If work done by hinge force on a hinged rod is 0, then why is there a hinge force at all?

Question

Consider a rod hinged at one end, and free in the other, initially, the rod makes a 30° angle with the horizontal axis through the hinge, it is then allowed to fall.

I was told, that energy could be conserved for the rod, because the hinge force does not do any work, as there is no displacement of the hinge point.

But, if there is no displacement of the hinge point, then there must be no acceleration of the hinge point as well, so if there is no acceleration of the hinge point, then how does the hinge force even exist?

Answer 1

Work done by a force is $0$ if $\vec F \cdot \vec d = 0$.

The hinge rotates but does not move. $\vec d = 0$

Forces can exist even when the object they act on is stationary. In that case, the total force is $0$. There may be multiple forces that cancel each other. For example, gravity pulls you down. The ground holds you up.

Answer 2

if there is no displacement of the hinge point, then there must be no acceleration of the hinge point as well, so if there is no acceleration of the hinge point, then how does the hinge force even exist?

Remember Newton’s 2nd law. It is not $$\vec F =m \vec a$$ Instead it is actually $$\vec F_{net}=m \vec a$$ where $\vec F_{net}=\sum \vec F_i$ is the sum of all the forces acting on the object.

Here the hinge force is not the only force on the door. The 0 acceleration of the hinge point does not imply that the hinge force is zero.

Answer 3

What would happen if there was no hinge?
The centre of mass of the rod and the rest of the rod would undergo a downward translational accelerate of $g$ and the angle of inclination of the rod would stay at $30^\circ$ to the horizontal.

What happens if a hinge is present?
One end of the rod suffers no downward translational acceleration and the angle the rod makes with the horizontal increases.
This means that there must be a force acting at the hinge as there is no other place another force could act.
That force at the hinge results in there being a torque acting on the rod because if there was no torque then the rod would not start to rotate.