#Rotational Work and Energy Transfer ⚙️

Hey there, future AP Physics 1 master! Let's dive into the world of rotational motion and energy transfer. This section focuses on how torque does work and how to calculate it. Get ready to make some connections!

#Energy Transfer by Torque

#How Torque Transfers Energy

• Torque is like the rotational version of force. It can transfer energy into or out of a system when it acts over an angular displacement. Think of it like pushing something to make it rotate. 💪
  • If the torque and angular displacement are in the same direction, energy is transferred into the system (it speeds up).
  • If the torque and angular displacement are in opposite directions, energy is transferred out of the system (it slows down).

Use the right-hand rule to visualize the direction of torque and angular displacement.

#Work-Torque Relationship

#The Equation

• The work done by a torque is determined by the magnitude of the torque and the angular displacement:

  $$W = \tau \Delta\theta$$

  - $W$ = work done (in joules, J) - $\tau$ = torque (in newton-meters, N⋅m) - $\Delta\theta$ = angular displacement (in radians, rad)

#Understanding the Variables

• Torque ($\tau$): The rotational equivalent of force. It's what causes an object to rotate.
• Angular Displacement ($\Delta\theta$): How much the object has rotated (in radians). Think of it as the angle through which the object has turned.

#Torque-Angle Graphs

#Visualizing Work

• Torque-angle graphs show how torque changes with angular position. The area under the curve of a torque vs. angular position graph represents the work done. 📈
  • Vertical axis (y-axis): Torque ($\tau$)
  • Horizontal axis (x-axis): Angular position ($\theta$)
  • Area under the curve: Total work done

The area under the curve represents the work done by the torque.

#Interpreting the Area

• Constant Torque: The area is a simple rectangle (base × height).
• Varying Torque: Approximate the area using smaller rectangles or triangles.
• Sign of Work:
  • Positive area (torque and displacement in the same direction) = positive work (energy into the system)
  • Negative area (torque and displacement in opposite directions) = negative work (energy out of the system)

#Final Exam Focus

#Key Areas to Review

• Work-Energy Theorem: How work done by torque relates to changes in rotational kinetic energy.
• Rotational Inertia: How an object's mass distribution affects its rotational motion.
• Conservation of Energy: Applying energy conservation principles to rotating systems.

#Common Question Types

• Multiple Choice: Conceptual questions about energy transfer and the sign of work.
• Free Response: Problems involving calculating work from torque and angular displacement, often with varying torques.

#Last-Minute Tips

• Time Management: Quickly identify the core concepts in the problem and focus on applying the relevant equations.
• Common Pitfalls: Watch out for unit conversions (especially radians!) and pay close attention to the direction of torque and angular displacement.
• Strategies: Draw free body diagrams, and always think about the energy flow in the system.

#Practice Questions

Practice Question

#Multiple Choice Questions

1. A wheel rotates through an angle of 10 radians under the influence of a constant torque of 5 N⋅m. How much work is done by the torque? (A) 2 J (B) 5 J (C) 50 J (D) 100 J

2. A torque-angle graph shows a triangular area with a base of 4 radians and a height of 10 N⋅m. What is the work done by the torque? (A) 10 J (B) 20 J (C) 40 J (D) 80 J

3. A rotating object experiences a torque that opposes its direction of rotation. What can be said about the work done by the torque? (A) The work is positive, and the object gains rotational kinetic energy. (B) The work is negative, and the object gains rotational kinetic energy. (C) The work is positive, and the object loses rotational kinetic energy. (D) The work is negative, and the object loses rotational kinetic energy.

#Free Response Question

A uniform disk of mass M = 2.0 kg and radius R = 0.5 m is initially at rest. A constant torque of τ = 4.0 N⋅m is applied to the disk, causing it to rotate about its central axis.

(a) Calculate the rotational inertia (moment of inertia) of the disk. (b) Calculate the angular acceleration of the disk. (c) Calculate the angular speed of the disk after it has rotated through an angle of 10 radians. (d) Calculate the work done by the torque as the disk rotates through an angle of 10 radians. (e) Calculate the rotational kinetic energy of the disk after it has rotated through an angle of 10 radians.

#Scoring Breakdown

(a) 2 points - 1 point for using the correct formula for the rotational inertia of a disk: I = (1/2)MR² - 1 point for calculating the correct numerical value: I = 0.25 kg⋅m²

(b) 2 points - 1 point for using the rotational version of Newton's second law: τ = Iα - 1 point for calculating the correct numerical value: α = 16 rad/s²

(c) 3 points - 1 point for using the correct kinematic equation: ω² = ω₀² + 2αΔθ - 1 point for recognizing that the initial angular speed is zero: ω₀ = 0 - 1 point for calculating the correct numerical value: ω = 17.9 rad/s

(d) 2 points - 1 point for using the correct work-torque equation: W = τΔθ - 1 point for calculating the correct numerical value: W = 40 J

(e) 2 points - 1 point for using the correct formula for rotational kinetic energy: KE = (1/2)Iω² - 1 point for calculating the correct numerical value: KE = 40 J