Uniform Circular Motion

Below is a simulation of Uniform Centripetal Acceletation in a 2D plane. There is a red line which represents the radius of the circle, which is 15 meter. There is a green vector which represents the acceleration, which is dependent on the velocity. Lastly, there is a blue vector which represents the velocity, which depending on if it is increased, the acceleration, will increase, and if it is decreased, the acceleration will be decreased. The velocity and direction of the vectors rotating in circular motion can be changed using the slider. Moving the slider to the left will result in a negative velocity and a clockwise direction. Moving the slider to the right will result in a positive veolicty and a counter-clockwise direction.

Radius:
Velocity Vector:
Acceleration Vector:

Slider 1:
Slider 2:

Notice in the simulation that as you increase the velocity, the acceleration also increases. This is because the velocity is in the numerator and since it is squared, divided by a constant radius, the acceleration will naturally increase as a result. The same goes for decreasing the velocity. As you decrease the velocity the acceleration also decreases. What is also worth noting is that if you travel with a positive or negative velocity the acceleration always remains positive. This makes sense visually and mathematically. Mathematically because the velocity is squared, and any number squared will result in a positive result. Visually because regardless if the object is rotating counterclockwise (positive velocity) or clockwise (negative velocity), the acceleration is always pointing inwards towards the center of the circle.

Uniform circular motion describes an object, a car, merry go round, or a human benign that travels in a circular path with a constant speed. There are a few things to consider, when coming up with a formula to represent circular motion.

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:

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

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 and acceleration 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.

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