Centripetal Circuit Slides AP Physics 1
Centripetal Circuit
Mastering the dynamics of uniform circular motion
The Kinematics
Uniform Circular Motion (UCM)
Motion in a circle at a constant speed . Even though the speed is constant, the velocity is changing because the direction is changing.
Key Variables:
T: Period (seconds per revolution)
f: Frequency (revolutions per second, Hz)
r: Radius (meters)
\[v = \frac{2\pi r}{T} = 2\pi rf\]
Tangential Velocity Formula
Acceleration "Center-Seeking"
If velocity is changing, there must be an acceleration. In UCM, this acceleration is always directed toward the center of the circle.
Key Fact
"Centripetal" is a direction, not a type of force. It means "center-seeking."
\[a_c = \frac{v^2}{r} = \frac{4\pi^2 r}{T^2}\]
v a_c
Velocity is Tangent | Accel is Radial
The Centripetal Force
"There is no such thing as a centripetal force... until you identify what physical force is providing it!"
\[\Sigma F_c = ma_c = \frac{mv^2}{r}\]
Who's the "Actor"?
Tension: A ball on a string
Friction: Car turning a corner
Gravity: A moon orbiting a planet
Normal Force: The walls of a Rotor ride
Vertical Loops
At the Top:
\[\Sigma F_c = F_g + F_N = \frac{mv^2}{r}\]
Critical Speed: Minimum speed to stay in the loop occurs when \(F_N = 0\).
At the Bottom:
\[\Sigma F_c = F_N - F_g = \frac{mv^2}{r}\]
You feel "heavier" here because \(F_N > F_g\).
F_g & F_N F_g F_N
Battle Plan
1
Draw FBD
Identify all physical forces acting on the object. Do NOT draw a "centripetal force" vector!
2
Set Axes
Set the positive direction toward the center of the circle. This is your radial axis.
3
Sum Forces
\[\Sigma F_{radial} = ma_c\]
Orbit Oddities Worksheet Orbit Oddities
AP Physics 1 • Circular Motion Dynamics
NAME:
DATE:
VELOCITY
\[v = \frac{2\pi r}{T}\]
ACCELERATION
\[a_c = \frac{v^2}{r}\]
DYNAMICS
\[\Sigma F_c = \frac{mv^2}{r}\]
1
The Vector Twist
A puck of mass \(m\) slides in a horizontal circle of radius \(R\) on a frictionless table, held by a string. If the speed of the puck is doubled, how does the tension in the string change? Justify your answer using equations.
2
The Highway Hero
A car of mass \(1500\text{ kg}\) travels at a constant speed of \(20\text{ m/s}\) around a flat, unbanked curve with a radius of \(50\text{ m}\).
A) Draw a Free-Body Diagram for the car from a rear-view perspective.
B) Calculate the minimum coefficient of static friction \(\mu_s\) required to keep the car from sliding.
3
The Rollercoaster Rush
A roller coaster car of mass \(M\) enters a vertical loop of radius \(R\). At the very top of the loop, the cart has a speed \(v\).
A) Draw the FBD for the cart at the top of the loop.
B) Derive an expression for the Normal Force \(F_N\) acting on the cart at the top in terms of \(M, v, R,\) and \(g\).
C) Calculate the "Critical Speed" \(v_{min}\) required to stay on the track at the top if \(R = 10\text{ m}\). (Assume \(g = 10\text{ m/s}^2\))
4
The Conical Twist
A ball of mass \(m\) is attached to a string of length \(L\). It swings in a horizontal circle such that the string makes an angle \(\theta\) with the vertical.Express the tension \(T\) in terms of \(m, g,\) and \(\theta\).
θ
Conical Pendulum
Orbit Oddities Teacher Guide Centripetal Circuit: Teacher Guide
Unit: Dynamics | Lesson: Centripetal Circuit
Learning Objectives
Determine the direction of velocity and acceleration vectors in UCM.
Identify physical forces providing the centripetal acceleration.
Solve for unknown variables (v, r, μ, T) using \(\Sigma F_c = ma_c\).
Analyze energy and force at the top/bottom of vertical loops.
Misconception Alert
"Centrifugal Force": Students often want to draw an outward-pointing force. Remind them that the "feeling" of being pushed outward is just inertia (Newton's 1st Law)—their body wants to travel in a straight line while the car pulls them inward.
Spin Cycle Lab Tips
Equipment Note
Ensure the glass or plastic tubes are smooth. Excessive friction in the tube will act as an unintended force, causing the calculated mass to be lower than actual.
Success Metric
A linear graph of \(F_c\) vs \(v^2\) should yield a slope of \(m/r\). Students should get within 10-15% of the stopper's mass.
Common Lab Errors
Vertical Angle: The string often dips below horizontal. Remind students that \(F_c = T \sin\theta\). In UCM labs, we assume \(\theta \approx 90^\circ\) for simplicity, but this is a source of error.Radius Control: Students struggle to keep the tape mark steady. Assign one group member as the "Safety Officer" to watch the mark exclusively.Reaction Time: Timing 10 revolutions helps minimize the % error from human reaction time with the stopwatch.
Orbit Oddities Solutions
1
The Vector Twist
Tension increases by a factor of 4.
\[T = \frac{mv^2}{R} \implies T_{new} = \frac{m(2v)^2}{R} = 4T\]
2
The Highway Hero
A) FBD: \(F_N\) (up), \(F_g\) (down), \(F_s\) (inward).
\[\mu_s = \frac{v^2}{gR} = \frac{20^2}{(10)(50)} = 0.8\]
3
The Rollercoaster Rush
FBD: \(F_g\) and \(F_N\) both point down.
\[F_N = \frac{Mv^2}{R} - Mg\] \[v_{min} = \sqrt{gR} = 10\text{ m/s}\]
4
The Conical Twist
\[T \cos\theta = mg \implies T = \frac{mg}{\cos\theta}\]
Two-Day Pacing Suggestion
Day 1: Intro
Slides & Vector Demos (20m), Worksheet P1-P2 (20m), Exit Ticket (10m)
Day 2: Application
Spin Cycle Lab Data Collection (30m), Analysis & Synthesis (20m), Debrief (10m)
Spin Cycle Lab Activity Experimental Physics
Spin Cycle Lab
AP Physics 1 • Investigating Centripetal Force
Group:
Date:
Objective
The goal of this investigation is to determine the relationship between the centripetal force (\(F_c\)) acting on a rotating object and its tangential velocity (\(v\)). You will verify the theoretical relationship \(\Sigma F_c = \frac{mv^2}{r}\) by manipulating the tension in a string.
Materials
Glass/Plastic Tube (handle)
Nylon String (~1.5m)
Rubber Stopper (rotating mass)
Hanging Mass Set (tension)
Stopwatch & Meter Stick
Procedure
Measure the mass of the rubber stopper (\(m\)) and record it in kilograms.
Thread the string through the tube. Attach the stopper to one end and the mass hanger to the other.
Mark the string with a piece of tape exactly \(0.50\text{ m}\) from the center of the stopper. This is your radius (\(r\)) . Keep this mark just below the tube while spinning.
Hang a known mass (\(M\)) from the bottom. The tension in the string is equal to the weight of this hanging mass (\(Mg\)). This provides the centripetal force (\(F_c\)).
Spin the stopper in a horizontal circle. Adjust your speed so the tape mark stays consistent.
Time 10 revolutions and record the time. Repeat for at least 5 different hanging masses.
m M F_c
Setup Diagram
Data Collection
Trial Hanging Mass (\(M\), kg) Force (\(F_c = Mg\), N) Time for 10 rev (s) Period (\(T\), s) Velocity (\(v\), m/s) \(v^2\) (\(m^2/s^2\)) 1 2 3 4 5
Data Analysis
1. Plot a graph of Centripetal Force (\(F_c\)) on the y-axis and \(v^2\) on the x-axis. Sketch your line of best fit below.
FORCE (N)
VELOCITY SQUARED (m²/s²)
2. Based on the equation \(F_c = \frac{m}{r}v^2\), what physical quantity does the slope of your graph represent?
