Here are your friendly physics notes on mechanical energy conservation – perfect for high school students! 🌟“`html
Understanding Mechanical Energy Conservation
🔑 Key Concepts for NEET
- Conservative forces (like gravity/springs) depend only on start/end positions, not path taken.
- Total mechanical energy (kinetic + potential) is conserved when only conservative forces act.
- At maximum height in projectile/pendulum motion, kinetic energy is minimal while potential energy peaks.
- Spring potential energy = \(\frac{1}{2}kx^2\) (Hooke’s law: \(F_s = -kx\)).
- Work done by conservative forces in a closed path is zero.
⚡ Conservative Forces & Potential Energy
- A force \(F(x)\) is conservative if you can define a potential energy \(V(x)\) where: \[ F(x) = -\frac{dV}{dx} \]
- Work done by conservative forces: \[ \int_{x_i}^{x_f} F(x) dx = V_i – V_f \] Example: Gravity on frictionless ramps 🎢 – object starting at height \(h\) always hits bottom at speed \(\sqrt{2gh}\), regardless of ramp angle!
- Potential energy change: \[ \Delta V = -F(x) \Delta x \quad \text{(Eq. 5.9)} \]
💫 Conservation of Mechanical Energy
- Total mechanical energy = Kinetic energy (\(K\)) + Potential energy (\(V\))
- For conservative forces: \[ \Delta K + \Delta V = 0 \quad \text{(Eq. 5.10)} \] \[ K_i + V(x_i) = K_f + V(x_f) \quad \text{(Eq. 5.11)} \] → Total energy never changes! 🪄
- Why? Work done by conservative forces converts \(K\) ↔ \(V\) perfectly.
🌍 Real-World Examples
1. Falling Ball (Gravity)
- Dropped from height \(H\) → energy converts:
- Top (\(y = H\)): All potential → \(E_H = mgH\)
- Middle (\(y = h\)): Mix → \(E_h = mgh + \frac{1}{2}mv_h^2\)
- Ground (\(y = 0\)): All kinetic → \(E_o = \frac{1}{2}mv_f^2\)
2. Pendulum Swing (Example 5.7)
- Bob mass \(m\), string length \(L\), horizontal speed \(v_o\) at bottom (A).
- At top (C): \[ \text{String slackens!} \quad ⇒ \quad T_c = 0 \] \[ mg = \frac{mv_c^2}{L} \quad ⇒ \quad v_c = \sqrt{gL} \]
- Energy conservation (A → C): \[ \frac{1}{2}mv_o^2 = \frac{1}{2}mv_c^2 + 2mgL \quad ⇒ \quad v_o = \sqrt{5gL} \]
- At B (mid-height): \[ v_B = \sqrt{3gL}, \quad \frac{K_B}{K_C} = \frac{\frac{1}{2}mv_B^2}{\frac{1}{2}mv_C^2} = 3 \]
- After point C: Bob behaves like a projectile! 🚀
3. Spring Potential Energy
- Spring force: \(F_s = -kx\) (Hooke’s law)
- Work done by spring when stretched/compressed by \(x_m\): \[ W_s = -\frac{1}{2}kx_m^2 \quad \text{(Eq. 5.15)} \] Why negative? Spring force opposes displacement.
- Potential energy stored: \[ V_{\text{spring}} = \frac{1}{2}kx^2 \]
💎 Key Takeaways
- Mechanical energy = \(K + V\) – stays constant with conservative forces.
- Energy freely converts between kinetic (motion) and potential (position).
- Non-conservative forces (e.g., friction) break conservation – they turn mechanical energy into heat.
