Back to the roller coaster 🎢. At the top, speed is zero but potential energy is maximum. At the bottom, speed is enormous but potential energy is low. Compute $K + U$ at any point on the ride — you get exactly the same number every time 🤯. This is one of the five deep laws of physics, and the single most powerful tool for solving classical mechanics problems.
The core idea in one paragraph 📌
Mechanical energy = $E = K + U$. In the absence of non-conservative forces (friction, air drag, permanent deformation), this quantity is constant: $E_i = E_f$. Energy just swaps between kinetic and potential. If friction is present, mechanical energy decreases, but total energy (mechanical + heat) is still conserved — the universal law of energy conservation, with no known exceptions.
The central equation 📐
$$\boxed{E = K + U = \tfrac{1}{2} m v^2 + U(x) = \text{constant}}$$
Between two states $i$ and $f$ in a conservative system:
$$\boxed{\tfrac{1}{2}m v_i^2 + U_i = \tfrac{1}{2}m v_f^2 + U_f}$$
Why it’s such a powerful tool 💎
With one equation, you can find speed at any point along a path without knowing the acceleration, the timing, or even the trajectory. You only need the endpoints.
Classical example — frictionless slide: a ball is released from rest at height $h = 5$ m. Speed at the bottom of the slide?
$$\underbrace{0 + mgh}{\text{top}} = \underbrace{\tfrac{1}{2}mv^2 + 0}}
\;\Rightarrow\; v = \sqrt{2gh} = \sqrt{98} \approx 9.9\ \text{m/s}$$
Mass drops out! The result is the same for any object (ball, child, pickup truck) — one of physics’ little magic tricks.
The pendulum — conservation in motion 🕰️
A pendulum of length $L$ released from angle $\theta_0$. Speed at the lowest point?
Height above the lowest point: $h = L(1 – \cos\theta_0)$
$$mgL(1 – \cos\theta_0) = \tfrac{1}{2}mv^2 \Rightarrow v = \sqrt{2gL(1 – \cos\theta_0)}$$
For $L = 1$ m, $\theta_0 = 30°$:
$$v = \sqrt{2 \times 9.8 \times 1 \times (1 – 0.866)} \approx 1.62\ \text{m/s}$$
Conservative vs. non-conservative forces ⚔️
| Property | Conservative | Non-conservative |
|---|---|---|
| Examples | Gravity, spring, electric | Friction, air drag |
| Work over a closed loop | Zero | Nonzero (negative) |
| Can be stored as $U$? | ✅ Yes | ❌ No |
| Effect on mechanical energy | Preserves | Converts to heat |
Deep insight: the very definition of a “conservative force” is that its work is path-independent — go from A to B directly or via wild loops, the work is the same. Only such forces let us define a potential energy $U$.
Handling friction 🔥
When friction is present:
$$E_i – E_f = |W_\text{friction}| = f_k \cdot d$$
That is, the drop in mechanical energy = heat generated by friction.
Example: a ball slides from a 5-m high ramp with $\mu = 0.1$, length $L = 10$ m. Speed at the bottom?
$$mgh – \mu m g L \cos\theta = \tfrac{1}{2}m v^2$$
(Assuming a gentle slope with $\cos\theta \approx 1$)
$$v = \sqrt{2g(h – \mu L)} = \sqrt{2 \times 9.8 \times (5 – 1)} = \sqrt{78.4} \approx 8.85\ \text{m/s}$$
(Less than the 9.9 m/s of the frictionless case — exactly as expected)
Python analysis 🐍
1) Ball drop — conservation visualized
import numpy as np
import matplotlib.pyplot as plt
m, g = 1.0, 9.8
h0 = 10.0 # initial height
dt = 0.001
t, y, v = [0], [h0], [0]
while y[-1] > 0:
v_new = v[-1] - g*dt
y_new = y[-1] + v[-1]*dt
t.append(t[-1] + dt)
v.append(v_new)
y.append(y_new)
t, y, v = np.array(t), np.array(y), np.array(v)
K = 0.5 * m * v**2
U = m * g * np.maximum(y, 0)
E = K + U
# Check conservation
print(f"E at t=0: {E[0]:.4f} J")
print(f"E at T/2: {E[len(E)//2]:.4f} J")
print(f"E at end: {E[-1]:.4f} J")
print(f"E variation: {(E.max() - E.min())/E.mean()*100:.4f}%")
# Conserved to numerical precision ~0.01%
plt.plot(t, K, label='K')
plt.plot(t, U, label='U')
plt.plot(t, E, '--', label='E_total', color='black')
plt.xlabel('t (s)'); plt.ylabel('Energy (J)')
plt.legend(); plt.grid()
plt.savefig('conservation.png', dpi=120)
2) With friction — energy leaks out
import numpy as np
m, g = 1.0, 9.8
mu = 0.1 # friction coefficient
h, L = 5.0, 10.0
E_i = m * g * h
W_friction = mu * m * g * L # work of friction (positive to subtract)
E_f = E_i - W_friction
v_f = np.sqrt(2 * E_f / m)
print(f"Initial energy: {E_i:.2f} J")
print(f"Friction heat: {W_friction:.2f} J")
print(f"Final mechanical E: {E_f:.2f} J")
print(f"Speed at bottom: {v_f:.2f} m/s")
# Check: total energy (including heat) is still conserved
print(f"\nTotal (K + heat): {E_f + W_friction:.2f} J = initial E ✓")
3) Pendulum — motion by conservation
import numpy as np
L = 1.0 # pendulum length (m)
theta0 = np.deg2rad(30) # initial angle
g = 9.8
# Speed at the lowest point
h = L * (1 - np.cos(theta0))
v_bottom = np.sqrt(2 * g * h)
# Period for small angles
T = 2 * np.pi * np.sqrt(L/g)
print(f"h at top: {h*100:.1f} cm")
print(f"v at bottom: {v_bottom:.2f} m/s")
print(f"Period: {T:.2f} s")
Take-home summary 🎁
$E = K + U =$ constant (in the absence of friction). A superpower tool for solving problems — speed at any point without needing acceleration or time. Only conservative forces can be assigned a $U$; non-conservative forces (like friction) dissipate energy as heat. Mass cancels out in most such problems — that’s why in a vacuum a feather and a bowling ball hit the ground at the same speed. Python confirms conservation to ~0.01% numerical precision 😎.
“Nice to know” box: the neutrino’s discovery 💡
Conservation of energy is so sacred that when energy seems to be missing, physicists don’t question the law — they question their instruments. In 1930, beta decay ($\beta$-decay) appeared to lose energy. Wolfgang Pauli refused to abandon the law and instead hypothesized a tiny, nearly invisible particle carrying the energy away. He called it a “neutrino” — “little neutral one.” Twenty-six years later, in 1956, Cowan and Reines actually detected the neutrino experimentally. Nobel Prize in 1995. A discovery made possible entirely by trusting a physical law 🎯.
Test yourself 📝
References and further exploration 📚
Articles and reference
- Wikipedia: Conservation of energy, Mechanical energy, Conservative force, Neutrino history
- HyperPhysics — Conservation of Energy
- Feynman Lectures — Vol. I, Ch. 4: Conservation of Energy (essential reading!)
Videos (YouTube)
- MIT OCW 8.01 Walter Lewin: Conservation of energy — pendulum experiment (Lewin risks his life on his own pendulum!)
- Veritasium: What is a neutrino?
- 3Blue1Brown: Why energy is conserved (Noether’s theorem)
- SmarterEveryDay: Pendulum experiments
External simulators
- PhET — Energy Skate Park (best tool for feeling conservation)
- PhET — Pendulum Lab
On this site 🔗
- نسخهی فارسی — پایستگی انرژی مکانیکی
- Previous section — Potential Energy
- Prerequisite — Kinetic Energy
In the next section: work and internal energy — where does the energy lost to friction actually go? 🔥 See you there 👋
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