#AP Physics C: Mechanics - Linear and Rotational Motion 🔄

Hey there, future physics master! Let's get you prepped for the exam with a super-focused review of linear and rotational motion. We'll make sure you're not just memorizing formulas, but truly understanding how everything connects. Let's dive in!

#Connecting Linear and Rotational Motion

This section is all about how linear motion (like moving in a straight line) and rotational motion (like spinning) are related. It’s like discovering the secret language that connects two seemingly different worlds of motion. Think of it as translating between two dialects of physics! 💡

#Key Relationships

• Displacement: Linear distance ($s$) is related to angular displacement ($\theta$) by: $s = r\theta$. This is your go-to for translating between how far something spins and how far it moves along a circle.
• Velocity: Linear velocity ($v$) is related to angular velocity ($\omega$) by: $v = r\omega$. This links how fast something spins to how fast it moves along a circle.
• Acceleration: Tangential acceleration ($a_T$) is related to angular acceleration ($\alpha$) by: $a_T = r\alpha$. This connects how quickly something's spin changes to how quickly its linear speed changes along the circle.

#Linear Motion of a Rotating Point

Let's zoom in on a single point as it rotates. This helps us understand how linear motion arises from circular motion.

#Linear Distance and Angle

• Concept: When a point rotates around a fixed axis, the linear distance it travels is directly proportional to the angle it sweeps out. Think of a point on a spinning wheel – the bigger the angle, the further it travels along the circle.
• Formula: $\Delta s = r \Delta \theta$ where:
  • $\Delta s$ is the linear distance traveled
  • $r$ is the radius
  • $\Delta \theta$ is the angular displacement (in radians)
• Analogy: Imagine a point on the edge of a spinning CD 💿. The further out from the center, the greater the linear distance covered for the same angle.

#Velocity and Acceleration Relationships

• Linear Velocity ($v$): The linear speed of a rotating point is given by $v = r\omega$, where $\omega$ is the angular velocity. The faster it spins, the faster the point moves along its circular path.
  • Example: A point 0.5 m from the axis rotating at 4 rad/s has a linear velocity of (0.5 m)(4 rad/s) = 2 m/s.
• Tangential Acceleration ($a_T$): The linear acceleration of a rotating point is given by $a_T = r\alpha$, where $\alpha$ is the angular acceleration. The faster its spin changes, the faster its linear speed changes along the circular path.
  • Example: A point 0.2 m from the axis with an angular acceleration of 5 rad/s² has a tangential acceleration of (0.2 m)(5 rad/s²) = 1 m/s².

#Angular Velocity and Acceleration

• Rigid Systems: In a rigid system (like a spinning top or a rolling wheel), all points have the same angular velocity ($\omega$) and angular acceleration ($\alpha$) at any given instant.
• Key Idea: Even though points further out have greater linear speeds, they all rotate through the same angle in the same time, meaning they share the same $\omega$.
• Same \alpha: Similarly, all points experience the same rate of change of angular velocity, or $\alpha$, regardless of their distance from the axis.

#Important Boundary Statements

• Vector Manipulation: On the AP exam, you'll manipulate the magnitudes of angular displacement, velocity, and acceleration using vector conventions, but you will not be assessed on the directions of the vectors.
• Direction Descriptions: Descriptions of the directions of rotational kinematics quantities are limited to clockwise and counterclockwise with respect to a given axis of rotation. This simplifies things a bit!

#Final Exam Focus

Okay, let's get down to the nitty-gritty of what you really need to focus on for the exam:

• High-Value Topics:
  • Understanding and applying the relationships between linear and angular quantities ($s = r\theta$, $v = r\omega$, $a_T = r\alpha$).
  • Analyzing the motion of rigid bodies, recognizing that all points share the same $\omega$ and $\alpha$.
• Common Question Types:
  • Problems that require converting between linear and angular quantities.
  • Questions involving objects rolling without slipping.
  • Scenarios where you need to analyze the motion of different points on a rotating object.
• Time Management:
  • Quickly identify whether a problem involves linear, rotational, or both types of motion.
  • Use the correct formulas and units, double-checking your work.
• Common Pitfalls:
  • Forgetting to convert degrees to radians when using rotational formulas.
  • Mixing up linear and angular quantities.
  • Not considering the radius when calculating linear quantities from angular ones.

Focus on problems that combine linear and rotational motion. These are common on the exam and show a deep understanding of the concepts. Practice makes perfect!

#Practice Questions

Practice Question

Multiple Choice Questions

1. A wheel with a radius of 0.4 m is rotating at a constant angular speed of 10 rad/s. What is the linear speed of a point on the edge of the wheel? (A) 2 m/s (B) 4 m/s (C) 8 m/s (D) 10 m/s (E) 25 m/s

2. A disc starts from rest and accelerates with a constant angular acceleration of 2 rad/s². What is the angular speed of the disc after 5 seconds? (A) 2 rad/s (B) 5 rad/s (C) 10 rad/s (D) 25 rad/s (E) 50 rad/s

3. A point on a rotating disk is 0.2 meters from the center, and it has a linear speed of 4 m/s. What is the angular speed of the disk? (A) 0.8 rad/s (B) 2 rad/s (C) 8 rad/s (D) 20 rad/s (E) 40 rad/s

Free Response Question

A solid cylinder of mass M and radius R is initially at rest on a horizontal surface. A string is wrapped around the cylinder and pulled with a constant force F, causing the cylinder to roll without slipping. The moment of inertia of a solid cylinder about its central axis is (1/2)MR^2. (a) Draw a free-body diagram of the cylinder, showing all forces acting on it.

(b) Write the net force equation in the horizontal direction.

(c) Write the net torque equation about the center of mass of the cylinder.

(d) Determine the linear acceleration of the center of mass of the cylinder in terms of given variables.

(e) Determine the angular acceleration of the cylinder in terms of given variables.

Answer Key and Scoring Rubric

Multiple Choice Answers

1. (B) $v = r\omega = (0.4 m)(10 rad/s) = 4 m/s$
2. (C) $\omega = \alpha t = (2 rad/s^2)(5 s) = 10 rad/s$
3. (D) $\omega = v/r = (4 m/s) / (0.2 m) = 20 rad/s$

Free Response Question Scoring

(a) (2 points) - 1 point for correct weight (mg) and normal force (N) acting on the center of the cylinder. - 1 point for correct tension (F) and friction (f) forces acting at the point of contact with the ground.

(b) (1 point) - $\sum F_x = F - f = Ma_{cm}$

(c) (2 points) - 1 point for correct torque equation $\tau = I \alpha$ - 1 point for correct torque due to friction $\tau = fR$ - $fR = (1/2)MR^2 \alpha$

(d) (2 points) - 1 point for recognizing the relationship $a_{cm} = R\alpha$ - 1 point for correct linear acceleration $a_{cm} = (2/3) F/M$

(e) (2 points) - 1 point for correct angular acceleration $\alpha = a_{cm}/R$ - 1 point for correct angular acceleration $\alpha = (2/3) F/(MR)$

Alright, you've got this! Remember, the key is to understand the relationships between linear and rotational motion and to practice applying these concepts. You're well on your way to acing that exam!