AP Physics – Conservative vs Non-Conservative Forces

AP Physics – Conservative vs Non?Conservative Forces

AP Physics: Conservative vs. Non-Conservative Forces – Exam-Ready Study Guide

What This Is

Conservative and non-conservative forces are two types of forces that affect how energy is stored and transferred in a system. Conservative forces (like gravity or springs) conserve mechanical energy—they don’t "waste" energy as heat or sound. Non-conservative forces (like friction or air resistance) convert mechanical energy into other forms (e.g., thermal energy), making them path-dependent. This matters on the AP exam because it’s key to solving energy problems, especially those involving work, potential energy, and the Work-Energy Theorem. Real-world example: A roller coaster’s first hill is always the tallest because friction (a non-conservative force) slowly drains energy from the system, while gravity (a conservative force) just converts potential energy to kinetic energy and back.

Key Terms & Concepts

• Conservative Force: A force where the work done depends only on the starting and ending positions, not the path taken. Examples: gravity, spring force, electrostatic force.

• Key property: Mechanical energy (KE + PE) is conserved if only conservative forces act.

• Non-Conservative Force: A force where the work done depends on the path taken. Examples: friction, air resistance, applied forces (like pushing a box).

• Key property: Mechanical energy is not conserved; some energy is "lost" (converted to thermal energy, sound, etc.).

• Work Done by a Force (W): ( W = \vec{F} \cdot \vec{d} = Fd \cos \theta )

• ( \vec{F} ): Force vector (N)
• ( \vec{d} ): Displacement vector (m)

• ( \theta ): Angle between force and displacement.

• Work-Energy Theorem: ( W_{net} = \Delta KE )

• Net work done on an object equals its change in kinetic energy.

• Mechanical Energy (E): ( E = KE + PE )

• ( KE = \frac{1}{2}mv^2 ) (kinetic energy)

• ( PE ) can be gravitational (( mgh )) or elastic (( \frac{1}{2}kx^2 )).

• Path Independence (Conservative Forces): The work done by a conservative force around a closed loop is zero (e.g., lifting a book and putting it back on the same shelf).

• Path Dependence (Non-Conservative Forces): The work done depends on the distance traveled, not just start/end points (e.g., pushing a box across a rough floor vs. a smooth one).

• Potential Energy (PE): Energy stored due to position or configuration. Only defined for conservative forces.

• Gravitational PE: ( PE_g = mgh )

• Elastic PE: ( PE_s = \frac{1}{2}kx^2 )

• Thermal Energy: Energy "lost" due to non-conservative forces (e.g., friction heats up surfaces).

• Power (P): Rate of work done: ( P = \frac{W}{t} ) (Watts, W).

• AP trap: Power is not the same as work or energy!

Step-by-Step: Solving Energy Problems with Conservative/Non-Conservative Forces

1. Identify the forces acting on the object. Label them as conservative (gravity, springs) or non-conservative (friction, applied pushes).
2. Draw a bar chart (if helpful) showing initial and final ( KE ), ( PE ), and work done by non-conservative forces (( W_{nc} )).
3. Write the energy equation:
4. If only conservative forces act: ( KE_i + PE_i = KE_f + PE_f ).
5. If non-conservative forces act: ( KE_i + PE_i + W_{nc} = KE_f + PE_f ).
6. Calculate work done by non-conservative forces (if any):
7. ( W_{nc} = F_{nc} \cdot d \cdot \cos \theta ) (e.g., ( W_{friction} = -fd )).
8. Plug in known values and solve for the unknown (e.g., final velocity, height, or work).
9. Check units and signs:
10. Work done by the system is negative (e.g., friction removes energy).
11. Work done on the system is positive (e.g., pushing a box adds energy).

Example: A 2 kg block slides down a 5 m ramp (? = 30°) with = 0.2. Find its speed at the bottom. - Step 1: Forces = gravity (conservative), friction (non-conservative). - Step 2: Initial energy = ( PE_i = mgh = 2 \cdot 9.8 \cdot (5 \sin 30°) = 49 ) J. - Step 3: ( PE_i + W_{friction} = KE_f ). - Step 4: ( W_{friction} = -fd = -\mu_k mg \cos \theta \cdot d = -0.2 \cdot 2 \cdot 9.8 \cdot \cos 30° \cdot 5 = -16.97 ) J. - Step 5: ( 49 - 16.97 = \frac{1}{2} \cdot 2 \cdot v^2 )? ( v = 5.66 ) m/s.

Common Mistakes

• Mistake: Assuming all forces are conservative.

• Correction: Friction, air resistance, and applied forces are non-conservative. Always check the problem for these!

• Mistake: Forgetting that work done by non-conservative forces can be positive or negative.

• Correction: Work done by the system (e.g., friction) is negative; work done on the system (e.g., pushing) is positive.

• Mistake: Using ( W = Fd ) for conservative forces without considering path independence.

• Correction: For conservative forces, use ( \Delta PE ) instead of ( W = Fd ) (e.g., ( \Delta PE_g = mgh )).

• Mistake: Ignoring thermal energy in non-conservative systems.

• Correction: Non-conservative forces convert mechanical energy to thermal energy (e.g., rubbing hands together).

• Mistake: Mixing up power and energy.

• Correction: Power is rate of energy transfer (( P = \frac{W}{t} )), not the total energy.

AP Exam Insights

1. FRQ Hot Topic: Expect a work-energy problem where you must distinguish between conservative and non-conservative forces. Example: A block sliding down a ramp with friction—you’ll need to calculate ( W_{friction} ) and use it in the energy equation.
2. Multiple-Choice Trap: Questions may ask, "Which force is conservative?" and include options like "tension" or "normal force." Answer: Only gravity, springs, and electrostatic forces are conservative.
3. Tricky Distinction: Path independence vs. path dependence.
4. Conservative forces: Work depends only on start/end points (e.g., lifting a book).
5. Non-conservative forces: Work depends on the path (e.g., pushing a box in a zigzag vs. straight line).
6. Lab-Based Questions: The AP exam may ask you to design an experiment to determine if a force is conservative (e.g., measure work done over different paths).

Quick Check Questions

1. Multiple Choice: A block slides down a frictionless incline. Which statement is true? (A) The work done by gravity depends on the angle of the incline. (B) The work done by gravity is the same for any path between the same start and end points. (C) The work done by gravity is zero because the displacement is perpendicular to the force. (D) The work done by gravity increases if the block takes a longer path. Answer: (B) Gravity is conservative, so work depends only on the vertical displacement.

2. Short FRQ: A 3 kg box is pushed 4 m across a rough floor ( = 0.3) with a constant force of 20 N. Calculate the work done by friction and the final speed of the box if it starts from rest. Answer:

3. ( W_{friction} = -fd = -\mu_k mgd = -0.3 \cdot 3 \cdot 9.8 \cdot 4 = -35.28 ) J.
4. Net work = ( W_{applied} + W_{friction} = 20 \cdot 4 + (-35.28) = 44.72 ) J.

5. ( W_{net} = \Delta KE )? ( 44.72 = \frac{1}{2} \cdot 3 \cdot v^2 )? ( v = 5.46 ) m/s.

6. Multiple Choice: Which of the following is not a conservative force? (A) Gravitational force (B) Spring force (C) Frictional force (D) Electrostatic force Answer: (C) Friction is non-conservative because it depends on the path.

Last-Minute Cram Sheet

1. Conservative forces: Gravity, springs, electrostatic. Path-independent work.
2. Non-conservative forces: Friction, air resistance, applied pushes. Path-dependent work.
3. Work-Energy Theorem: ( W_{net} = \Delta KE ).
4. Energy equation (non-conservative): ( KE_i + PE_i + W_{nc} = KE_f + PE_f ).
5. Friction work: ( W_{friction} = -fd ) (always negative).
6. Gravitational PE: ( PE_g = mgh ). Depends only on height, not path.
7. Elastic PE: ( PE_s = \frac{1}{2}kx^2 ).
8. Work done by normal force is usually zero (displacement is perpendicular to force).
9. Thermal energy is "lost" energy from non-conservative forces.
10. Power = Work/Time (( P = \frac{W}{t} )). Not the same as energy!