Linear and Rotational Motion: A Last-Minute Review 🚀

Hey there, future AP Physics 1 master! Let's get these concepts locked down for your exam. Remember, linear and rotational motion aren't separate worlds—they're two sides of the same coin. We'll explore how they connect, making those tricky circular motion problems much easier. Let's dive in!

Rotational motion is a high-value topic on the AP exam. Expect to see it in both multiple-choice and free-response questions. Understanding the relationship between linear and angular quantities is crucial.

🔗 Linking Linear and Rotational Motion

Distance Traveled During Rotation

• Key Idea: When something rotates, points on that object travel a linear distance. This distance depends on how far the point is from the center and how much the object has rotated.

• Formula: $Delta s = rDelta\theta$ 📏

  • $Delta s$ is the linear distance traveled.
  • $r$ is the radius (distance from the axis of rotation).
  • $Delta\theta$ is the angle of rotation (in radians).

• Analogy: Imagine a point on a spinning bike wheel. As the wheel turns, the point moves along the edge. The distance it travels is the arc length, $\Delta s$.

• Example: If a point is 0.2 m from the center of a wheel and the wheel rotates by $\pi$ radians (180°), the point travels $\Delta s = (0.2 \text{ m})(\pi \text{ rad}) = 0.2\pi \text{ m}$.

Linear vs. Angular Quantities

• Key Idea: Linear and angular quantities are related by the radius of rotation. These equations help us translate between the two.

• Formulas:

  • $s = r\theta$ (arc length)
  • $v = r\omega$ (linear velocity)
  • $a_t = r\alpha$ (tangential acceleration)
  • Where:
    • $s$ is the arc length (linear distance).
    • $v$ is the linear velocity.
    • $a_t$ is the tangential acceleration.
    • $\theta$ is the angular displacement (in radians).
    • $\omega$ is the angular velocity.
    • $\alpha$ is the angular acceleration.

• Analogy: Think of a merry-go-round. Points farther from the center travel a greater linear distance and have a higher linear velocity than points closer to the center, even though they all have the same angular velocity.

• Examples:

  • If you walk 1/4 of the way around a circle with a radius of 2 m, you've gone $s = (2 \text{ m})(\frac{\pi}{2} \text{ rad}) = \pi \text{ m}$.
  • A point 0.5 m from the center of a disk spinning at 10 rad/s has a linear velocity of $v = (0.5 \text{ m})(10 \text{ rad/s}) = 5 \text{ m/s}$.
  • If a wheel is speeding up at 2 rad/s², a point 0.3 m from the axis feels a tangential acceleration of $a_t = (0.3 \text{ m})(2 \text{ rad/s}^2) = 0.6 \text{ m/s}^2$.

Uniform Angular Motion

• Key Idea: In a rigid rotating system, all points share the same angular velocity and angular acceleration.

• Rigid System: A system that maintains its shape and size during rotation.

• Uniform Angular Velocity: Every point has the same $\omega$, regardless of its distance from the center. They all complete one full rotation in the same amount of time.

• Uniform Angular Acceleration: Every point has the same $\alpha$, meaning they all experience the same rate of change in angular velocity.

• Visual: Imagine a spinning CD. All points on the CD have the same angular velocity and acceleration, even though points near the edge travel much faster than those near the center.

🚫 Boundary Statements:

• Rotational directions are limited to clockwise and counterclockwise with respect to a given rotational axis on the exam.

Final Exam Focus 🎯

• Highest Priority Topics:

  • Relationships between linear and angular displacement, velocity, and acceleration.
  • Uniform circular motion and its connection to rotational motion.
  • Applying the formulas $s = r\theta$, $v = r\omega$, and $a_t = r\alpha$.
  • Understanding that angular velocity and angular acceleration are uniform in a rigid system.

• Common Question Types:

  • Multiple-choice questions testing your understanding of the relationships between linear and angular quantities.
  • Free-response questions involving calculations of arc length, linear velocity, and tangential acceleration in rotating systems.
  • Problems that combine rotational motion with other concepts, such as energy conservation or Newton's laws.

• Last-Minute Tips:

  • Double-check that your angles are in radians before plugging them into formulas.
  • Clearly label your variables and units. This will help you avoid errors and earn partial credit if you make a mistake.
  • If you get stuck on a problem, try drawing a diagram. This can help you visualize the situation and identify the relevant quantities.
  • Don't panic! Take a deep breath, read the question carefully, and trust your preparation. You've got this!

Practice Questions

Practice Question

Multiple Choice Questions

1. A bicycle wheel with a radius of 0.4 m rotates at a constant angular speed of 5 rad/s. What is the linear speed of a point on the edge of the wheel? (A) 0.8 m/s (B) 2 m/s (C) 2.5 m/s (D) 12.5 m/s

2. A disc rotates about a fixed axis. Which of the following statements is true about the angular velocity and angular acceleration of different points on the disc? (A) Both angular velocity and angular acceleration are different for different points. (B) Angular velocity is the same for all points, but angular acceleration is different. (C) Angular acceleration is the same for all points, but angular velocity is different. (D) Both angular velocity and angular acceleration are the same for all points.

3. A particle moves in a circle of radius 2 m with an angular velocity given by $\omega(t) = 3t^2 - 2t + 1$ (in rad/s). What is the magnitude of the tangential acceleration of the particle at t=1 s? (A) 2 m/s² (B) 4 m/s² (C) 8 m/s² (D) 10 m/s²

Free Response Question

A uniform disk of radius $R = 0.5 \text{ m}$ and mass $M = 2 \text{ kg}$ is initially at rest. A constant tangential force $F = 10 \text{ N}$ is applied to the edge of the disk, causing it to rotate about its center. Assume the disk rotates in the counter-clockwise direction. The moment of inertia of the disk is given by $I = \frac{1}{2}MR^2$.

(a) Calculate the moment of inertia of the disk. (b) Calculate the torque applied to the disk. (c) Calculate the angular acceleration of the disk. (d) Calculate the angular velocity of the disk after 3 seconds. (e) Calculate the tangential speed of a point on the edge of the disk after 3 seconds.

Scoring Breakdown:

(a) (2 points)

• 1 point for correct formula: $I = \frac{1}{2}MR^2$
• 1 point for correct calculation: $I = \frac{1}{2}(2 \text{ kg})(0.5 \text{ m})^2 = 0.25 \text{ kg m}^2$

(b) (2 points)

• 1 point for correct formula: $\tau = rF$
• 1 point for correct calculation: $\tau = (0.5 \text{ m})(10 \text{ N}) = 5 \text{ N m}$

(c) (2 points)

• 1 point for correct formula: $\tau = I\alpha$
• 1 point for correct calculation: $\alpha = \frac{\tau}{I} = \frac{5 \text{ N m}}{0.25 \text{ kg m}^2} = 20 \text{ rad/s}^2$

(d) (2 points)

• 1 point for using the kinematic equation: $\omega = \omega_0 + \alpha t$
• 1 point for correct calculation: $\omega = 0 + (20 \text{ rad/s}^2)(3 \text{ s}) = 60 \text{ rad/s}$

(e) (2 points)

• 1 point for correct formula: $v = r\omega$
• 1 point for correct calculation: $v = (0.5 \text{ m})(60 \text{ rad/s}) = 30 \text{ m/s}$

Keep up the great work, and you'll ace this exam! You've got this! 💪