Calculating Centripetal Force: Diagramming and Solving for X and Y Forces

[kdburns]

How would you go about proving that centripetal F=(mv^2)/r by diagramming and solving for the magnitude of the forces in the x and y direction?

For example, suppose I wanted to calculate the work done in the radial direction by centripetal force.

 

[A.T.]

http://en.m.wikipedia.org/wiki/Centripetal_force#Uniform_circular_motion

 

[kdburns]

Thanks so much! Really helpful

 

[daqddyo1]

Work is the product of the applied force magnitude x the magnitude of the displacement (D) in the same direction as the applied force. As a result, this is sometimes expressed as W = F x Dcosθ where θ is the angle between the two vectors (F and D). Since the F and D vectors are always perpendicular for circular motion of a mass at a constant speed, the angle between the two vectors is always 90°. So, the work done in circular motion at constant speed is 0 since cos90° is 0.

 

[PJC01]

To prove that centripetal force is equal to (mv^2)/r, we can use the laws of motion and basic trigonometry to diagram and solve for the magnitude of the forces in the x and y direction.

First, we need to understand that centripetal force is the net force acting on an object moving in a circular path. This force is always directed towards the center of the circle and is responsible for keeping the object moving in a curved path.

To begin, we can draw a diagram of the situation with the object moving in a circular path, with a radius of r. The centripetal force, Fc, is acting towards the center of the circle, and we can break this force into its x and y components, Fcx and Fcy, respectively.

Using trigonometry, we can see that the x component, Fcx, is equal to Fc multiplied by the cosine of the angle between the force and the x-axis. Similarly, the y component, Fcy, is equal to Fc multiplied by the sine of the angle.

Now, we can use Newton's second law, which states that the net force on an object is equal to its mass multiplied by its acceleration. In this case, the acceleration is towards the center of the circle and is equal to v^2/r, where v is the velocity of the object.

Therefore, we can set up the following equation for the x direction:

Fcx = (mv^2)/r * cosθ

And for the y direction:

Fcy = (mv^2)/r * sinθ

Next, we can calculate the magnitude of the centripetal force by using the Pythagorean theorem:

Fc = √(Fcx^2 + Fcy^2)

Substituting the values for Fcx and Fcy, we get:

Fc = √((mv^2)/r)^2 * (cos^2θ + sin^2θ)

Simplifying, we get:

Fc = (mv^2)/r * √(cos^2θ + sin^2θ)

Since cos^2θ + sin^2θ = 1, the equation becomes:

Fc = (mv^2)/r

Thus, we have proven that the centripetal force is equal to (mv^2)/r by diagramming and solving for the magnitude of the forces in the x and y direction. This equation is a fundamental principle