Circular Motion Overview
The words circular and centripetal are used interchangeably.
Circular motion can be split into two part: rotational kinematics and centripetal forces.
Circular kinematics utilizes the same big 5 linear kinematic equations, but with rotational variables.
Centripetal forces explains why objects can move in a circular path.
Note – Linear velocity and tangential velocity are used interchangeably.
Centripetal Acceleration
Acceleration is a change in velocity. You can change velocity via speed OR direction.
In AP Physics 1 we cover uniform circular motion. This is when tangential velocity is changing in direction, but not speed. Hence the term “uniform.”
For example, imagine a yo-yo spinning clockwise with a constant linear velocity of \(15 \, \text{m/s}\) .
- at the top of the motion, the velocity vector points right
- at the bottom of the motion, the velocity vector points left.
Without a centripetal acceleration, an object would just continue moving in a straight line.
Circular Kinematics
Circular kinematics is the same as linear kinematics, but with rotational variables listed below.
| Linear Variables | Rotational Variables |
|---|---|
| \(\Delta x\) displacement in \(\text{meters}\) | \(\Delta \theta \) angular displacement in \(\text{radians}\) |
| \(v\) velocity in \(\frac{m}{s}\) | \(\Delta \omega \) angular velocity in \(\frac{rad}{s}\) |
| \(a\) acceleration in \(\frac{m}{s^2}\) | \(\alpha\) angular acceleration in \(\frac{rad}{s^2}\) |
To find the circular kinematic equation, take the linear kinematic equation and replace the linear variables with rotational variables.
For example, take the linear kinematic equation:
\[ (v_f)^2 =(v_0)^2 + 2a \Delta x \]
Swap out the linear variables for the corresponding rotational variables. The resulting rotational kinematic equation would be:
\[ (\omega_f)^2 =(\omega_0)^2 + 2\alpha \Delta \theta \]
Solving Circular Kinematic Problems
- Identify all the given variables. You will need atleast 3 known variables and 1 unknown variable.
- Pick your rotational kinematic formula that fits the given variables.
- Lastly plug everything in and solve for the unknown.
This is pretty much how you would solve linear kinematics (but slightly easier). If you still need practice with linear kinematics here are 50 AP style questions to master kinematics.
Using Radians
It’s important to understand that angular displacement ([katex] \Delta \theta [/katex]) is the change in radians.
If given a displacement in revolutions, convert it to radians.
1 revolution = \(2 \pi \) radians
So if an object has \(\Delta \theta = 30 \, \text{radians}\) it has made \(\dfrac{30}{2\pi}\) revolutions.
Converting from Rotational to Linear Motion
The value of \(r\) (radius of the circle) is the bridge between rotational and linear motion. See the table below for all the conversions.
| Rotational to Linear | Linear to Rotational |
|---|---|
| \[ \Delta x = \Delta \theta \times r\] | \[\Delta \theta = \dfrac{x}{r}\] |
| \[v = \Delta \omega \times r\] | \[ \Delta \omega = \dfrac{v}{r}\] |
| \[a = \alpha \times r\] | \[\alpha = \dfrac{a}{r}\] |
Period vs Frequency
The period \(T\) is the time taken to complete 1 full revolution. For example, a fan rotating 1 second PER revolution.
Frequency \(f\) is the inverse of period \(f = \frac{1}{T}\). It is the number of revolutions in 1 second in units of hertz \(Hz\). For example, a fan makes 10 revolutions in 1 second (\(10 Hz\)).
Centripetal Forces
A centripetal force accelerates an object.
\(F_c = ma_c \)
\(a_c \) is the centripetal acceleration that describes how the velocity, tangent to the circular path, changes direction.
\[a_c = \frac{v^2}{r} \]
\(a_c\) and thus \(F_c \) both point towards the center of the circle.
Note: “centripetal force” is just a placeholder. The actual “force” is the sum of forces directed towards the center.
For example, imagine a car driving around a flat curve. The centripetal forces comes from only from the friction of the tires.
Common Examples of Centripetal Acceleration
- Satellites rotating around the earth.
- Roller coaster at the top vs bottom of a loop
- Planes flying in a circle
- Cars going around a flat curve
- Cars going around a banked curve
- Swinging on a rope and pendulums
Solving Centripetal Force Problems
Solve these problems like any regular force problem. If you need a refresher on solving linear forces, skim through Forces Speed Review.
- Make an FBD. The net force should always point towards the center of the circle.
- Use Newton’s second law to make an equation for the net force that is directed to center of the circle.
- Replace any unknowns and solve the equation
Below are 10 questions to help you apply circular motion. For even more questions check out the Centripetal Motion UBQ.
Extra Help
That wraps up circular motion! If you still need help with on the topic, try out Elite Physics Tutoring or other physics programs. We guarantee that we can improve your understanding and scores in less than 3 lessons!
Practice – Try these 10 Questions
What is the magnitude and direction of the net force of the ball when it is at the top?
What is the velocity of the ball at the top?
The string is cut exactly when the ball is at the top. How long does it take to reach the ground?
How far does the ball travel horizontally before hitting the ground?
