#AP Physics C: Mechanics - Unit 3: Work, Energy, and Power 🚀

Hey there, future physicist! Let's get you prepped for the AP exam with a supercharged review of Unit 3. We're diving into work, energy, and power – the core concepts that tie everything together. Remember, energy is your friend; it's the secret weapon for tackling those tricky FRQs!

#Overview

Energy is everywhere in physics. It's the common thread that links all the units. A wise physics teacher once said, "Almost every FRQ can be at least partially solved using energy principles!" Keep that in mind as we go through this.

#Big Ideas

• Force Interactions: Why is no work done when you push against a wall, but work is done when you coast down a hill? 🤔
• Conservation: Why does a stretched rubber band return to its original length? Why is it easier to walk up a flight of stairs, rather than run, when the gravitational potential energy of the system is the same?

#Exam Impact

Unit 3 is a big deal, making up about 14-17% of your exam. You'll want to spend roughly 10-20 class periods (45 minutes each) mastering this material.

Don't forget to hit up the AP Classroom for practice! There are 20 MCQs and 1 FRQ waiting for you.

#Work-Energy Theorem 💪

This theorem is your go-to for relating work and energy. Here’s the breakdown:

• Work: The result of a force causing displacement. Mathematically, $W = Fd\cos\theta$, where $F$ is force, $d$ is displacement, and $\theta$ is the angle between them.
• Kinetic Energy (KE): The energy of motion. It's calculated as $KE = \frac{1}{2}mv^2$, where $m$ is mass and $v$ is velocity.
• The Theorem: $W = \Delta KE$. This means that work done on an object changes its kinetic energy. Speed it up, and you've done positive work; slow it down, and you've done negative work.

#Key Implications

• Conservative vs. Non-conservative Forces: The theorem applies to both! Conservative forces (like gravity and springs) depend only on position, while non-conservative forces (like friction and air resistance) depend on velocity. The work-energy theorem accounts for all of them.
• Multiple Forces: If multiple forces act on an object, add their work contributions to find the total change in kinetic energy.
• Calculations: You can use the theorem to find work done or changes in kinetic energy. If you know force and displacement, you can find the change in KE, and vice versa.
• Constant Mass: The theorem is valid only for objects with constant mass. If the mass changes, you'll need more advanced techniques.

#Equation Form

Here's the Work-Energy Theorem in equation form:

$$W = \Delta K$$

Where W is work and K is kinetic energy. Kinetic energy is:

$$KE = \frac{1}{2}mv^2$$

where m is mass and v is velocity.

#Derivation

Here's how we get to the Work-Energy Theorem (don't worry, you don't need to memorize this for the exam, but it's cool to see):

1. Start with Newton's Second Law: $F = ma = m\frac{dv}{dt}$
2. Use the chain rule: $\frac{dv}{dt} = \frac{dv}{dx} \frac{dx}{dt} = v\frac{dv}{dx}$
3. Substitute into Newton's Second Law: $F = mv\frac{dv}{dx}$
4. Work is $W = F \cdot d$, so $W = \int F dx = \int mv \frac{dv}{dx} dx = \int mv dv$
5. Integrate from initial to final velocity: $W = m\int_{v_0}^{v_f} v dv = m[\frac{1}{2}v^2]_{v_0}^{v_f} = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_0^2$
6. This gives us: $W = \Delta KE$ 🎉

#Work Done by a Variable Force

#What Exactly is Work?

Here's the classic formula for work (when the force is constant):

$$W = Fd\cos\theta$$

#Practice Questions:

Practice Question

1. (a) Calculate the work done on a 1500-kg elevator car by its cable to lift it 40.0 m at constant speed, assuming friction averages 100 N.

   (b) What is the work done on the lift by the gravitational force in this process?

   (c) What is the total work done on the lift?

   Answers:

   (a) Start with a free body diagram. Tension force (T) is upward, and gravity (mg) and friction (f) are downward.

   The net force is zero since the elevator moves at constant speed, so $T = mg + f$. Work done by tension: $W_T = Td = (mg+f)d = (1500 \cdot 9.8 + 100) \cdot 40 = 592000 J$

   (b) Work done by gravity: $W_g = -mgd = -1500 \cdot 9.8 \cdot 40 = -588000 J$ (negative because gravity opposes the motion)

   (c) Total work is the sum of work done by all forces: $W_{total} = W_T + W_g + W_f = 592000 - 588000 -100 * 40 = 0J$. Note that the work done by friction is negative because it opposes the motion

2. (a) Using energy considerations, calculate the average force a 60.0-kg sprinter exerts backward on the track to accelerate from 2.00 to 8.00 m/s in a distance of 25.0 m, if he encounters a headwind that exerts an average force of 30.0 N against him.

   Answer:

   Start with a free-body diagram. The forces are the sprinter's force (F), and the headwind (F_wind).

   The net work done equals the change in kinetic energy:

   $W_{net} = \Delta KE$

   $(F - F_{wind})d = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$

   $(F - 30) \cdot 25 = \frac{1}{2} \cdot 60 \cdot (8^2 - 2^2)$

   $(F - 30) \cdot 25 = 30 \cdot (64 - 4)$

   $25F - 750 = 1800$

   $25F = 2550$

   $F = 102 N$

#Final Exam Focus 🎯

Okay, it's crunch time! Here's what to focus on for the exam:

• Work-Energy Theorem: Know it inside and out. It's the most versatile tool in your energy toolkit.
• Conservative vs. Non-conservative Forces: Understand how they affect energy calculations. Remember, non-conservative forces like friction cause energy loss.
• Variable Forces: Be ready to calculate work from a force vs. position graph (area under the curve).
• Problem-Solving Strategy: Always start with a free-body diagram. Then, use the Work-Energy Theorem to connect forces, work, and changes in kinetic energy.

#Last-Minute Tips

• Time Management: Don't spend too long on any one question. If you're stuck, move on and come back later.
• Common Pitfalls: Watch out for angles! Make sure you're using the component of force parallel to displacement. Also, remember that work is a scalar quantity.
• FRQ Strategy: Show all your work clearly. Even if you don't get the final answer, you can earn partial credit for correct steps.

You've got this! Go into the exam confident and ready to apply your physics knowledge. You've put in the work, and now it's time to shine! ✨