Equilibrium position of a pendulum is defined as the position where no external force acts on the body and if no force is applied, it remains at rest. In the image, isn't gravity acting everywhere, then how will you define the middle point in the pendulum's motion as equilibrium position? Even if you remove gravity, initial push is required for it's motion.
Yes, gravity acts everywhere. However, in the center (equilibrium position) gravity is exactly balanced by the force acting along the string, which holds the ball. Therefore, this is the only point where we do not obtain a resulting acceleration $\vec a = \sum_i \vec F_i/m$. Therefore, the center point is the only equilibrium position.
PS: If we use a bar instead of a string we obtain a second equilibrium position. This second equilibrium position is the point where the pendulum is "upside down". As a minimal force is sufficient to imbalance this second equilibrium position it is called instable.
Equilibrium position of a pendulum is defined as the position where no external force acts on the body and if no force is applied, it remains at rest.
No, the equilibrium position is defined where there is no net external forces acting on the pendulum. At the bottom position the tension in the string acting upwards equals the force of gravity acting downwards for a net force of zero.
Hope this helps.
Equilibrium position of a pendulum is defined as the position where no external force acts on the body and if no force is applied, it remains at rest.
This is incorrect. The equilibrium position, in general, is characterized by the object having no net external force on the object or system in consideration (along with a few other characteristics). Furthermore, the object (system) need not be at rest to be at equilibrium, it could also move with constant velocity, (this follows from $ {\bf{F}_{ext}} = m.{\bf\ddot{R}}$, so ${\bf{F}_{ext}}$ depends on the derivative of the velocity)
At the bottom of the pendulum, tension and gravitational force cancel each other out (when the Bob isn't moving), and since there are no horizontal forces (or forces in any other direction), the position can be classified as an equilibrium point.
More specifically, the term equilibrium implies that the object (system) moves with a constant velocity and constant angular velocity, relative to an observer in an inertial frame (both of them can be zero; a constant, like in this case).
Hope this helps.
When we talk about the equilibrium position of a pendulum, we're referring to the spot where the pendulum would come to rest if there were no other forces acting on it. This position occurs when the pendulum is hanging straight down, with gravity acting vertically downward and providing the tension force in the string or rod that keeps the pendulum in place. At this position, the pendulum is not swinging; it is in a stable equilibrium.
Now, let's consider the swinging motion of the pendulum. When the pendulum is set into motion, it swings back and forth due to the force of gravity pulling it downward and the tension in the string or rod counteracting it. At the bottom of the swing, the tension is at its maximum, and gravity is still acting downward. At this point, the pendulum is in its lowest point and again in a stable equilibrium.
At the top of the swing, the pendulum momentarily comes to a stop before starting to fall back down due to gravity. At this highest point, the tension in the string or rod is at its minimum, but gravity is still acting downward. Therefore, this point is not an equilibrium position since it's an unstable equilibrium. The slightest disturbance will cause the pendulum to start moving down again.
Here's an image illustrating the concept of a swinging pendulum:
As for the initial push, you are correct; it is required to set the pendulum in motion. Once the pendulum is set into motion, it will keep swinging back and forth between the two stable equilibrium points - the bottom of the swing and the position where the pendulum hangs straight down. In the absence of any external forces like air resistance, the pendulum's motion would continue indefinitely.
So, in the motion of a pendulum, there are two equilibrium positions: one is the lowest point of the swing, and the other is the position where the pendulum hangs straight down. The highest point of the swing is not an equilibrium position, as it is an unstable equilibrium. The initial push is required to start the motion, and once the pendulum is in motion, it will continue swinging between the two stable equilibrium points.
I highly recommend that you read about vertical circular motion as it would clear your doubt completely and analyze how the formulas came about based on the location of the bob.
Image Source: https://study.com/academy/lesson/objects-moving-in-vertical-circles-analysis-practice-problems.html
In short:
The motion of a simple pendulum is governed by the restoring force, which can be described using the equation: F = -m * g * sin(θ), where F is the restoring force, m is the mass of the pendulum bob, g is the acceleration due to gravity, and θ is the angle of displacement from the equilibrium position. When the pendulum is at its equilibrium position (θ = 0), the restoring force becomes zero, keeping the bob at rest. As the pendulum swings, it oscillates between its lowest point (stable equilibrium) and its highest point (unstable equilibrium). At the lowest point, the restoring force is maximum, pulling the pendulum back to equilibrium. At the highest point, the restoring force is zero, making it an unstable position where any disturbance causes the pendulum to swing back.
Comment: I went all out on this answer so I hope that it helped you with your understanding!
This question needs to be considered in a Hamiltonian or Lagrangian formulation.
In the latter, theta is the singular generalized coordinate, and the generalized force at zero or pi is zero, hence they are equilibrium positions.
