How Is the Formula for Centripetal Acceleration Derived?

[Lindsey]

Can anyone please tell me how the formula for centripetal acceleration (a=v2/r) is derived?

 

[Doc Al]

You should be able to find this in any number of physics books. In any case, the basic idea is this:

An object in circular motion has at any point a tangential speed V= (omega)r. To find the acceleration, take two points separated by d(theta). Draw the vectors representing these two velocities. The difference between them (which points towards the center) is dV = Vd(theta). The acceleration a = dV/dt = Vd(theta)/dt = V(omega)= V(V/r) = V²/r.

Hope this helps a little.

Just to be clear: omega is the angular speed, V is linear speed.

 

[M98Ranger]

The formula for centripetal acceleration (a=v2/r) can be derived from the basic principles of circular motion. Let's consider an object moving in a circular path with a constant speed (v). Since the object is moving in a circle, it is constantly changing its direction, which means it is accelerating towards the center of the circle.

The acceleration towards the center of the circle is known as centripetal acceleration (a). Now, let's draw a diagram to understand the forces acting on the object.

We can see that there are two forces acting on the object - the centripetal force (F) towards the center of the circle and the tangential force (T) in the direction of motion. According to Newton's second law of motion, the net force acting on an object is equal to its mass (m) multiplied by its acceleration (a). So, we can write the following equation:

ΣF = ma

Since the object is moving at a constant speed, the tangential force (T) is balanced by an equal and opposite force (T) in the opposite direction. Therefore, the net force (ΣF) acting on the object is only the centripetal force (F). Thus, we can rewrite the equation as:

F = ma

Now, let's consider the centripetal force (F). According to Newton's law of universal gravitation, the force between two objects is directly proportional to their masses (m1 and m2) and inversely proportional to the square of the distance (r) between them. In this case, the centripetal force (F) is provided by the gravitational force between the object and the center of the circle. So, we can write the following equation:

F = Gm1m2/r2

where G is the universal gravitational constant. Now, we can substitute this value of F in the previous equation:

Gm1m2/r2 = ma

We can rearrange this equation to get the value of centripetal acceleration (a):

a = Gm2/r2

But, we know that the mass of the object (m2) is constant and the radius (r) is the distance between the object and the center of the circle. So, we can write:

a = Gm2/r2 = m2v2/r2

Finally, we can cancel out the mass of the object (m2) from both sides of the equation