#AP Physics C: Mechanics - Work & Energy in Rotational Motion 🚀

Hey there! Let's get you prepped for the exam with a super-focused review of rotational work and energy. We'll break down the concepts, highlight key points, and make sure you're feeling confident and ready to ace this section!

#1. Work Done by Torque

#1.1 Energy Transfer by Torque

• Torque, a rotational force, can transfer energy to or from an object when applied over an angular displacement. Think of it like pushing a door open 🚪—you're applying torque, and the door gains energy.

• The direction of the torque relative to the angular displacement determines whether energy is added to or removed from the system. Clockwise torque with clockwise displacement adds energy; clockwise torque with counterclockwise displacement removes energy.

• Example: Applying a clockwise torque to a door handle while opening the door transfers energy into the door, causing it to rotate.

#1.2 Work-Torque Relationship

• The work done on a rigid system by a torque depends on the magnitude of the torque and the angular displacement the system rotates through while the torque is applied.

• Calculating work involves integrating the torque with respect to the angular displacement over the interval of interest.

• Equation: $$W=\int_{\theta_{1}}^{\theta_{2}} \tau d \theta$$

  • $W$ represents the work done by the torque.
  • $\tau$ represents the magnitude of the torque as a function of angular position.
  • $\theta_{1}$ and $\theta_{2}$ represent the initial and final angular positions, respectively.

#1.3 Graphical Work Analysis

• Graphing torque as a function of angular position provides a visual representation of the work done by the torque.

• Positive areas represent positive work (energy added to the system), while negative areas represent negative work (energy removed from the system).

• Graphical analysis is particularly useful when the torque varies with angular position, as it allows for the calculation of work without explicitly integrating the torque function.

#2. Connecting Concepts & Exam Focus

• Energy Conservation: Remember that work done by torque can change the rotational kinetic energy of a system. This ties directly into the conservation of energy principles.

• Relationship to Linear Motion: Just like force does work in linear motion, torque does work in rotational motion. The concepts are parallel, so use your understanding of linear work to help with rotational work.

• Pay close attention to problems that combine rotational motion with linear motion. These are common on the exam.

• Graphical Analysis: Always be ready to interpret torque vs. angular position graphs. The area under the curve is your friend! 📈

#3. Final Exam Focus

• High-Priority Topics:

  • Work-energy theorem for rotational motion
  • Calculating work from torque and angular displacement
  • Graphical analysis of torque vs. angular position
  • Conservation of energy in rotational systems

• Common Question Types:

  • Multiple-choice questions involving the calculation of work done by a constant or variable torque.
  • Free-response questions requiring the application of the work-energy theorem in rotational systems.
  • Problems that combine linear and rotational motion, often involving rolling objects.

• Last-Minute Tips:

  • Time Management: Don't spend too long on a single question. If you're stuck, move on and come back later.
  • Common Pitfalls: Be careful with units (radians vs. degrees). Make sure you're using the correct sign conventions for work and torque.
  • Strategies: Draw free-body diagrams for rotational motion problems. Visualize the direction of torque and rotation. Always check your answers for reasonableness.

#4. Practice Questions

Practice Question

#Multiple Choice Questions:

MCQ 1: A wheel with a radius of 0.2 m has a constant torque of 10 Nm applied to it. If the wheel rotates through an angle of 2π radians, the work done by the torque is:

(A) 2π J (B) 4π J (C) 10 J (D) 20π J

MCQ 2: A torque is applied to a rotating object causing its angular displacement to change. Which of the following is true about the work done by the torque?

(A) It is always positive. (B) It is always negative. (C) It can be positive, negative, or zero depending on the direction of torque and angular displacement. (D) It is only positive when the object is speeding up.

#Free Response Question:

A uniform solid cylinder of mass M and radius R is initially at rest on a horizontal surface. A constant force F is applied tangentially to the edge of the cylinder, causing it to roll without slipping. The cylinder has a moment of inertia $I = \frac{1}{2}MR^2$.

(a) Calculate the torque applied by the force F about the center of the cylinder. (2 points)

(b) Determine the angular acceleration of the cylinder. (3 points)

(c) Calculate the work done by the force F as the cylinder rotates through an angle θ. (3 points)

(d) Determine the final rotational kinetic energy of the cylinder after it has rotated through the angle θ. (2 points)

#Solutions:

MCQ 1 Solution:

(D) The work done by a constant torque is given by $W = \tau \Delta \theta$. In this case, $W = (10 \text{ Nm})(2\pi \text{ rad}) = 20\pi \text{ J}$.

MCQ 2 Solution:

(C) The work done by torque can be positive, negative, or zero depending on the relative direction of the torque and the angular displacement.

FRQ Solution:

(a) The torque is given by $\tau = rF$, where r is the radius and F is the force. Since the force is applied tangentially, the lever arm is equal to the radius. So, $\tau = RF$ (2 points)

(b) Using Newton's second law for rotation, $\tau = I\alpha$, where I is the moment of inertia and \alpha is the angular acceleration. So, $RF = \frac{1}{2}MR^2\alpha$. Solving for \alpha, we get $\alpha = \frac{2F}{MR}$ (3 points)

(c) The work done by the force is given by $W = \tau \theta$. Since $\tau = RF$, we have $W = RF\theta$ (3 points)

(d) The rotational kinetic energy is equal to the work done, so $KE_{rot} = W = RF\theta$ (2 points)

Keep up the great work! You've got this! 💪