zero.
Why it works
A uniform rod has its weight at the centre; a non-uniform one does not — it may be thicker at one end, or loaded unevenly. Its whole weight still acts at a single point, the centre of mass, but you don't know where that is in advance. Moments let you find it.Model the rod's weight as a single force acting at an unknown distance from one end. Each piece of information (a known reaction, or a balance point) gives a moment equation, and solving it pins down or . The method is the same as for a uniform beam — only the weight's position is now an unknown to find rather than a known centre.
On the point of tilting. A beam on two supports is held flat by both reactions. If you load one end heavily enough, the beam is about to tip up over the nearer support — and at that instant the other support is about to be left, so its reaction has dropped to zero: Setting that reaction to zero turns "on the point of tilting" into an ordinary moments problem. The body is pivoting about the support it's tilting on, so taking moments about that support is the clean choice.
The trickiest part is just deciding which way it tilts (and so which reaction vanishes): it tilts towards the overloaded end, lifting off at the far support.