AP Physics 1- 3.4 Conservation of Energy- Study Notes- New Syllabus
AP Physics 1-3.4 Conservation of Energy – Study Notes
AP Physics 1-3.4 Conservation of Energy – Study Notes -AP Physics 1 – per latest Syllabus.
Key Concepts:
- Energies Present in a System
- Conservation of Mechanical Energy
- System Selection and Energy Changes
Energies Present in a System
Single-Object System:
A system composed of only a single object can only have kinetic energy, since there are no internal conservative interactions to store potential energy.
Multi-Object or Deformable System:
A system that contains objects interacting via conservative forces (e.g., gravity, springs, electric forces), or a system that can change its shape reversibly, may possess both kinetic energy (KE) and potential energy (PE).
Total Mechanical Energy:
For such systems, the total energy is the sum: \( E_{total} = KE + PE \).
Examples
- Single Object: A ball rolling freely on a frictionless surface → only KE.
- Two-Object Gravitational System: Earth and Moon → KE (motion) + PE (gravity).
- Elastic System: Mass attached to an ideal spring → KE (mass motion) + PE (spring stretch).
Example
A \( 1.0 \, \text{kg} \) mass is attached to a spring (\( k = 50 \, \text{N/m} \)) and stretched by \( 0.2 \, \text{m} \). Find the total energy of the system at this instant if the mass is released from rest.
▶️Answer/Explanation
Step 1: Kinetic Energy (KE) = 0 (since released from rest).
Step 2: Potential Energy (spring) = \( U_s = \tfrac{1}{2} k x^2 \).
\( U_s = \tfrac{1}{2} (50)(0.2^2) = \tfrac{1}{2}(50)(0.04) = 1.0 \, \text{J} \).
Step 3: Total Energy = KE + PE = 0 + 1.0 = 1.0 J.
Conservation of Mechanical Energy
Mechanical energy is the sum of a system’s kinetic energy (KE) and potential energy (PE): \( E_{mech} = KE + PE \).
Energy transformation within the system:
Any decrease in one form of energy must be matched by an equal increase in another form. Example: As a pendulum swings down, PE decreases while KE increases, keeping total energy constant.
Closed system (no external work or nonconservative forces):
The total mechanical energy remains constant: \( KE_i + PE_i = KE_f + PE_f \).
Open system (energy transfer with surroundings): If the total energy of a system changes, that change equals the net energy transferred into or out of the system (work or heat).
Key Ideas
- Energy can transform between KE and PE within a system but the total remains constant (if no external work).
- When nonconservative forces (like friction or air resistance) act, they remove mechanical energy from the system, converting it into other forms (e.g., thermal energy).
- A system can be chosen so that its total energy is constant, simplifying analysis.
Example:
A \( 2.0 \, \text{kg} \) block slides down a smooth frictionless ramp of height \( 5.0 \, \text{m} \). Find its speed at the bottom.
▶️Answer/Explanation
Step 1: Apply CME → \( KE_i + PE_i = KE_f + PE_f \).
At the top: \( KE_i = 0 \), \( PE_i = mgh = (2)(9.8)(5) = 98 \, \text{J} \).
At the bottom: \( PE_f = 0 \), so \( KE_f = 98 \, \text{J} \).
Step 2: Solve for velocity: \( KE_f = \tfrac{1}{2} m v^2 \) → \( 98 = \tfrac{1}{2}(2) v^2 \).
\( v^2 = 98 \) → \( v \approx 9.9 \, \text{m/s} \).
Final Answer: The block’s speed at the bottom is 9.9 m/s.
Example:
If the same block slides down but loses \( 20 \, \text{J} \) to friction, find its speed at the bottom.
▶️Answer/Explanation
Total initial energy = \( 98 \, \text{J} \).
Energy lost to friction = \( 20 \, \text{J} \).
Remaining energy = \( 98 – 20 = 78 \, \text{J} \) → this is the final KE.
\( \tfrac{1}{2} (2) v^2 = 78 \) → \( v^2 = 78 \) → \( v \approx 8.8 \, \text{m/s} \).
Final Answer: With friction, the block’s speed at the bottom is 8.8 m/s.
System Selection and Energy Changes
Energy is always conserved in all interactions, but whether the energy of a chosen system remains constant depends on how the system is defined.
Case 1:
- No external work, no nonconservative forces If the work done on a selected system is zero and there are no nonconservative interactions inside it, then the total mechanical energy of that system is constant. \( E_{mech} = KE + PE = \text{constant} \).
Case 2:
- External work is done If the work done on a selected system is nonzero, then energy is transferred between the system and the environment. The system’s total energy changes by the amount of work done.
System boundary choice matters:
- If you include all interacting objects, internal forces are conservative and total energy is constant.
- If you choose only part of the system, external forces may do work, changing the system’s energy.
Example:
A ball of mass \(m\) is dropped from height \(h\). Describe the energy changes if:
- (a) the system = Ball only
- (b) the system = Ball + Earth
▶️Answer/Explanation
(a) System = Ball only:
- Gravity acts as an external force.
- Work done by Earth’s gravity increases the ball’s kinetic energy (KE).
- The ball’s total energy is not conserved because we ignored Earth.
(b) System = Ball + Earth:
- Gravity is now an internal conservative force.
- As the ball falls, gravitational potential energy (PE) decreases and kinetic energy (KE) increases.
- Total mechanical energy KE + PE remains constant.
Example
A block of mass \(m\) is pulled across a rough horizontal surface with a constant force \(F\). Describe the energy changes if:
- (a) the system = Block only
- (b) the system = Block + Surface
▶️Answer/Explanation
(a) System = Block only:
- Pulling force does positive work → increases block’s kinetic energy.
- Friction does negative work → decreases block’s kinetic energy.
- Net energy change depends on whether \( F > f_\text{friction} \) or not.
- Energy of block changes.
(b) System = Block + Surface:
- Friction is now internal.
- Work by applied force transfers energy into the system.
- Some energy becomes block’s KE, some becomes thermal energy of the surface.
- Total energy is conserved, but mechanical energy decreases (converted to heat).