The Centripetal Force

Below is a simulation of Centripetal Force in a 2D plane. There is an adjustable radius, mass and speed for the rotating object. There is a green and black vector which represent the acceleration (green) and centripetal force (black). There is also a blue vector tangent to the circular curve representing the velocity vector. Then lastly, there is a red line that presents the radius of the object away from the origin of the circle. Depending on how you as the user change the radius, mass of the object or velocity, all of these factors will play in changing the centripetal acceleration and centripetal force in different ways. The velocity of the rotating object can also be adjusted to rotate in either the positive or negative direction (counterclockwise or clockwise).

Radius:
Velocity Vector:
Acceleration Vector:
Force Vector:

Slider 1:
Slider 2:
Slider 3:

The centripetal force can be calulated using Newton's Secidn Law which states that the acceleration of an object is dependent upon two variables: the net force acting upon the object and the mass of the object: F = ma.

Since we are dealing with centripetal circular motion, the formula would look slightly different, like so: F_c = ma_c.

We can solve for this equation by first solving for the acceleration:

First we need to consider the distance from the axis of rotation. This is the radius (r). The distance from the center of the circle to where the object is.

\[ r \]

Second we need to consider the distance the object needs to travel to complete one revolution or loop in the circle. This is the circumference (C).

\[ C = 2 \pi r \]

Now when we look at the simulation what can be noticed is that the velocity vector is tangent along the circle, whereas the acceleration vector is pointing towards the center of the circle opposite the direction of the radius. Let us separate the velocity and acceleration vectors and place them on separate circles.

From the velocity formula we know that velocity is distance over time or simply:

\[ v = \frac{d}{t} \]

Since the distance for an object to travel a circle once is the circumference, we write that in for d. Now:

\[ v = \frac{2 \pi r}{t} \]

Now if we solve for time we get an equation:

\[ t = \frac{2 \pi r}{v} \]

(multiply t on both sides and divide by v)

Similarly, from the acceleration formula we know that acceleration is the change in velocity over time or simply:

\[ a = \frac{v_f - v_i}{t} \]

We can rewrite this as:

\[ a = \frac{2 \pi v - 0}{t} \]

\[ a = \frac{2 \pi v}{t} \]

Notice in the previous equation it was 2πr. There we considered distance. Here we are considering velocity, and since we are applying this to the acceleration formula, but the acceleration is rotating tangentially because of the velocity, we need to consider adding the 2π as well.

Solving for time we obtain an equation:

\[ t = \frac{2 \pi v}{a} \]

Now that we have two formulas for time:

\[ t = \frac{2 \pi r}{v} \text{ and } t = \frac{2 \pi v}{a} \]

We can solve for acceleration. We know that t = t so that means that:

\[ \frac{2 \pi r}{v} = \frac{2 \pi v}{a} \]

Now applying algebraic properties we obtain the equation for uniform centripetal acceleration:

\[ a = \frac{v^2}{r} \]

Or formally:

\[ a_c = \frac{v^2}{r} \]

We can now write the formula for centripetal force using the equation we have calculated for centripetal acceleration:

\[ F = ma \]

\[ F_c = m \frac{v^2}{r} \]

\[ F_c = m a_c \]

In summary, it is important to consider different factors when formulating a physics equation. You should ask questions, like what other physics formulas can I apply here? Here we applied the linear velocity, acceleration and Newton's 2nd Law for Force equations. We set both equations equal to time, because no matter how much time passes as an object rotates about a circle, the object still has a constant radius following the circular path. Then when solving for acceleration, we used algebraic rules to obtain the equation. From the derived acceleration, we then used that acceleration to formulate the centripetal force equation.

With all of these terms combined we get the formula for calculating the centripetal force.

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