## Circular Motion

Circular motion requires an inward force, centripetal force, to create a change in direction resulting in a circular pattern.

### Two Types of Circular Motion

A **rotation** is when an object spins around an internal axis. A** revolution** is when an object turns around an external point beyond the axis. See how the stickman is revolving around the central axis. The **axis** is a central point around which rotation takes place.

### Tangential Velocity

The tangent is a straight line that would just touch the curve of the circle as seen in the animation.

Because velocity is a vector, it includes magnitude and direction. Since direction changes, circular motion is considered accelerated. So will use the term tangential velocity to represent what the velocity would be if released. If you lost the centripetal force the object would continue in a straight line tangent to the circle as seen.

**Circular Motion is the Result of Centripetal Force**

Newton's First Law of Motion, **inertia**, states that an object moving would continue in a straight line path. To have circular motion an inward force is necessary to create the continuous turning. **Centripetal Force** is a force towards the center of a circle keeping it in a circular path.

### Centripetal Force Equation Variables

**Variable**: Name (Unit)

**F**: Centripetal Force (N)_{C}**m**: Mass (kg)**v**: Tangential Velocity (m/s)**r:**Radius (m)

What could provide the centripetal force causing circular motion?

- A rope:
**F**tension in the rope_{c}= F_{T} - Gravity:
**F**gravity_{c}= F_{g} - Friction:
**F**friction_{c}= F_{f}

**Important Note:** The problems in this unit may not state centripetal force directly. They may ask what is the tension, gravity, or friction keeping an object in circular motion. If stated this way, solve for centripetal force (**F _{c}**).

**Centripetal Acceleration (**a_{c})

Acceleration is a vector which includes magnitude and direction. **Changing either direction, speed, or both is considered acceleration**. So going from 10 m/s north to 10 m/s east would require an acceleration.

In circular motion it’s centripetal force (F_{c}) that causes a centripetal acceleration (a_{c}) seen in the equation (F_{C} = ma_{C}).

The variable for **centripetal acceleration is a _{c}** and its standard unit like other accelerations is

**m/s**.

^{2}When you plug in the centripetal acceleration equation a_{C} = (v^{2}/r) into (F_{C} = ma_{C}) it becomes our standard centripetal force equation (F_{C} = mv^{2}/r) as seen on most equation sheets.

### Tangential Velocity Equation

Tangential velocity is calculated by taking the circular distance divided by the time it takes to go around the circle or **period** (**T**). The distance around the circle is found by calculating the circumference of a circle.

**2πr** = circle circumference (unit: meters (m))

**T** = Period: Time it takes for one rotation (unit: seconds (s))

### What is a Period?

Period (T) is the time it takes for a specific motion to occur. See in the animation how this could be one swing, rotation, wave, segment, or osculation. In this unit it will be a rotation.

### Period and Frequency

A **period (T)** is the amount of time per cycle.

A **frequency (f)** is the number of cycles per time. The unit is the **hertz (Hz) **which means cycles per second.

You can find either with cycles and time or the other. If you have frequency just take the inverse of it. If you have period you can take the inverse of it to find frequency as seen in the animation.

A greater radius gives an object a greater velocity when having the same period. You can analyze the equations in this unit along with any other using the **rule of ones**. **Click here** if you want a reminder on how to use this. Notice in the image that the fastest circle is green which is the furthest out and the slowest is red.

### Centripetal Force Example Problems

1. What would happen to the centripetal force required to keep an object going in a circle if the radius of a circle was doubled?

2. How many times the centripetal force would you have if a car slowed from 60 mph to 20 mph going around a curve?

3. Kent spins in his chair with a frequency of 0.5 Hz. What is the period of his spin?

4. A record takes 1.3 s to make one complete rotation. An object on this record is 0.12 m from the center. What is its velocity?

5a. The pilot of a 60,500 kg jet plane is flying in circles whose radius is 5.00 x 10^{4 }m. It takes 1.8 x 10^{3} s to make one rotation. What is the velocity of the plane?

5b. How much centripetal force would there be?

6. What is the centripetal acceleration of a bike traveling a tangential speed of 8 meters per second in a circle that has a radius of 5 meters?

### Circular Motion Practice Quiz

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