Newton's laws - Simple pendulum

Problem Statement:

Draw the forces that act on the mass m of the pendulum of the figure and calculate the components of its acceleration.

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Solution:

When you draw the forces that act on an object, the first thing you need to do is analyze what interactions it is undergoing. In the case of the mass hanging from the rope of the figure, it is the tension that the rope exerts on the mass and its weight if we are close to the Earth.

In the following figure we have represented both forces together with the axes that we are going to use to calculate the components of the acceleration (tangential acceleration and normal or centripetal acceleration).

As you can see, the axes have been chosen so that one of them is tangent to the trajectory and the other one is perpendicular (normal) to it. In this way, as you will see below, the projection of Newton’s second law on the tangent axis will give us the tangential acceleration and the projection of Newton’s second law on the perpendicular axis will give us the normal or centripetal acceleration.