Mechanical Energy Conservation and Losses

Why It Matters

Many JC mechanics problems are solved by deciding whether mechanical energy is conserved or whether non-conservative work must be included.

Definition

Mechanical energy is the sum of kinetic energy and potential energy:

$$E_{\text{mech}}=K+U$$

If only conservative forces do work, mechanical energy is conserved:

$$K_i+U_i=K_f+U_f$$

Key Representations

• Mechanical energy: $E_{\text{mech}}=K+U$
• Conservative-only motion: $K_i+U_i=K_f+U_f$
• With non-conservative work: $K_i+U_i+W_{\text{nc}}=K_f+U_f$
• With dissipated energy: $E_{\text{initial}}=E_{\text{final}}+E_{\text{dissipated}}$

Including Non-Conservative Work

When non-conservative forces such as friction, drag, or an applied motor force do work, mechanical energy changes:

$$K_i+U_i+W_{\text{nc}}=K_f+U_f$$

where $W_{\text{nc}}$ is the total work done by non-conservative forces on the system.

An equivalent energy-accounting form is:

$$E_{\text{initial}}=E_{\text{final}}+E_{\text{dissipated}}$$

when mechanical energy is transferred into thermal energy, sound, or internal energy.

Ideal Systems

An ideal mechanical system assumes no dissipative losses. Typical examples include:

• A smooth track with negligible friction.
• A spring obeying Hooke’s law with negligible damping.
• A projectile where air resistance is ignored.
• A vertical circular motion problem where only gravity changes the speed.

Real Systems with Losses

Real systems often lose useful mechanical energy through:

• Friction between surfaces.
• Air resistance or fluid drag.
• Internal deformation.
• Sound and heating.

The phrase “energy loss” means loss from the mechanical or useful energy store, not loss from the universe.

Common Exam Points

• Check whether the question states “smooth”, “frictionless”, or “air resistance negligible”.
• Use force equations when acceleration or contact forces are needed.
• Use energy equations when relating speeds and positions without time.
• Do not assume mechanical energy is conserved if an engine, friction, or drag is doing work.

Links

• Related: work energy and power
• Related: energy forms and conservation
• Related: energy forms and conservation
• Related: kinetic energy and work energy theorem
• Related: potential energy and conservative forces
• Related: power and efficiency