#AP Physics C: Mechanics - Energy Review 🚀

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

#Conservation of Energy: The Big Picture

#The Core Idea

At its heart, the Law of Conservation of Energy states that:

In a closed system, the total energy remains constant. Energy can transform from one form to another, but it cannot be created or destroyed.

Think of it like this: you have a fixed amount of money, you can exchange it for different currencies, but the total amount remains the same. 💡

• Key Concept: Energy is a conserved quantity. It's all about transfers and transformations, not creation or destruction.
• Forms of Energy: Kinetic (motion), potential (position), thermal (heat), chemical, nuclear, etc. All are part of the total energy.

#Mathematical Representation

The total mechanical energy (ME) of a system is the sum of its kinetic energy (K) and potential energy (U):

$$ME = K + U$$

• Kinetic Energy (K): Energy of motion. $K = \frac{1}{2}mv^2$
• Potential Energy (U): Stored energy due to position or configuration. Examples include gravitational potential energy ($U_g = mgh$) and spring potential energy ($U_s = \frac{1}{2}kx^2$).

#Work and Nonconservative Forces

When nonconservative forces (like friction or air resistance) do work on a system, the total mechanical energy changes. The work done by these forces is:

$$\Delta ME = W_{nonconservative}$$

• Important Note: This work can either add or remove energy from the system.
• Conservative vs. Nonconservative: Conservative forces (gravity, spring) conserve total mechanical energy; nonconservative forces do not.

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In a conservative system (where only conservative forces act), you can use conservation of energy to solve for unknowns in various scenarios like:

• Rollercoasters
• Ramps
• Pendulums
• Springs

#Conservative vs. Nonconservative Forces

#Conservative Forces

• Definition: Work done is independent of the path taken. It only depends on the initial and final positions.
• Examples: Gravitational force, elastic force (springs), electrostatic force.
• Potential Energy: Can be described by a scalar potential energy function.
• Energy Conservation: Total mechanical energy of a closed system is conserved.

#Nonconservative Forces

• Definition: Work done depends on the path taken.
• Examples: Friction, air resistance, viscosity.
• Potential Energy: Cannot be described by a scalar potential energy function.
• Energy Conservation: Total mechanical energy of a closed system is NOT conserved. Energy is often lost as heat or sound.

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• Always State Conservation: Start energy problems by explicitly stating whether energy is conserved. This often earns you a point!
• Identify Initial and Final States: Determine what's happening at the beginning and end of the problem to track energy transfers.
• Proportions: When comparing the same variable, look for equations and focus on changing coefficients rather than plugging in values.
• Differential Equations: AP won't ask you to solve complex differential equations due to time constraints. Focus on setting them up correctly.
• Variable Awareness: You might not need all the variables given in a problem. Be aware of connections between different physics units like forces and energy.
• Separation of Variables: If you see a differential equation, separate variables first, then integrate. Remember to use correct limits for definite integrals.

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• Forgetting Nonconservative Work: Always account for work done by nonconservative forces like friction, which changes the total mechanical energy.
• Incorrectly Applying Conservation: Make sure you're only using conservation of energy when only conservative forces are at play. If nonconservative forces are present, use the work-energy theorem.
• Mixing up Kinetic and Potential Energy: Double-check that you're using the correct formulas for each type of energy.
• Ignoring Initial Conditions: Always consider the initial potential and kinetic energies of the system.

#Final Exam Focus

• Conservation of Energy: This is huge! Expect to see it in various forms (mechanical, rotational, etc.).
• Work-Energy Theorem: Understand how work changes a system's energy, especially when nonconservative forces are present.
• Potential Energy Functions: Be comfortable with gravitational and spring potential energies, and how they relate to conservative forces.
• Problem-Solving: Practice setting up energy equations and solving for unknowns. Focus on identifying the initial and final states.
• FRQ Focus: Expect a mix of conceptual and calculation-based questions. Be ready to explain your reasoning.

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Practice Question

Practice Questions

#Multiple Choice Questions

1. A block of mass m is released from rest at the top of a frictionless incline of height h. What is the kinetic energy of the block at the bottom of the incline? (A) mgh/2 (B) mgh (C) 2mgh (D) 4mgh

2. A spring with spring constant k is compressed by a distance x. What is the potential energy stored in the spring? (A) kx/2 (B) kx (C) kx^2/2 (D) kx^2

3. A ball is thrown vertically upwards. Neglecting air resistance, which of the following is true about its mechanical energy? (A) It increases as the ball goes up. (B) It decreases as the ball goes up. (C) It remains constant. (D) It is zero at the maximum height.

#Free Response Question (FRQ)

A block of mass m is released from rest at the top of a ramp of height h and slides down the ramp. The ramp is inclined at an angle θ with the horizontal. The coefficient of kinetic friction between the block and the ramp is μ.

(a) Derive an expression for the work done by friction as the block slides down the ramp. (b) Derive an expression for the kinetic energy of the block at the bottom of the ramp. (c) If the block is observed to have half the kinetic energy at the bottom of the ramp compared to the frictionless case, derive an expression for the coefficient of kinetic friction μ in terms of h, θ, and g.

#FRQ Scoring Breakdown

(a) Work done by friction (3 points)

• 1 point: Correctly identifying the normal force as mgcos(θ).
• 1 point: Correctly calculating the frictional force as μmgcos(θ).
• 1 point: Correctly calculating the work done by friction as -μmghcos(θ)/sin(θ), or -μmgdcos(θ), where d is the distance along the ramp.

(b) Kinetic energy at the bottom (3 points)

• 1 point: Correctly stating the conservation of energy principle: mgh = K + Wfriction.
• 1 point: Correctly substituting the work done by friction from part (a).
• 1 point: Correctly solving for the kinetic energy at the bottom as K = mgh - μmghcos(θ)/sin(θ), or K = mgh - μmgdcos(θ).

(c) Coefficient of kinetic friction (4 points)

• 1 point: Correctly stating the kinetic energy at the bottom in the frictionless case as K = mgh.
• 1 point: Correctly stating that the kinetic energy at the bottom with friction is mgh/2. * 1 point: Correctly setting up the equation mgh/2 = mgh - μmghcos(θ)/sin(θ).
• 1 point: Correctly solving for the coefficient of kinetic friction as μ = (1/2)tan(θ).