#AP Physics C: Mechanics - Rotational Dynamics 🚀

Hey! Let's get you prepped for the AP Physics C: Mechanics exam with a focus on rotational dynamics. We'll break down Newton's Second Law for rotation, explore how mass distribution affects rotational inertia, and make sure you're ready to tackle any problem they throw at you. Let's dive in!

#Torque and Angular Acceleration

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• Newton's Second Law for Rotation 🌀: This is your go-to for rotational motion. It states that the net torque on an object is equal to the product of its rotational inertia and angular acceleration.

  $$\sum \tau = I \alpha$$

• Rotational Inertia ($I$): Think of this as the rotational version of mass. It measures how much an object resists changes in its rotational motion. The bigger the $I$, the harder it is to start or stop it spinning.

• Torque ($\tau$): This is the rotational equivalent of force. It's what causes an object to start rotating or change its rotation. Remember, torque isn't just about force, but also where that force is applied!

#Direction of Angular Acceleration

• Torque and Angular Acceleration Vectors: The net torque vector and the resulting angular acceleration vector always point in the same direction. This means that if you apply a torque in a certain direction, the object will accelerate its rotation in that direction.
• Clockwise vs. Counterclockwise: A counterclockwise torque will cause a counterclockwise angular acceleration, and a clockwise torque will cause a clockwise angular acceleration. Simple as that!
• Vector Addition: To find the direction of the angular acceleration, you'll need to add up all the individual torques acting on the object as vectors. This will give you the net torque vector, which points in the direction of the angular acceleration.

#Mass Distribution Effects

• Mass Distribution Matters: An object's rotational inertia depends not only on its total mass but also on how that mass is distributed relative to the axis of rotation. This is super important!

• Farther = More Inertia: The farther the mass is from the axis of rotation, the larger the rotational inertia ($I$). This means it's harder to change the object's rotational speed.

• Hollow vs. Solid: A hollow cylinder has a greater $I$ than a solid cylinder of the same mass and radius because its mass is distributed farther from the center 🎡. This is why a hollow pipe rolls down a ramp slower than a solid cylinder.

  

  Caption: A hollow cylinder has a greater rotational inertia than a solid cylinder of the same mass and radius.

• Spinning Skater: When a figure skater pulls their arms in while spinning, they reduce their rotational inertia ($I$), which causes them to spin faster. This is due to the conservation of angular momentum.

• Calculating Rotational Inertia: For a system of point masses, use $I = \sum mr^2$. For continuous mass distributions, use $I = \int r^2 dm$. Don't worry, you'll usually be given the formula for the rotational inertia of common shapes.

#Connecting Linear and Rotational Motion

• Tangential Speed: The tangential speed ($v$) of a point on a rotating object is related to its angular speed ($\omega$) by $v = r\omega$, where $r$ is the distance from the axis of rotation. This is a crucial link between linear and rotational motion.
• Tangential Acceleration: Similarly, the tangential acceleration ($a$) is related to the angular acceleration ($\alpha$) by $a = r\alpha$. This shows how angular acceleration translates to linear acceleration at a point on the rotating object.
• Rolling Without Slipping: When an object rolls without slipping, the linear speed of its center of mass is related to its angular speed by $v_{cm} = R\omega$, where $R$ is the radius of the object. This is a key concept for analyzing rolling motion.

#Final Exam Focus

#Key Topics to Nail:

• Newton's Second Law for Rotation: $\sum \tau = I \alpha$ is your bread and butter. Know it inside and out.
• Rotational Inertia: Understand how mass distribution affects $I$ and be able to calculate it for simple shapes.
• Torque: Master the concept of torque and how to calculate it, including the direction of the torque vector.
• Rolling Motion: Be comfortable with the relationships between linear and angular motion, especially for rolling without slipping.

#Common Question Types:

• Multiple Choice: Expect conceptual questions about how mass distribution affects rotational inertia and how torques cause angular acceleration.
• Free Response: Look for problems involving rolling objects, systems with multiple torques, and situations where you need to apply both linear and rotational dynamics. Be ready to derive equations and show your work clearly.

#Last-Minute Tips:

• Time Management: Don't spend too long on a single question. If you get stuck, move on and come back to it later.
• Show Your Work: Even if you don't get the final answer, you can get partial credit for showing your work and using the correct equations.
• Check Units: Make sure your units are consistent throughout your calculations.
• Draw Diagrams: A free body diagram can be a lifesaver when dealing with torques and forces.

#Practice Questions

Practice Question

#Multiple Choice Questions

1. A solid cylinder and a hollow cylinder of the same mass and radius are released from rest at the top of an incline. Which cylinder reaches the bottom first? (Assume they roll without slipping). (A) The solid cylinder (B) The hollow cylinder (C) They reach the bottom at the same time (D) It depends on the angle of the incline

2. A torque is applied to a rotating object. If the rotational inertia of the object is doubled, what happens to the angular acceleration if the torque remains constant? (A) It doubles (B) It quadruples (C) It is halved (D) It is quartered

3. A figure skater pulls her arms inward during a spin. Which of the following quantities decreases? (A) Angular speed (B) Rotational inertia (C) Angular momentum (D) Kinetic energy

#Free Response Question

A uniform solid disk of mass $M$ and radius $R$ is mounted on a frictionless horizontal axle. A string is wrapped around the disk, and a block of mass $m$ is attached to the free end of the string. The block is released from rest and falls, causing the disk to rotate.

(a) On the diagram below, draw and label all the forces acting on the block and on the disk.

(b) Derive an expression for the linear acceleration of the block in terms of $M$, $m$, $R$, and $g$.

(c) Derive an expression for the tension in the string in terms of $M$, $m$, and $g$.

(d) If the disk was replaced with a hollow cylinder of the same mass and radius, would the linear acceleration of the block be greater, smaller, or the same? Explain your reasoning.

#Scoring Rubric

(a) Free Body Diagrams (4 points)

• 1 point for correct forces on the block (tension upwards, weight downwards)
• 1 point for correct forces on the disk (tension tangential, weight downwards, normal force upwards)
• 1 point for correct direction of forces
• 1 point for labeling forces correctly

(b) Linear Acceleration (6 points)

• 2 points for correct application of Newton's second law on the block: $mg - T = ma$
• 2 points for correct application of Newton's second law for rotation on the disk: $TR = I\alpha$ and $I = \frac{1}{2}MR^2$
• 1 point for using the relationship between linear and angular acceleration: $a = R\alpha$
• 1 point for correct final answer: $a = \frac{mg}{m + \frac{M}{2}}$

(c) Tension in the String (3 points)

• 1 point for using the correct expression for acceleration from part (b)
• 1 point for substituting the acceleration into the force equation from part (b)
• 1 point for correct final answer: $T = \frac{Mmg}{2m+M}$

(d) Comparison with Hollow Cylinder (2 points)

• 1 point for stating that the acceleration would be smaller
• 1 point for correct reasoning that the hollow cylinder has a larger rotational inertia

Alright, you've got this! Keep reviewing, stay calm, and you'll do great on the exam. Good luck! 🌟