The extra term on the left-hand side indicates that there is a force directed along the string that you failed to account for. This is the centripetal acceleration, that is the force that keeps the bob's moving in a circular path.

Your error occurs when you say that the tension "the tension on the string and the component of gravitational force parallel to the position vector negate one another". If this were the case then the bob would move in a straight line. Thus we see that the tension force must be greater than the force of gravity perpendicular to the motion of the bob. The difference between tension and gravity is the centripetal force.

The extra term on the left-hand side indicates that there is a force directed along the string that you failed to account for. This is the centripetal acceleration, that is the force that keeps the bob moving in a circular path.

Your error occurs when you say that "the tension on the string and the component of gravitational force parallel to the position vector negate one another". If this were the case then the bob would move in a straight line. Thus we see that the tension force must be greater than the force of gravity perpendicular to the motion of the bob. The difference between tension and gravity is the centripetal force.