The centripetal acceleration formula is derived from the principles of circular motion. The proof involves using Newton's laws of motion and some basic trigonometry.
Let's consider an object moving in a circle of radius 'r' with a constant speed 'v'. The centripetal acceleration, denoted by 'a', is the acceleration directed toward the center of the circle that keeps the object in circular motion.
To derive the formula for centripetal acceleration, we start with Newton's second law of motion, which states that the net force acting on an object is equal to the product of its mass and acceleration (F = ma).
In circular motion, the force responsible for keeping the object in the circular path is called the centripetal force. For an object moving in a circle, the centripetal force is provided by the tension in a string, the force of gravity, or any other force directed toward the center of the circle.
Now, let's break down the forces acting on the object. There are two forces involved: the centripetal force (Fc) and the force of inertia (F∥), which acts tangentially to the circular path.
The force of inertia is the force that an object tends to continue in a straight line due to its inertia. It is given by F∥ = m × tangential acceleration.
Considering the object's motion in the circular path, we know that the acceleration is always directed toward the center of the circle. This means that the tangential acceleration is equal to the centripetal acceleration (a).
Now, using trigonometry, we can relate the tangential acceleration to the angular velocity (ω) and the linear velocity (v). The tangential acceleration is given by a∥ = r × α, where α is the angular acceleration.
Since the object is moving at a constant speed, the angular velocity is constant, and hence, the angular acceleration is zero (α = 0). Therefore, the tangential acceleration (a∥) becomes zero.
Now, using the Pythagorean theorem, we can relate the centripetal acceleration (a) and the tangential acceleration (a∥) as follows:
a² = a∥² + a⊥²
Since a∥ = 0, the equation simplifies to:
a² = a⊥²
Substituting a⊥ with v²/r (from the equation for tangential acceleration), we get:
a² = (v²/r)²
Taking the square root of both sides, we obtain:
a = v²/r
This is the formula for centripetal acceleration, which states that the centripetal acceleration (a) is equal to the square of the linear velocity (v) divided by the radius of the circular path (r).
This proof demonstrates that for an object moving in a circular path with a constant speed, the centripetal acceleration is directly proportional to the square of the linear velocity and inversely proportional to the radius of the circle.