#AP Physics 1: Circular Motion & Kepler's Laws 🚀

Hey there, future physicist! Let's get you prepped for the AP exam with a super-focused review of circular motion and Kepler's laws. We'll break down the concepts, highlight key formulas, and get you feeling confident. Let's dive in!

#Circular Motion: The Basics

#Centripetal Acceleration 🎡

• Definition: Acceleration directed towards the center of a circular path. It's what keeps objects moving in a circle, constantly changing their direction.

• Formula: $$a_{c} = \frac{v^{2}}{r}$$ where:

  • $a_c$ is centripetal acceleration
  • $v$ is tangential speed
  • $r$ is the radius of the circular path

• Direction: Always points towards the center of the circle, perpendicular to the object's velocity.

• Cause: Can be caused by various forces (or components of forces), like tension, friction, or gravity.

*Caption: Centripetal acceleration (ac) is always directed toward the center of the circle, while the velocity (v) is tangential to the circle.*

• Vertical Loops: At the top of a loop, the minimum speed to maintain circular motion is when gravity provides the entire centripetal force: $$v = \sqrt{gr}$$. Think roller coasters!

• Banked Curves: Static friction and normal forces combine to create the centripetal force on banked turns (like race tracks). Remember, we only analyze these quantitatively when friction isn't needed for uniform circular motion.

• Conical Pendulums: Tension in the string has a component that contributes to the centripetal force (think amusement park swings).

#Forces in Circular Motion

• Forces cause centripetal acceleration. The net force always points to the center of the circle.

• Examples include tension in a string, gravity for orbiting bodies, and friction for a car turning.

#Tangential Acceleration

• Definition: Acceleration along the circular path that changes an object's speed.

• Effect: It increases or decreases the tangential speed of the object.

• Net Acceleration: The vector sum of centripetal and tangential accelerations. It points at an angle to the circular path when both are present.

  *Caption: Tangential acceleration (at) changes the speed of the object, while centripetal acceleration (ac) changes the direction.*

#Net Acceleration in Circles 📐

• Centripetal: Changes direction, not speed (perpendicular to velocity).

• Tangential: Changes speed, not direction (parallel to velocity).

• Net: Vector sum of both, angled relative to the circular path.

#Period and Frequency

• Uniform Circular Motion: Object moves at constant speed around a circle.

• Period (T): Time for one complete revolution. Measured in seconds (s). Example: Earth's orbital period around the Sun is 1 year.

• Formula: $$T = \frac{2\pi r}{v}$$

• Frequency (f): Number of revolutions per unit time. Measured in Hertz (Hz) or cycles per second. Example: A spinning tire at 600 revolutions per minute has a frequency of 10 Hz

• Relationship: Frequency is the reciprocal of the period: $$f = \frac{1}{T}$$ or $$T = \frac{1}{f}$$

#Kepler's Third Law

#Circular Orbits 🌎

• Centripetal Force: For satellites in circular orbits, gravity provides the centripetal force.

• Kepler's Third Law: Relates a satellite's orbital period to its orbital radius and the mass of the central body.

• Formula: $$T^{2} = \frac{4\pi^{2}}{GM}R^{3}$$ where:

  • $T$ is the orbital period
  • $R$ is the orbital radius
  • $M$ is the mass of the central body
  • $G$ is the universal gravitational constant

• Applications: Applies to planets orbiting the Sun, moons orbiting planets, and artificial satellites orbiting Earth.

#🚫 Boundary Statements:

• Banked Curves: AP Physics 1 only requires quantitative analysis of banked curves when friction is not needed for uniform circular motion. Qualitative descriptions are sufficient for scenarios involving friction.

• Kepler's Laws: The exam does not require knowledge of Kepler's first or second laws. Focus on the third law!

#Final Exam Focus

Okay, you've made it through the key concepts! Here's what to focus on for the exam:

• Centripetal Acceleration: Understanding the formula and direction is crucial. Be ready to apply it in various scenarios.

• Forces in Circular Motion: Identify forces that cause centripetal acceleration.

• Period and Frequency: Know the relationship and how to use them in calculations.

• Kepler's Third Law: Understand the relationship between period, radius, and mass. Be able to use the formula correctly.

• Problem Solving: Practice combining concepts from different units, as AP questions often do.

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Practice Question

Practice Questions

#Multiple Choice Questions:

1. A car is moving at a constant speed around a circular track. What is the direction of the car's acceleration? (A) Tangent to the circle (B) Radially inward (C) Radially outward (D) Zero

2. A satellite orbits the Earth in a circular path. If the orbital radius is doubled, what happens to the orbital period? (A) It is halved (B) It is doubled (C) It is increased by a factor of 2√2 (D) It is increased by a factor of 4

3. A ball is swung in a vertical circle. At which point is the tension in the string the greatest? (A) At the top of the circle (B) At the bottom of the circle (C) At the sides of the circle (D) Tension is constant throughout the motion

#Free Response Question:

A small ball of mass m is attached to a string of length L and swung in a horizontal circle at a constant speed v. The string makes an angle θ with the vertical, as shown in the figure below.

(a) Draw a free-body diagram for the ball, labeling all forces acting on it.

(b) Derive an expression for the tension T in the string in terms of m, g, and θ.

(c) Derive an expression for the speed v of the ball in terms of g, L, and θ.

(d) If the length of the string is 1.0 m and the angle θ is 30°, calculate the speed of the ball.

#FRQ Scoring Breakdown:

(a) Free-Body Diagram (3 points):

• 1 point for correctly drawing the tension force (T) along the string.
• 1 point for correctly drawing the gravitational force (mg) vertically downward.
• 1 point for not including any extraneous forces.

(b) Tension in the String (3 points):

• 1 point for correctly resolving the tension force into vertical and horizontal components: $T_y = T \cos\theta$ and $T_x = T \sin\theta$.
• 1 point for recognizing that the vertical forces are balanced: $T \cos\theta = mg$.
• 1 point for solving for tension: $T = \frac{mg}{\cos\theta}$.

(c) Speed of the Ball (4 points):

• 1 point for recognizing that the horizontal component of tension provides the centripetal force: $T\sin\theta = \frac{mv^2}{r}$.
• 1 point for relating the radius of the circle to the length of the string and the angle: $r = L\sin\theta$.
• 1 point for substituting the expression for tension from part (b) into the centripetal force equation: $(\frac{mg}{\cos\theta})\sin\theta = \frac{mv^2}{L\sin\theta}$.
• 1 point for solving for the speed: $v = \sqrt{gL\sin\theta\tan\theta}$.

(d) Calculation of Speed (2 points):

• 1 point for correctly substituting the given values: $v = \sqrt{(9.8 \text{ m/s}^2)(1.0 \text{ m})\sin(30^\circ)\tan(30^\circ)}$.
• 1 point for calculating the correct speed: $v \approx 1.68 \text{ m/s}$.

You've got this! Remember to stay calm, trust your preparation, and tackle each question methodically. You're ready to rock the AP Physics 1 exam! 🎉