AP Physics – Conservation of Mechanical Energy

AP Physics – Conservation of Mechanical Energy

AP Physics: Conservation of Mechanical Energy – Exam-Ready Study Guide

What This Is

Conservation of Mechanical Energy states that in a system with only conservative forces (like gravity or springs), the total mechanical energy (kinetic + potential) stays constant. This is a cornerstone of AP Physics because it lets you solve problems without tracking every force or acceleration—just compare energy at two points. Real-world example: A roller coaster at the top of a hill has maximum gravitational potential energy (GPE). As it descends, GPE converts to kinetic energy (KE), but the total energy (GPE + KE) stays the same (ignoring friction). If the coaster starts at 50 m, it can’t go higher than 50 m later—energy can’t be created or destroyed, only transformed.

Key Terms & Concepts

• Mechanical Energy (ME): The sum of kinetic energy (KE) and potential energy (PE) in a system.

• ME = KE + PE

• Kinetic Energy (KE): Energy of motion.

• KE = ½mv² (m = mass, v = velocity)

• Gravitational Potential Energy (GPE): Energy due to an object’s height in a gravitational field.

• GPE = mgh (m = mass, g = 9.8 m/s², h = height relative to a reference point)

• Elastic Potential Energy (EPE): Energy stored in a stretched or compressed spring.

• EPE = ½kx² (k = spring constant, x = displacement from equilibrium)

• Conservative Force: A force where work done is independent of the path taken (e.g., gravity, springs). Work done by conservative forces only depends on initial and final positions.

• Example: Lifting a book straight up vs. carrying it up stairs—work done by gravity is the same.

• Non-Conservative Force: A force where work done depends on the path (e.g., friction, air resistance). These forces remove mechanical energy from a system (convert it to heat, sound, etc.).

• Example: A sliding box slows down due to friction—ME decreases.

• Work-Energy Theorem: Work done on an object equals its change in kinetic energy.

• W = ?KE = KE_final – KE_initial

• Conservation of Mechanical Energy (for conservative forces only):

• ME_initial = ME_final

• KE_i + PE_i = KE_f + PE_f

• Power: Rate of energy transfer or work done.

• P = W/t = ?E/t (W = work, t = time)

• Units: Watts (W) = Joules/second (J/s)

• Reference Level (for GPE): The arbitrary "zero" height where GPE = 0. Always define this in problems! (e.g., "Let h = 0 at the bottom of the ramp.")

• Energy Bar Charts: Visual tool to track energy changes in a system (see Step-by-Step).

Step-by-Step / Process Flow

How to solve conservation of energy problems on the AP exam:

1. Identify the system and forces:
2. Is the system isolated (no external work)? Are there only conservative forces (gravity, springs)? If yes, ME is conserved.

3. If non-conservative forces (friction, applied force) are present, use W_nc = ?ME (work done by non-conservative forces = change in ME).

4. Define the initial and final states:

5. Pick two points in the problem (e.g., top of a hill-bottom of a hill).

6. Label knowns/unknowns (heights, velocities, spring displacements).

7. Choose a reference level for GPE:

8. Set h = 0 at the lowest point in the problem (simplifies calculations).

9. Write the conservation equation:

10. KE_i + PE_i = KE_f + PE_f

11. Plug in formulas (½mv² for KE, mgh for GPE, ½kx² for EPE).

12. Solve for the unknown:

13. Cancel out mass if it appears in every term (common in projectile problems).

14. Check units (e.g., height in meters, velocity in m/s).

15. Check for non-conservative forces (if applicable):

16. If friction is present, W_friction = ?ME (work done by friction = force × distance × cos?).
17. If an external force does work, W_ext = ?ME.

Example Problem: A 2 kg block slides down a frictionless ramp from a height of 5 m. What is its speed at the bottom? - Step 1: Only gravity (conservative) acts-ME conserved. - Step 2: Initial state = top (h = 5 m, v = 0); final state = bottom (h = 0, v = ?). - Step 3: Set h = 0 at the bottom. - Step 4: mgh_i + ½mv_i² = mgh_f + ½mv_f² -(2)(9.8)(5) + 0 = 0 + ½(2)v_f² - Step 5: Solve for v_f: v_f = ?(2gh) = ?(2 × 9.8 × 5)-9.9 m/s.

Common Mistakes

• Mistake: Forgetting to set a reference level for GPE.

• Correction: Always define h = 0 (e.g., "Let h = 0 at the ground"). GPE is relative, not absolute.

• Mistake: Including non-conservative forces in the conservation equation.

• Correction: If friction or applied forces are present, use W_nc = ?ME instead of ME_i = ME_f. Conservation only applies to systems with only conservative forces.

• Mistake: Mixing up work and energy.

• Correction: Work is energy transfer (Joules), while energy is a property of the system. Work done by conservative forces changes the form of energy (e.g., GPE-KE), but not the total ME.

• Mistake: Ignoring rotational kinetic energy (for rolling objects).

• Correction: For rolling objects (e.g., a ball down a ramp), include rotational KE (½I?²) in the conservation equation. AP often tests this in FRQs!

• Mistake: Assuming energy is conserved when it’s not.

• Correction: Check for non-conservative forces (e.g., friction, air resistance). If present, ME isn’t conserved—use W_nc = ?ME.

AP Exam Insights

1. FRQs love energy bar charts:

2. You’ll often be asked to draw a bar chart showing KE, GPE, and EPE at different points. Practice this! (See Quick Check Question #2.)

3. Tricky distinction: Conservative vs. non-conservative forces:

4. Conservative: Gravity, springs (work depends only on initial/final positions).
5. Non-conservative: Friction, applied forces (work depends on path).

6. AP trap: A problem might say "frictionless" to hint that ME is conserved.

7. Common FRQ setup:

8. A block slides down a ramp, compresses a spring, or launches off a cliff. You’ll need to:

   • Write the conservation equation.
   • Solve for velocity, height, or spring compression.
   • Bonus: Calculate work done by friction if it’s present.

9. Multiple-choice traps:

10. Trap 1: Giving a problem with friction but asking for "conservation of energy." Answer: ME isn’t conserved—use W_friction = ?ME.
11. Trap 2: Mixing units (e.g., height in cm instead of m). Always convert to SI units!
12. Trap 3: Forgetting that mass cancels in many problems (e.g., projectile motion).

Quick Check Questions

1. Multiple Choice

A 0.5 kg ball is dropped from a height of 10 m. Ignoring air resistance, what is its speed when it hits the ground? (A) 5 m/s (B) 10 m/s (C) 14 m/s (D) 20 m/s

Answer: (C) 14 m/s Explanation: Use ME_i = ME_f: mgh = ½mv²-v = ?(2gh) = ?(2 × 9.8 × 10)-14 m/s.

2. FRQ-Style (Energy Bar Chart)

A block slides down a frictionless ramp, compresses a spring at the bottom, and briefly comes to rest. Sketch an energy bar chart for the block at: - Point A: Top of the ramp (h = 4 m, v = 0). - Point B: Bottom of the ramp (h = 0, v = max). - Point C: Maximum spring compression (h = 0, v = 0).

Answer: - Point A: Only GPE (bar at max height, KE = 0, EPE = 0). - Point B: Only KE (GPE = 0, KE at max, EPE = 0). - Point C: Only EPE (GPE = 0, KE = 0, EPE at max).

3. Multiple Choice

A 1 kg block slides down a 5 m ramp with a coefficient of kinetic friction-= 0.2. What is the block’s speed at the bottom? (Assume g = 10 m/s²) (A) 5 m/s (B) 7 m/s (C) 9 m/s (D) 10 m/s

Answer: (B) 7 m/s Explanation: Friction does work: W_friction = ?mgd (d = ramp length). Use W_nc = ?ME: ?mgd = ½mv² – mgh-0.2 × 1 × 10 × 5 = ½ × 1 × v² – 1 × 10 × 5-v-7 m/s.

Last-Minute Cram Sheet

1. ME = KE + PE (only conserved if only conservative forces act).
2. KE = ½mv² (velocity squared—double speed = 4× KE!).
3. GPE = mgh (h is relative to a reference level—define it!).
4. EPE = ½kx² (x = displacement from equilibrium, k = spring constant).
5. Conservative forces: Gravity, springs (work depends only on initial/final positions).
6. Non-conservative forces: Friction, applied forces (work depends on path).
7. W_nc = ?ME (if non-conservative forces do work).
8. Mass cancels in energy problems (e.g., mgh = ½mv²-v = ?(2gh)).
9. Always check for friction/air resistance—ME isn’t conserved if they’re present!
10. Rolling objects? Include rotational KE (½I?²) in the conservation equation.