Students will learn that in circular motion there is always an acceleration (and hence a force) that points to the center of the circle defined by the objects motion. This force changes the direction of the velocity vector of the object but not the speed. Students will also learn how to calculate that speed using the period of motion and the length of its path (circumference of the circle it traces out).

### Key Equations

\begin{align*} v = \frac{2 \pi r} {T} \end{align*}

If a particle travels a distance \begin{align*}2 \pi r\end{align*} in an amount of time \begin{align*}T\end{align*}, then its speed is distance over time or \begin{align*}\frac{2 \pi r}{T}\end{align*}

The Earth-Sun distance is about \begin{align*}1.5 \times 10^{11} \ m\end{align*} The Earth-Moon distance is about \begin{align*}3.84 \times 10^8 \ m\end{align*}

- An orbital period, \begin{align*}T\end{align*}, is the time it takes to make one complete rotation.

- If a particle travels a distance \begin{align*}2\pi r\end{align*} in an amount of time \begin{align*}T\end{align*}, then its speed is distance over time or \begin{align*}2 \pi r /T\end{align*}.

- An object moving in a circle has an instantaneous velocity vector
*tangential*to the circle of its path. The force and acceleration vectors point to the center of the circle.

- Net force and acceleration
*always*have the same direction.

- Centripetal acceleration is just the acceleration provided by centripetal forces.

#### Example 1

### Watch this Explanation

### Explore More

- When you make a right turn at constant speed in your car what is the force that causes
*you*(not the car) to change the direction of*your*velocity? Choose the best possible answer.- Friction between your butt and the seat
- Inertia
- Air resistance
- Tension
- All of the above
- None of the above

- A pendulum consisting of a rope with a ball attached at the end is swinging back and forth. As it swings downward to the right the ball is released at its lowest point. Decide which way the ball attached at the end of the string will go at the moment it is released.
- Straight upwards
- Straight downwards
- Directly right
- Directly left
- It will stop

- A ball is spiraling outward in the tube shown to the right. Which way will the ball go after it leaves the tube?
- Towards the top of the page
- Towards the bottom of the page
- Continue spiraling outward in the clockwise direction
- Continue in a circle with the radius equal to that of the spiral as it leaves the tube
- None of the above

- Explain using Newton’s Second Law why an object moving in a circle must experience a net force towards the center of the circle.
- Using the known distance Earth is from the sun, calculate the speed that Earth is moving through space in relation to the sun.
- Using the known distance the moon is from Eath, calculate the speed that the moon is moving through space in relation to Earth.

**Answers**

- .
- .
- .
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- 30,000 m/s
- 1,020 m/s