# Circular Motion

Circular Motion and Centripetal Force.

Tip: This isn't the place to ask a question because the teacher can't reply.

## Key Questions

• The moving of an object around a circular path is called circular motion .

Circular Motion can be discussed in a couple of different ways. The first way is linear and the second is rotational (also known as angular).

Linear speed (also known as tangent speed) is measured using the circumference of the circular path. The larger the path, the greater the linear speed. The larger the circle, the greater the distance the object must travel. This creates a greater tangent velocity as shown in the diagram.

Tangent velocity is the direction the object would move if the circular path were broken. For example, twirl an object on a string in a circle and then let go. The object will fly off tangent to the path. So, the velocity is dependent on the circle.

Rotational speed is not dependent on the size of the circle.

In this diagram, the 0° and the 60° lines are different size circles on the Earth. However, both are turning at the same rate. A person standing on each circle would both take 24 hours to go around one time.

This leads to more in-depth topics within circular motion such as centripetal acceleration, centripetal force, angular acceleration, and more.

See below.

#### Explanation:

Common classroom examples of uniform circular motion include, for obvious reasons, the simplest model: a ball tied to a string being swung in a horizontal circle. Since the net force is in the horizontal direction, the tension force is the only force we are concerned with.

$T = \frac{m \cdot {v}_{T}^{2}}{r}$

If we then further complicate this, we can write an expression for a vertical circle, which is easiest at the top or bottom of the circle

$\sum F = T - m g = \frac{m \cdot {v}_{T}^{2}}{r}$

at the bottom of the circle and

$\sum F = T + m g = \frac{m \cdot {v}_{T}^{2}}{r}$

at the top of the circle.

More complicated examples include gravitation problems, where we estimate that a planet or the moon revolves around something in uniform circular motion. Here we would equate the gravitation force to the centripetal force

$\frac{G {m}_{1} {m}_{2}}{r} _ {12}^{2} = \frac{m \cdot {v}_{T}^{2}}{r}$

We can use this to derive Kepler's ${T}^{2} \propto {r}^{3}$ law with a few substitutions based on our rotational kinematics rules.

Other more complex examples include loops in roller coasters (combining uniform circular motion with conservation laws) and other creative examples.

One example I saw dropped a pendulum that swung into a screw and entered uniform circular motion around the screw.

Creativity will allow a professor or instructor to create very difficult problems, but the above examples are the ones that a student in physics 1 should definitely see.

• It depends what information you begin with.

1. If frequency is known: $T = \frac{1}{f}$
2. If speed is known: T=(2πr)/v
3. If angular frequency is known: T=(2π)/ω
4. If you know the centripetal force, object's mass and radius of motion: Use $F = \frac{m {v}^{2}}{r}$ to find $v$ then use T=(2πr)/v
5. If you know the centripetal acceleration and radius of motion: Use $a = \frac{{v}^{2}}{r}$ to find $v$ then use T=(2πr)/v

## Questions

• · 3 weeks ago