AP Physics 1: Unit 3 - Circular Motion and Gravitation 🚀

Hey there, future physicist! Let's get you prepped for Unit 3. This unit is all about how things move in circles and how gravity plays a role. It might seem a bit abstract, but we'll break it down step-by-step. Remember, you've got this! 💪

This unit makes up 4-6% of the AP exam, so it's not the biggest chunk, but mastering these concepts will help you in other areas too! Think of it as a bridge to more complex physics.

Image courtesy of makeameme.org

Big Ideas: The Foundation 🧱

• Big Idea #1: Systems - Objects have mass and structure. Think of planets, cars, even tiny particles as systems.
• Big Idea #2: Fields - Invisible forces (like gravity) act through fields. It's like an invisible hand that pulls things together.
• Big Idea #3: Force Interactions - Forces describe how objects interact. Every action has a reaction (thanks, Newton!).
• Big Idea #4: Change - Interactions cause changes in systems. Motion, speed, and direction can all change through these interactions.

Key Concepts: What You Need to Know 🧠

• Vector - Magnitude and direction (like velocity and force).
• Vector Field - A map of vectors (like gravitational field).
• Uniform Circular Motion - Moving in a circle at a constant speed.
• Centripetal Force - The force that keeps things moving in a circle.
• Centrifugal Force - Spoiler alert: This is not a real force! (More on this later).
• Gravitational Force - Attraction between masses.
• Newton’s Universal Law of Gravitation - The formula for gravitational force.
• Gravitational Field - The field created by masses that exerts gravitational force.
• Gravitational Mass - How strongly an object interacts with gravity.
• Inertial Mass - How much an object resists acceleration.
• Frame of Reference - Your perspective on motion.

Key Equations: Your Toolkit 🧰

3.1 Vector Fields

• Velocity Vector Field: In circular motion, the velocity vector is always tangent to the circle. Its magnitude (speed) is constant, but its direction is always changing.

• Other Vector Fields: Vector fields can represent force, acceleration, and magnetic fields. They're a powerful tool for visualizing how these quantities change in space.

• Simplifying: In AP Physics 1, we often simplify vector fields into single vectors. For instance, the gravitational force between Earth and the moon is represented by two vectors, each pointing towards the other object.

  

  Image courtesy of School for Champions.

  • The Earth pulls on the moon (vector on the moon pointing left).
  • The moon pulls on the Earth (vector on the Earth pointing right).
  • The Earth's pull is stronger, so the moon orbits it.

3.2 Fundamental Forces

• Gravitational Force: The force of attraction between masses. It's the weakest force but dominates at large scales (planets, galaxies).

• Electromagnetic Force: Interactions of charged particles. Responsible for electricity, magnetism, and light.

• Weak Force: Responsible for radioactive decay.

• Strong Force: Holds the nucleus of an atom together.

• Long-Range Force: Gravity is a long-range force, meaning its effects are felt over large distances. It never completely disappears, just gets weaker with distance.

• Dominance: Gravity dominates at large mass and distance scales because its strength is proportional to the mass of the objects involved. This is why you're not pulled towards your coffee cup, but the Earth pulls you down.

3.3 Gravitational Force

The equation for the gravitational force is Newton's Universal Law of Gravitation:

$$F = G \frac{m_1 m_2}{r^2}$$

• F is the force of gravity.

• m1 and m2 are the masses of the two objects.

• r is the distance between the centers of the objects.

• G is the gravitational constant (a tiny number).

• Always Attractive: Gravity always pulls objects together. It's what keeps us on Earth and planets in orbit.

• Universal: This law applies to every object in the universe, from your notebook to the most distant galaxy!

3.4 Gravitational Field/Acceleration Due to Gravity

• Gravitational Acceleration (g): The acceleration experienced by an object due to gravity. On Earth, it's about 9.8 m/s². But it's different on other planets.

• Surface Gravity Equation: The equation for gravitational acceleration on any planet is:

$$g = G \frac{M}{R^2}$$

*   `G` is the gravitational constant.
*   `M` is the mass of the planet.
*   `R` is the radius of the planet.

• Gravitational Field: A region around an object where another object experiences a gravitational force. The strength of the field is represented by g.

3.5 Inertial vs. Gravitational Mass

• Inertial Mass: Measures an object's resistance to acceleration. Think of it as how "lazy" an object is to change its motion. A bowling ball has a higher inertial mass than a feather.

• Gravitational Mass: Measures how strongly an object interacts with gravity. It's what determines the force of gravity on an object.

• The Same Value: Amazingly, the inertial and gravitational mass of an object are always the same. This is a fundamental principle of physics.

• Bowling Ball vs. Feather: In a vacuum, both fall at the same rate because the ratio of gravitational force to inertial mass is the same for all objects. Air resistance is what makes them fall differently on Earth.

3.6 Centripetal Acceleration and Centripetal Force

• Centripetal Acceleration (a_c): The acceleration of an object moving in a circle at a constant speed. It's always directed towards the center of the circle.

  $$a_c = \frac{v^2}{r}$$

  • v is the velocity of the object.
  • r is the radius of the circle.

• Velocity and Acceleration: In circular motion, velocity is always tangent to the circle, and acceleration is always perpendicular to the velocity, pointing towards the center.

• Centripetal Force (F_c): The net force that causes centripetal acceleration. It's not a new force, but rather the net force in circular motion. It could be tension, gravity, normal force, or friction.

  $$F_c = m a_c = m \frac{v^2}{r}$$

  • m is the mass of the object.

3.7 Free-Body Diagrams for Objects in Uniform Circular Motion

• Free-Body Diagrams (FBDs): Visual representation of forces acting on an object. Essential for analyzing motion, including circular motion.

• Coordinate System: Choose a coordinate system where the positive direction is towards the center of the circle. This makes calculations easier.

• Forces: Identify all forces acting on the object. Break forces into components if they don't align with your coordinate system.

  

  Image courtesy of APlusPhysics.

  • Example: A rollercoaster at the top of a loop. Both weight (mg) and normal force (N) point downwards, towards the center of the circle. The net force is the centripetal force.

3.8 Applications of Circular Motion and Gravitation

• Inertial Frame of Reference: A frame where an object is at rest or moving at a constant velocity. In an inertial frame, the laws of physics are the same for all observers.

• Non-Inertial Frame of Reference: A frame where an object is accelerating. In non-inertial frames, you might experience "fictitious" forces, like the feeling of being pushed to the side in a turning car.

• Rotational Kinematics: Linear quantities (position, velocity, acceleration) have rotational analogs:

  • Position: Angle (θ) in radians
  • Velocity: Angular velocity (ω) in radians per second
  • Acceleration: Angular acceleration (α) in radians per second squared

• Rotational Kinematics Equations: Similar to linear kinematics equations, but use rotational quantities. Only work for constant angular acceleration.

  

Final Exam Focus 🎯

• High-Priority Topics: Gravitational force, centripetal force, and free-body diagrams for circular motion. These concepts are frequently tested.
• Common Question Types:
  • Calculating gravitational force and acceleration.
  • Analyzing circular motion scenarios with FBDs.
  • Applying centripetal force equations.
  • Conceptual questions about inertial vs. gravitational mass and frames of reference.
• Time Management: Don't spend too long on a single question. If you're stuck, move on and come back later. Remember to show all your work for FRQs.
• Common Pitfalls:
  • Confusing centripetal force with a new force (it's the net force).
  • Forgetting that centripetal acceleration points towards the center of the circle.
  • Incorrectly drawing FBDs.
  • Not converting units to SI units.
• Strategies for Challenging Questions:
  • Draw a clear FBD.
  • Write down all knowns and unknowns.
  • Use the appropriate equations.
  • Check your units and answers for reasonableness.

Practice Questions

Practice Question

Multiple Choice Questions

1. A satellite is orbiting Earth in a circular path. If the radius of the orbit is doubled, what happens to the gravitational force between the Earth and the satellite? (A) It is doubled. (B) It is halved. (C) It is quadrupled. (D) It is reduced to one-fourth.

2. A car is moving around a circular track at a constant speed. Which of the following statements is true about the car's acceleration? (A) The car's acceleration is zero. (B) The car's acceleration is constant and in the direction of motion. (C) The car's acceleration is constant and directed towards the center of the circle. (D) The car's acceleration is changing in both magnitude and direction.

3. A ball is swung in a vertical circle at the end of a string. At which point in the circle is the tension in the string the greatest? (A) At the top of the circle. (B) At the bottom of the circle. (C) When the string is horizontal. (D) The tension is the same at all points.

Free Response Question

A small block of mass m is placed on a horizontal turntable that is rotating at a constant angular speed ω. The block is located a distance r from the center of the turntable. The coefficient of static friction between the block and the turntable is μs.

(a) On the dot below, which represents the block, draw and label the forces (not components) that act on the block when the turntable is rotating. Each force must be represented by a distinct arrow starting on, and pointing away from, the dot.

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(b) Derive an expression for the maximum speed the block can have without slipping off the turntable in terms of m, g, μs and r.

(c) If the turntable is now tilted at an angle θ with respect to the horizontal, how does this affect the maximum speed the block can have without slipping? Explain your reasoning.

Scoring Guidelines

(a) 3 points

• 1 point for correctly drawing and labeling the normal force (N) pointing upwards.
• 1 point for correctly drawing and labeling the weight (mg) pointing downwards.
• 1 point for correctly drawing and labeling the static friction force (fs) pointing towards the center of the circle.

(b) 4 points

• 1 point for recognizing that the static friction force provides the centripetal force: fs = Fc
• 1 point for using the correct formula for centripetal force: Fc = mv²/r
• 1 point for using the correct formula for maximum static friction: fs = μsN
• 1 point for setting the centripetal force equal to the static friction and solving for v: v = √(μsgr)

(c) 2 points

• 1 point for stating that the maximum speed will decrease.
• 1 point for explaining that the normal force will decrease, which reduces the maximum static friction force available to provide the centripetal force.

Answer Key

Multiple Choice:

1. (D)
2. (C)
3. (B)

Free Response: (a) A dot with three arrows: Normal force (up), weight (down), and static friction (towards the center). (b) v = √(μsgr) (c) The maximum speed will decrease. Tilting the turntable reduces the normal force, which in turn reduces the maximum static friction force, which is needed to provide the centripetal force.

Alright, you've made it through Unit 3! You're now equipped with the knowledge and tools to tackle circular motion and gravitation. Go get 'em! 🌟