AP Physics C Mechanics- 3.4 Conservation of Energy- Study Notes- New Syllabus
AP Physics C Mechanics- 3.4 Conservation of Energy – Study Notes
AP Physics C Mechanics- 3.4 Conservation of Energy – Study Notes – per latest Syllabus.
Key Concepts:
- Energies Present in a System
- Conservation of Mechanical Energy
- Effect of System Selection on Energy Changes
Energies Present in a System
The total energy of a system depends on the nature of its components and the types of interactions between them. Systems may possess kinetic energy (due to motion), potential energy (due to position or configuration), or a combination of both, depending on whether they include one or multiple interacting objects.
Single-Object System:
A system composed of only a single object can possess only kinetic energy.
- It cannot have potential energy, since potential energy arises from the relative configuration between interacting parts.
- The kinetic energy of a moving object of mass \( \mathrm{m} \) and velocity \( \mathrm{v} \) is given by:
\( \mathrm{K = \dfrac{1}{2}mv^2} \)
- Kinetic energy is a scalar quantity and always positive.
- It represents the energy associated with the object’s motion relative to the chosen reference frame.
Multi-Object or Interacting System:
A system that includes two or more interacting objects or can change shape reversibly may possess both:
- Kinetic Energy — due to motion of the components.
- Potential Energy — due to the configuration or internal interactions among the components (via conservative forces).
- Such systems can store energy internally and exchange it between kinetic and potential forms while conserving total mechanical energy (if no nonconservative forces act).
Examples of Systems with Both Energies:
- A mass oscillating on a spring (elastic potential + kinetic).
- A pendulum swinging (gravitational potential + kinetic).
- A planet orbiting a star (gravitational potential + kinetic).
System Energy Overview:
| Type of System | Energies Present | Description |
|---|---|---|
| Single object | Kinetic only | Energy due to motion relative to reference frame. |
| Two or more interacting objects | Kinetic + Potential | Energy due to motion and configuration (e.g., spring, gravity). |
| Deformable or reversible system | Kinetic + Elastic Potential | Energy stored through temporary deformation (e.g., rubber band, spring). |
Example:
A \( \mathrm{0.5\,kg} \) block attached to an ideal horizontal spring ( \( \mathrm{k = 200\,N/m} \) ) is displaced \( \mathrm{0.1\,m} \) from equilibrium and released from rest. Describe the types of energy present in the system at (a) the instant of release and (b) when the block passes through equilibrium.
▶️ Answer / Explanation
(a) At the instant of release:
- The block is momentarily at rest (\( \mathrm{v = 0} \)), so \( \mathrm{K = 0} \).
- The spring is stretched, storing elastic potential energy:
\( \mathrm{U_s = \dfrac{1}{2}k(\Delta x)^2 = \dfrac{1}{2}(200)(0.1)^2 = 1.0\,J} \)
(b) When passing through equilibrium:
- The spring is at its natural length (\( \mathrm{\Delta x = 0} \)), so \( \mathrm{U_s = 0} \).
- All energy is now kinetic as the block moves fastest through equilibrium:
- \( \mathrm{K = 1.0\,J} \)
Conclusion: The system continually exchanges energy between potential and kinetic forms while the total mechanical energy remains constant.
Conservation of Mechanical Energy
Mechanical energy is the sum of a system’s kinetic energy and potential energy. It represents the total energy associated with the motion and configuration of a system’s components. According to the principle of conservation of mechanical energy, the total mechanical energy of a system remains constant when only conservative forces act within it.
Expression for Total Mechanical Energy:
\( \mathrm{E_{mech} = K + U} \)
- \( \mathrm{K} \): kinetic energy of the system.
- \( \mathrm{U} \): potential energy due to conservative forces.
- In the absence of nonconservative forces (like friction or air resistance), \( \mathrm{E_{mech}} \) is constant:
\( \mathrm{E_{mech,\,i} = E_{mech,\,f}} \quad \Rightarrow \quad K_i + U_i = K_f + U_f} \)
Energy Transformation Within a System:
Energy may be transformed between kinetic and potential forms within the system.
- As potential energy decreases, kinetic energy increases by the same amount, and vice versa:
\( \mathrm{\Delta K + \Delta U = 0} \)
- This means no net gain or loss of total energy — only redistribution between forms.
Systems with Energy Transfer to or from the Surroundings:
- If the total energy of a system changes, the change equals the energy transferred into or out of the system.
- This can occur through work or heat transfer across the system boundary:
\( \mathrm{\Delta E_{total} = W_{ext} + Q} \)
- \( \mathrm{W_{ext}} \): work done on the system by external forces.
- \( \mathrm{Q} \): energy transferred as heat (if applicable).
Choosing the System:
- A system can be defined so that the total mechanical energy remains constant (isolated system), if external energy transfer is negligible.
- Expanding or restricting the system boundary changes whether energy is considered internal (potential) or external (transferred).
Example:
A \( \mathrm{1.0\,kg} \) ball is released from rest at a height of \( \mathrm{5.0\,m} \) above the ground. Assuming no air resistance, find its speed when it has fallen to a height of \( \mathrm{2.0\,m} \). Take \( \mathrm{g = 9.8\,m/s^2} \).
▶️ Answer / Explanation
Step 1: Apply conservation of mechanical energy:
\( \mathrm{K_i + U_i = K_f + U_f} \)
Step 2: Substitute known terms:
- At the top: \( \mathrm{K_i = 0}, \; U_i = mgh_i = (1.0)(9.8)(5.0) = 49.0\,J} \)
- At height \( \mathrm{2.0\,m} \): \( \mathrm{U_f = mgh_f = (1.0)(9.8)(2.0) = 19.6\,J} \)
Step 3: The difference in potential energy equals the gain in kinetic energy:
\( \mathrm{\Delta U = -\Delta K} \Rightarrow K_f = 49.0 – 19.6 = 29.4\,J} \)
Step 4: Find the speed:
\( \mathrm{K_f = \dfrac{1}{2}mv^2 \Rightarrow v = \sqrt{\dfrac{2K_f}{m}} = \sqrt{\dfrac{2(29.4)}{1.0}} = 7.67\,m/s} \)
Result: The ball’s speed at \( \mathrm{2.0\,m} \) height is \( \mathrm{7.7\,m/s} \).
Effect of System Selection on Energy Changes
The selection of a system determines whether its total energy appears to change or remain constant. While energy is always conserved in the universe, the apparent change in the energy of a chosen system depends on how the system boundary is defined and whether energy crosses that boundary as work or heat.
Energy Conservation Principle:
- Energy cannot be created or destroyed — only transformed or transferred.
- In any interaction, the total energy of the universe remains constant:
\( \mathrm{\Delta E_{universe} = 0} \)
- However, the total energy of a selected system may appear to change if energy enters or leaves the system boundary.
Isolated System (No Energy Transfer):
- If no external work is done on the system and there are no nonconservative forces within it, the total mechanical energy remains constant:
\( \mathrm{W_{ext} = 0 \;\; \Rightarrow \;\; E_{mech,\,i} = E_{mech,\,f}} \)
- This means the system’s internal energy changes only through transformation between kinetic and potential forms.
- Examples: a freely falling object (neglecting air resistance), or a mass–spring oscillator in ideal conditions.
Non-Isolated System (Energy Transfer Occurs):
- If external work is done on the system or by the system, then energy is transferred between the system and its surroundings.
- This results in a change in the system’s total mechanical energy:
\( \mathrm{\Delta E_{system} = W_{ext} + Q} \)
- \( \mathrm{W_{ext}} \): work done on the system by external forces.
- \( \mathrm{Q} \): energy transferred as heat (if relevant).
- If \( \mathrm{W_{ext} > 0} \), energy enters the system; if \( \mathrm{W_{ext} < 0} \), energy leaves the system.
Importance of System Definition:
- The choice of system boundary determines whether an energy transfer is internal (conservative) or external (nonconservative).
- By including or excluding certain objects, one can simplify analysis and decide which forces are considered internal vs. external.
Example:
A block slides down a frictionless incline. Compare the energy change for two different system selections:
- System A: the block alone
- System B: the block + Earth
▶️ Answer / Explanation
Case 1: System A (block only)
- The block’s potential energy decreases as it slides down, and its kinetic energy increases.
- However, from the block’s perspective, gravity is an external force doing work on the system.
- Thus, \( \mathrm{W_{ext} \neq 0} \), and the block’s energy changes.
Case 2: System B (block + Earth)
- Now, gravitational interaction is internal to the system.
- Energy only transforms between potential and kinetic forms.
- No energy crosses the boundary, so total mechanical energy is conserved.
Result:
- System A → non-isolated → \( \mathrm{E_{system}} \) changes (due to external work).
- System B → isolated → \( \mathrm{E_{system}} \) constant.