#AP Physics C: Mechanics - Energy Conservation Study Guide 🚀

Hey! Let's get you prepped for the exam with a super-focused review of energy conservation. We'll break down the key concepts, make connections, and get you feeling confident. Let's do this!

#1. Introduction to Energy Conservation

#System Energies 🔋

#Single Object Kinetic Energy

• If a system has only one object, its energy is purely kinetic. Think of a lone skater gliding on ice ⛸️.
• Kinetic energy (KE) depends on mass and velocity.
• Formula: $$KE = \frac{1}{2}mv^2$$ where:
  • $m$ = mass
  • $v$ = velocity
• Example: A ball rolling on a flat surface is a perfect example of single-object KE.

#Kinetic and Potential Energies

• When objects interact via conservative forces (like gravity or springs), or when there are reversible shape changes, both KE and potential energy (PE) are in play.
• Conservative forces allow energy to switch between KE and PE without loss.
• Reversible shape changes (like stretching a rubber band) store PE that can become KE.
• Example: A pendulum swinging back and forth is a classic example of KE and PE exchange.

Practice Question

Multiple Choice:

1. A 2 kg mass is moving at 3 m/s. What is its kinetic energy? (A) 3 J (B) 6 J (C) 9 J (D) 12 J
2. A spring with a spring constant of 100 N/m is compressed by 0.2 m. What is the potential energy stored in the spring? (A) 2 J (B) 4 J (C) 8 J (D) 10 J

Free Response: A 0.5 kg block is released from rest at the top of a frictionless ramp that is 2 m high. At the bottom of the ramp, it slides onto a horizontal surface with a coefficient of kinetic friction of 0.2. (a) What is the speed of the block at the bottom of the ramp? (2 points) (b) How far does the block travel on the horizontal surface before coming to rest? (3 points)

Answer Key: Multiple Choice:

1. (C) 9 J
2. (A) 2 J

Free Response: (a) Using conservation of energy: $$mgh = \frac{1}{2}mv^2$$; $$v = \sqrt{2gh} = \sqrt{2 * 9.8 * 2} = 6.26 m/s$$ (2 points) (b) Using work-energy theorem: $$W = \Delta KE = -f_k d = -\mu_k mgd$$; $$d = \frac{v^2}{2 \mu_k g} = \frac{6.26^2}{2 * 0.2 * 9.8} = 10 m$$ (3 points)

#2. Conservation of Mechanical Energy 💡

#Sum of Kinetic and Potential

• Mechanical Energy (ME) is the sum of KE and PE: $$ME = KE + PE$$
• KE is the energy of motion, and PE is stored energy.
• Example: A roller coaster at the top of a hill has high PE, which transforms into KE as it goes down.

#Energy Changes and Transfers

• Changes in energy forms are balanced within a system or by transfers with the environment.
• Energy isn't created or destroyed, only converted or transferred.
• Changes in KE are offset by changes in PE or by work done on/by the system.
• Heat and sound are energy forms that can transfer between a system and its environment.

#Constant Total Energy

• If you choose your system carefully, the total energy remains constant. No energy enters or leaves the system 🔒.
• All energy conversions are internal, keeping the total constant.
• Example: An ideal frictionless pendulum is a great example of constant total energy.

#Energy Transfer Equivalence

• Changes in a system's total energy directly match the energy transferred in or out.
• Energy entering increases total energy, while energy leaving decreases it.
• The change in total energy equals the energy transferred.
• Example: Pushing a box transfers energy into the box-floor system, increasing its total energy.

Practice Question

Multiple Choice:

1. A 1 kg ball is dropped from a height of 10 m. Ignoring air resistance, what is its kinetic energy just before it hits the ground? (A) 10 J (B) 49 J (C) 98 J (D) 196 J
2. A block slides down a frictionless ramp. Which of the following is true about its mechanical energy? (A) Increases (B) Decreases (C) Remains constant (D) Cannot be determined

Free Response: A 2 kg block is attached to a spring with a spring constant of 200 N/m. The block is pulled 0.1 m from its equilibrium position and released. The surface is frictionless. (a) What is the total mechanical energy of the system? (2 points) (b) What is the maximum speed of the block? (3 points)

Answer Key: Multiple Choice:

1. (C) 98 J
2. (C) Remains constant

Free Response: (a) $$ME = \frac{1}{2}kA^2 = \frac{1}{2} * 200 * (0.1)^2 = 1 J$$ (2 points) (b) $$ME = \frac{1}{2}mv^2$$; $$v = \sqrt{\frac{2ME}{m}} = \sqrt{\frac{2 * 1}{2}} = 1 m/s$$ (3 points)

#3. System Selection and Energy

#Conservation in Interactions

• Energy is always conserved, no matter the interaction. This is a universal principle 🌍.
• Energy can change forms or transfer between objects, but the total amount is constant.

#Zero Work and Constant Energy

• When no work is done on a system and no nonconservative forces act, the total mechanical energy stays constant.
• No external work means no energy is added or removed.
• Absence of nonconservative forces prevents energy dissipation.
• Example: A satellite orbiting Earth experiences no net work and conserves its total mechanical energy.

#Nonzero Work and Energy Transfer

• If work is done on a system, energy transfers between the system and its surroundings.
• Positive work increases total energy; negative work decreases it.
• The energy transferred equals the work done.
• Example: Lifting an object transfers energy into the object-Earth system through the work done by the lifter.

Energy conservation is a high-value topic because it's often combined with other concepts like work, power, and momentum. Mastering this will help you tackle complex problems!

#Boundary Statements

• Nonconservative forces like friction and air resistance can dissipate mechanical energy as thermal energy or sound. Be aware of these in real-world scenarios.

Practice Question

Multiple Choice:

1. A box is pushed across a rough surface. Which of the following best describes the energy transformation? (A) Kinetic energy is converted to potential energy (B) Potential energy is converted to kinetic energy (C) Mechanical energy is converted to thermal energy (D) Thermal energy is converted to mechanical energy
2. A system does 50 J of work on its surroundings. What is the change in the system's energy? (A) +50 J (B) -50 J (C) 0 J (D) Cannot be determined

Free Response: A 1 kg block is pushed up a 30-degree incline with a force of 10 N parallel to the incline. The coefficient of kinetic friction between the block and the incline is 0.2. The block is pushed a distance of 2 m. (a) How much work is done by the applied force? (2 points) (b) How much work is done by friction? (2 points) (c) What is the change in the block's total mechanical energy? (1 point)

Answer Key: Multiple Choice:

1. (C) Mechanical energy is converted to thermal energy
2. (B) -50 J

Free Response: (a) $$W_{applied} = Fd = 10 * 2 = 20 J$$ (2 points) (b) $$f_k = \mu_k mg cos(30) = 0.2 * 1 * 9.8 * cos(30) = 1.7 N$$; $$W_{friction} = -f_k d = -1.7 * 2 = -3.4 J$$ (2 points) (c) $$\Delta ME = W_{applied} + W_{friction} = 20 - 3.4 = 16.6 J$$ (1 point)

#Final Exam Focus 🎯

Okay, you've made it! Here's what to focus on for the exam:

• High-Priority Topics:
  • Conservation of Mechanical Energy: Understand when it applies and when it doesn't (i.e., when non-conservative forces are present).
  • Work-Energy Theorem: Know how work relates to changes in kinetic energy. $$W = \Delta KE$$
  • Potential Energy: Be comfortable with gravitational and spring potential energy. $$PE_g = mgh$$ and $$PE_s = \frac{1}{2}kx^2$$
  • System Selection: Carefully define your system to determine if energy is conserved or if work is involved.
• Common Question Types:
  • Problems involving a mix of kinetic and potential energy.
  • Questions where you have to calculate work done by different forces.
  • Scenarios with friction or other non-conservative forces.
• Last-Minute Tips:
  • Time Management: Quickly identify the core concepts in each problem. Don't get bogged down in unnecessary calculations.
  • Common Pitfalls: Watch out for the sign of work (positive if energy is added to the system, negative if energy is removed). Remember that friction does negative work.
  • Challenging Questions: Break down complex problems into smaller, manageable parts. Draw free-body diagrams to help visualize forces and energy transfers.

You've got this! Go into the exam calm and confident. You're well-prepared. Good luck! 🎉