As a pendulum moves closer to the equilibrium position, how do the velocity, acceleration, and force change?

- The force acting on the pendulum is the weight of the mass. We are only concerned about the component of the weight tangential to the motion of the pendulum, since the other component (perpendicular) is equal to the tension of the rope.
The tangential component of the weight is given by:

where  is the angle between the pendulum and the vertical. We see from the formula that when the pendulum moves closer to the equilibrium position, the magnitude of the force decreases. Eventually, the force is zero when the pendulum reaches the equilibrium position.

- the acceleration is directly proportional to the force: in fact, Newton's second law states that

where F is the force and a the acceleration. Therefore, as the pendulum gets closer to the equilibrium position, its force decreases, and the acceleration decreases as well.

- The velocity has a completely different behaviour. We can think about it in terms of conservation of energy: at the point of maximum displacement of the pendulum, the mass attached to the pendulum is located at maximum height, so its gravitational potential energy is maximum while its kinetic energy is minimum (zero). As the pendulum moves closer to the equilibrium position, the height decreases, so the potential energy converts into kinetic energy, and the velocity increases. Eventually, when the pendulum reaches the equilibrium position, the velocity will be maximum.

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