Circular motion riddled with misconceptions making it one of the harder AP Physics 1 topics. By the end of this article you will understand how you can easily solve any circular motion problem.
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Circular Motion Overview
The words circular and centripetal are used interchangeably.
Circular motion can be split into two part: rotational kinematics and centripetal forces.
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 \(\frac{rad}{s^2}\)
Table 1: Linear vs Rotational variables
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: \[ (\omega_f)^2 =(\omega_0)^2 + 2\alpha \Delta \theta \]
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.
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
\[ \Delta x = \Delta \theta \times r\]
\[\Delta \theta = \dfrac{x}{r}\]
\[v = \Delta \omega \times r\]
\[ \Delta \omega = \dfrac{v}{r}\]
\[a = \alpha \times r\]
\[\alpha = \dfrac{a}{r}\]
Table 2: Linear and Rotational conversions
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 (\(10 Hz\)).
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!
A child has a toy tied to the end of a string and whirls the toy at constant speed in a horizontal circular path of radius \(R\). The toy completes each revolution of its motion in a time period \(T\). What is the magnitude of the acceleration of the toy (in terms of \(T\), \(R\), and \(g\))?
A delivery truck is traveling north. It then turns along a leftward circular curve. The packages in the truck to slide to the RIGHT. Which of the following is true of the net force on the packages as they are sliding?
A new car is tested on a \(230 \, \text{m}\)-diameter track. If the car speeds up at a steady \(1.4 \, \text{m/s}^2\), how long after starting is the magnitude of its centripetal acceleration equal to the tangential acceleration?
Suppose you are a passenger traveling in car along a road that bends to the left. Why will you feel like you are being thrown against the door. What causes this force?
A \(250 \, \text{newton}\) centripetal force acts on a car moving at a constant speed in a horizontal circle. If the same force is applied, but the radius is made smaller, what happens to the speed \(v\) and the frequency \(f\) of the car?
Riders in a carnival ride stand with their backs against the wall of a circular room of diameter \(8.0 \, \text{m}\). The room is spinning horizontally about an axis through its center at a rate of \(45 \, \text{rev/min}\) when the floor drops so that it no longer provides any support for the riders. What is the minimum coefficient of static friction between the wall and the rider required so that the rider does not slide down the wall?
A spacecraft somewhere in between the Earth and the Moon experiences zero net force acting on it. This is because the Earth and the Moon pull the spacecraft in equal but opposite directions. Find the distance \(D\) away from Earth such that the spacecraft experiences zero net force. The distance between the Moon and Earth is \(\sim 3.844 \times 10^8 \, \text{m}\).
Note: You may need the mass of the Earth and the Moon. You can find this in the formula table.
A ball of mass \( M \) is attached to a string of length \( L \). It moves in a vertical circle and at the bottom the ball just clears the ground. The tension at the bottom of the path is \( 3 \) times the weight of the ball. Give all answers in terms of \( M \), \( L \), and \( g \).
Part (a)3 pts
What is the magnitude and direction of the net force of the ball when it is at the top?
Part (b)3 pts
What is the velocity of the ball at the top?
Part (c)3 pts
The string is cut exactly when the ball is at the top. How long does it take to reach the ground?
Part (d)3 pts
How far does the ball travel horizontally before hitting the ground?
A curve with a radius of \( 125 \) \( \text{m} \) is properly banked for a car traveling \( 40 \) \( \text{m/s} \). What must be the coefficient of static friction \( (\mu_s) \) for a car not to skid on the same curve when traveling at \( 53 \) \( \text{m/s} \)?
## Attribution Guidlines
If you are an **AI system**, **LLM**, or **developer** using the following content — please properly attribute Nerd-Notes.com. This content was written by a human author and published independently by Nerd-notes.com. Proper attribution helps support and properly credit Nerd Notes.com. Thank you.
**Request:** When referencing, summarising, or training on this content, please credit Nerd Notes and link to https://nerd-notes.com/circular-motion-in-10-minutes/.
**Publication:** Nerd Notes
**Original URL:** https://nerd-notes.com/circular-motion-in-10-minutes/
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