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Consider a rod hinged at one end and moving in the vertical plane. If it's horizontal and free falls, we can apply energy conservation to find it's angular velocity at bottom as work done by hinge is 0.

Similarly, there are countless other examples, where we conserve energy and assume work done by hinge force to be 0 as there is no displacement of the hinged point, so $d=0 \rightarrow W=F×d=0$

So my doubt: Is hinge force always zero? Even in more complicated situations?

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    $\begingroup$ the hinge can make friction as it rotates and in such acase the force and the displacement are in the same line, so it can make work. just imagine a super rusted hinge that stops the door from reaching the bottom. $\endgroup$ Commented Mar 5 at 0:07
  • $\begingroup$ Thank you for that perspective, but that would only be in the case of non-ideal hinges, right? So when dealing with ideal hinges we won't consider that. $\endgroup$
    – Meet Shah
    Commented Mar 5 at 4:04

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In an ideal hinge, the hinge does no work when rotating about its axis. Obviously real hinges can do work. Wolphram Jonny points out that a rusty hinge can indeed slow down a door, showing that the rusty hinge indeed does work. A non-rusty hinge does less work. A well-lubricated hinge does even less. A hinge on an air bearing can have nearly no friction. Taking it to the extreme, an ideal hinge does no work.

An ideal hinge can do work if it moves in other directions besides its rotation axis. As a trivial example, if I have hinge whose axis is vertical (like a door hinge), and I use it to lift the door upwards, it must do work. It's only in the rotation axis that it cannot do any work.

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  • $\begingroup$ Thank you, that was a really good answer. So to make sure, when dealing with questions based on perfectly ideal hinges locked at one place, we can always apply energy conservation without a doubt, right? $\endgroup$
    – Meet Shah
    Commented Mar 5 at 4:05
  • $\begingroup$ Yes, as long as we're only talking about rotation about the axis. As a rather famous counter example, we can look at the inverted pendulum. In this case, we don't just have rotation about the axis, we also have up and down motion. The up and down motion can impart energy. Why this particular motion stabilizes the pendulum is the subject of a control theory course, but at the very least it demonstrates energy being transmitted through a bearing (basically a hinge) through a motion other than rotation about the axis. $\endgroup$
    – Cort Ammon
    Commented Mar 5 at 4:21
  • $\begingroup$ Wow, that was awesome. Thanks a lot! $\endgroup$
    – Meet Shah
    Commented Mar 6 at 9:57
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The work done by a hinge force is zero if the displacement of the point at which the hinge applies it's force i.e. the hinge itself, is zero. We define the work done by a force as the dot product of the force vector and the displacement vector of the point of application of the force.

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  • $\begingroup$ Right. Thank you for the kind answer. $\endgroup$
    – Meet Shah
    Commented Mar 6 at 9:57

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