#AP Physics C: Mechanics - Work & Energy Study Guide

Hey there! Let's get you prepped for the exam with a deep dive into work and energy. This guide is designed to be your go-to resource, especially the night before the test. Let's make sure you're feeling confident and ready to ace it! 💪

#1. Work: The Energy Transfer Mechanism

#1.1. Energy Transfer Through Work

• Work is all about how energy moves into or out of a system when a force acts over a distance. Think of it as the energy currency of motion! 🔋

• #Conservative forces (like gravity and springs) are path-independent. Only the start and end points matter, not the route taken.

- Potential energy is the energy associated *only* with conservative forces. - **Nonconservative forces** (like friction and air resistance) are path-dependent. The amount of work done changes depending on the path taken. -

#1.2. Path Independence of Conservative Forces

• Work is a scalar quantity: it has magnitude but no direction. It's all about the amount of energy transferred, not the direction of transfer.
• Work can be:
  • Positive: Energy added to the system.
  • Negative: Energy removed from the system.
  • Zero: No energy change.

#1.3. Work as a Scalar Quantity

#1.4. Work by Variable Forces

• When forces aren't constant, we need calculus to find the work done: $W=\int_{a}^{b} \vec{F}(r) \cdot d \vec{r}$. This integral sums up the work done over tiny displacements.

#1.5. Dot Product in Work Calculations

• The dot product helps us find the component of force that's parallel to the displacement: $\vec{A} \cdot \vec{B} = AB \cos \theta$. Only this parallel component does work.
• For constant parallel forces, work simplifies to: $W = F_{\parallel} d = Fd \cos \theta$

#1.6. Work-Energy Theorem

• The work-energy theorem is your best friend: it says the net work done on an object equals its change in kinetic energy. 💡
• Formula: $W_{\text{net}} = \Delta KE = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$

- It's a shortcut! You can solve problems by focusing on initial and final kinetic energies, without worrying about the details of the forces.

#2. Connecting the Concepts

• Work and Energy are interconnected: Work is the process, and energy is the state. Work changes the energy of a system.
• Conservative vs. Nonconservative: Remember, conservative forces store energy (potential), while nonconservative forces dissipate energy (usually as heat).
• Dot Product: It's not just a math trick; it's about finding the part of the force that actually does work by changing the object's speed.

#3. Final Exam Focus

• Work-Energy Theorem: This is huge! It's used in almost every mechanics problem. Make sure you understand it inside and out.
• Conservative vs. Nonconservative: Be able to identify them and understand their implications.
• Dot Product: Practice calculating work using the dot product. It's a common point of error if you don't get it right.
• Variable Forces: Review integration for work calculations. It might show up in FRQs.

#3.1. Last-Minute Tips

• Time Management: Don't spend too long on one question. If you're stuck, move on and come back later.
• Common Pitfalls: Watch out for the sign of work (positive or negative). Double-check your dot product calculations.
• FRQ Strategies: Show all your work! Even if you don't get the final answer, you can get partial credit for the correct steps.

#4. Practice Questions

Practice Question

#Multiple Choice Questions

1. A block of mass m is pulled along a rough horizontal surface by a force F at an angle θ above the horizontal. The block moves a distance d. What is the work done by the force of friction?

   (A) −μmgd (B) −μ(mg − Fsinθ)d (C) −μ(mg + Fsinθ)d (D) −μFcosθd

2. A particle moves along the x-axis from x = 0 to x = 5 m under the influence of a force F(x) = 7 - 2x, where F is in newtons and x is in meters. What is the work done by this force?

   (A) 10 J (B) 12.5 J (C) 20 J (D) 25 J

3. A 2 kg ball is dropped from a height of 10 m. If air resistance does 10 J of work on the ball as it falls, what is the ball's kinetic energy just before it hits the ground?

   (A) 10 J (B) 100 J (C) 190 J (D) 200 J

#Free Response Question

A 2.0 kg block is pushed against a spring with a spring constant of 500 N/m, compressing it by 0.20 m. The block is then released, and the spring propels the block along a horizontal surface. The coefficient of kinetic friction between the block and the surface is 0.25. (a) Calculate the potential energy stored in the spring when it is compressed. (b) Calculate the work done by friction as the block slides 1.0 m along the horizontal surface. (c) Use the work-energy theorem to calculate the speed of the block after it has moved 1.0 m. (d) How far will the block travel before it comes to rest, assuming the same frictional force?

Scoring Breakdown

(a) 2 points - 1 point for using the correct formula: $U = \frac{1}{2}kx^2$ - 1 point for the correct answer: $U = \frac{1}{2}(500 N/m)(0.20 m)^2 = 10 J$

(b) 2 points - 1 point for calculating the frictional force: $f = μmg = (0.25)(2.0 kg)(9.8 m/s^2) = 4.9 N$ - 1 point for calculating the work done by friction: $W = -fd = -(4.9 N)(1.0 m) = -4.9 J$ (negative sign is important)

(c) 3 points - 1 point for using the work-energy theorem: $W_{net} = \Delta KE$ - 1 point for identifying the initial energy as potential energy: $10 J$ - 1 point for the correct final speed: $10 J - 4.9 J = \frac{1}{2}(2.0 kg)v^2$, $v = 2.26 m/s$

(d) 3 points - 1 point for recognizing that the final kinetic energy is 0 - 1 point for setting up the work-energy equation: $0 - \frac{1}{2}mv^2 = -fd$ - 1 point for the correct distance: $d = 0.52 m$

Remember, you've got this! Focus on the key concepts, practice, and trust your preparation. You're going to do great! 🎉