vela said:Newton's second law says
$$\sum \vec{F}_i = m\vec{a}.$$ The centripetal acceleration is not a force. It's an acceleration. It goes into the righthand side of F=ma. The tension and the weight go into the lefthand side. Try again and pay attention to the sign of T and mg when summing the forces.
vela said:Closer. The lefthand side is correct, but the righthand side isn't. You can't add up a bunch of forces and then set the result to something that isn't a force. ##a_c## is the centripetal acceleration; it's not a force. It's the ##a## in ##ma## on the righthand side.
Centripetal acceleration is the acceleration towards the center of a circular path that an object experiences when it is moving in a circular motion.
The formula for centripetal acceleration is a = v^2/r, where a is the centripetal acceleration, v is the velocity of the object, and r is the radius of the circular path.
Regular acceleration, or linear acceleration, is the change in velocity of an object in a straight line. Centripetal acceleration, on the other hand, is always directed towards the center of a circular path and keeps the object moving in a circular motion.
The factors that affect centripetal acceleration include the speed of the object, the radius of the circular path, and the mass of the object. As the speed and radius increase, the centripetal acceleration also increases. A larger mass will also result in a larger centripetal acceleration.
Some real-life examples of centripetal acceleration include a car turning around a curve, a rollercoaster moving around a loop, and a satellite orbiting around the earth. Any object that moves in a circular path experiences centripetal acceleration.