§8.4 — Energy Conservation

One of the most powerful principles in physics is the conservation of mechanical energy: when only conservative forces act on a system, the total mechanical energy (kinetic plus potential) remains constant. This section derives this principle from the work–energy theorem, explores its scope and limitations, and demonstrates its extraordinary utility in solving problems that would be intractable using Newton's laws alone.

The mechanical energy

Define the total mechanical energy as:

\[ E = KE + U = \frac{1}{2}m\mathbf{v}^2 + U(\mathbf{r}) \]

where \( KE \) is kinetic energy and \( U \) is the sum of all potential energies (gravitational, elastic, etc.).

Derivation from the work–energy theorem

Recall from §7.3 that the work–energy theorem states:

\[ W_{\text{net}} = \Delta KE \]

If only conservative forces act, the net work is:

\[ W_{\text{net}} = W_{\text{conservative}} = -\Delta U \]

(from the definition of potential energy in §8.1).

Combining these:

\[ -\Delta U = \Delta KE \]

\[ \Delta KE + \Delta U = 0 \]

\[ \Delta(KE + U) = 0 \]

Conclusion: The total mechanical energy is constant:

\[ E_i = E_f \]

\[ KE_i + U_i = KE_f + U_f \]

Or equivalently:

\[ E = KE + U = \text{constant} \]

This is one of the most important results in classical mechanics.

Conditions for mechanical energy conservation

Mechanical energy is conserved if and only if:

1. Only conservative forces act (gravity, springs, electric forces)
2. Non-conservative forces are absent or do no work (friction, air drag)
3. The system is isolated (no external agents do work on it)

If any of these conditions is violated, mechanical energy is not conserved.

Example: Ball thrown vertically

A ball of mass \( m = 2 \) kg is thrown upward with initial velocity \( v_0 = 10 \) m/s from ground level. Assuming air resistance is negligible (only gravity acts), what is the maximum height reached?

At ground level (initial state):

• Height: \( h_i = 0 \)
• Velocity: \( v_i = 10 \) m/s
• \( KE_i = \frac{1}{2}mv_i^2 = \frac{1}{2} \times 2 \times 10^2 = 100 \) J
• \( U_i = mgh_i = 0 \) J
• \( E_i = 100 \) J

At maximum height (final state):

• Height: \( h_f = ? \)
• Velocity: \( v_f = 0 \) (momentarily at rest)
• \( KE_f = 0 \) J
• \( U_f = mgh_f = 2 \times 9.8 \times h_f = 19.6 h_f \) J
• \( E_f = 19.6 h_f \) J

By energy conservation:

\[ E_i = E_f \]

\[ 100 = 19.6 h_f \]

\[ h_f = \frac{100}{19.6} \approx 5.1 \text{ m} \]

The ball reaches a maximum height of 5.1 m.

Compare to kinematics (§2.5): Using \( v_f^2 = v_i^2 - 2gh_f \) with \( v_f = 0 \):

\[ h_f = \frac{v_i^2}{2g} = \frac{100}{2 \times 9.8} = \frac{100}{19.6} \approx 5.1 \text{ m} \]

Same result — but the energy method requires no knowledge of intermediate kinematics. We jump directly to the answer.

Example: Roller coaster with friction

A roller coaster car of mass \( 500 \) kg starts at height \( h_1 = 40 \) m with zero velocity. It rolls down to height \( h_2 = 10 \) m, where its velocity is measured to be \( v_2 = 20 \) m/s. How much energy was dissipated by friction?

Initial energy: \[ E_i = KE_i + U_i = 0 + mgh_1 = 500 \times 9.8 \times 40 = 196{,}000 \text{ J} \]

Final energy: \[ E_f = KE_f + U_f = \frac{1}{2}mv_2^2 + mgh_2 \]

\[ = \frac{1}{2} \times 500 \times 20^2 + 500 \times 9.8 \times 10 \]

\[ = 100{,}000 + 49{,}000 = 149{,}000 \text{ J} \]

Energy dissipated by friction:

\[ E_{\text{dissipated}} = E_i - E_f = 196{,}000 - 149{,}000 = 47{,}000 \text{ J} \]

About 24% of the initial energy was lost to friction and heat.

The work–energy theorem with non-conservative forces

If non-conservative forces (friction, air drag) act on the system, mechanical energy is not conserved. Instead:

\[ W_{\text{non-conservative}} = \Delta E = \Delta(KE + U) \]

or equivalently:

\[ E_f = E_i + W_{\text{non-conservative}} \]

If friction does negative work (\( W_{\text{friction}} < 0 \)), mechanical energy decreases:

\[ E_f < E_i \]

This is exactly what happened in the roller coaster example: friction did \( W_{\text{friction}} = -47{,}000 \) J of work.

Energy diagram method

A powerful graphical technique is the energy diagram: plot \( U(x) \) and horizontal lines representing the total energy \( E \) for different scenarios. The kinetic energy at any point is:

\[ KE(x) = E - U(x) \]

Turning points: Where \( KE = 0 \), the object momentarily stops. These are the classical turning points:

\[ E = U(x) \implies v = 0 \]

An object with total energy \( E \) cannot access regions where \( U > E \) (classically forbidden regions).

Example: Spring potential

For a spring, \( U(x) = \frac{1}{2}kx^2 \). If an object has total energy \( E = 10 \) J and the spring constant is \( k = 100 \) N/m, what is the range of motion?

At the turning points, \( E = U \):

\[ 10 = \frac{1}{2} \times 100 \times x^2 \]

\[ x^2 = 0.2 \implies x = \pm 0.447 \text{ m} \]

The object oscillates between \( x = -0.447 \) m and \( x = +0.447 \) m. The amplitude is \( A = 0.447 \) m.

Connection to Chapter 7

§7.3–§7.5 developed the work–energy theorem for arbitrary forces and introduced potential energy for conservative forces. §8.4 now systematizes this: energy conservation is the culmination of the work–energy theorem applied to conservative force systems. It provides an alternative (often simpler) path to solving dynamics problems.

What you should be able to do

After this section, you should be able to:

• State the condition for mechanical energy conservation (only conservative forces)
• Apply conservation of mechanical energy to solve for speeds, heights, or other quantities
• Distinguish between mechanical energy conservation (conservative forces) and the work–energy theorem (arbitrary forces)
• Calculate energy dissipation by friction using \( W_{\text{friction}} = E_i - E_f \)
• Use energy diagrams to identify allowed regions of motion and turning points
• Recognize when a problem can be solved by energy methods (preferred) vs. when kinematics or Newton's laws are necessary

Preview of §8.5: Non-Conservative Forces and Energy Dissipation

Friction, air drag, and other velocity-dependent forces dissipate mechanical energy into heat. §8.5 formalizes the concept of dissipation, explores how much energy is lost in typical systems, and develops techniques to account for non-conservative forces in energy-based problems.

📚 See also: Halliday Vol 1, Ch 8, §8.5 — Conservation of Mechanical Energy. 📖 Open reference: OpenStax University Physics Vol 1 — Chapter 8.5: Conservation of Mechanical Energy. 🎓 Video: MIT 8.01 Lecture 14 (Walter Lewin) — "Energy Conservation and Non-Conservative Forces".

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