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A moving particle has more energy than an identical particle at rest. The kinetic energy (K) is defined as:

• K = (1/2)mv², where m is mass and v is velocity.
• For a system of particles, K = Σ (1/2)m_iv_i²
• The change in kinetic energy (dK) is related to the force (F) and displacement (dr) by: dK = F ⋅ dr.
• Kinetic energy changes when the speed changes, which occurs when the resultant force has a tangential component.

Work (W) is defined as the integral of the dot product of force (F) and displacement (dr):

• W = ∫ F ⋅ dr = ∫ F cosθ dr, where θ is the angle between F and dr.
• The work-energy theorem states that the work done on a particle by the resultant force equals the change in its kinetic energy: W = K₂ - K₁
• The work done by the resultant force is the sum of the work done by individual forces: W = ∫ F₁ ⋅ dr + ∫ F₂ ⋅ dr + ∫ F₃ ⋅ dr + ...

The work-energy theorem states that the work done on a particle by the resultant force is equal to the change in its kinetic energy.

Definition of Potential Energy:
The change in potential energy (U) of a system corresponding to a conservative internal force is defined as:
U_f - U_i = -W = -∫ F ⋅ dr
where W is the work done by the internal conservative force as the system goes from initial (i) to final (f) configuration. Potential energy is not defined for non-conservative forces.

Conservation of Mechanical Energy:
If only conservative internal forces operate and external forces do no work, then the total mechanical energy (K + U) is conserved:
U_f + K_f = U_i + K_i
This is the principle of conservation of mechanical energy.

Non-Conservative Forces:
If non-conservative forces (like friction) are present, mechanical energy is not conserved. The work-energy theorem still applies.

Zero Potential Energy:
Only changes in potential energy are defined. The zero potential energy point can be chosen arbitrarily.

Work-Energy Theorem with Conservative and Non-Conservative Forces:
W_c + W_nc + W_ext = K_f - K_i
where W_c is work done by conservative internal forces, W_nc is work done by non-conservative internal forces, and W_ext is work done by external forces.
Since W_c = -(U_f - U_i), we have:
W_nc + W_ext = (K_f + U_f) - (K_i + U_i) = E_f - E_i
where E is the total mechanical energy.

• Consider a block of mass m near the surface of the Earth raised through a height h. The "earth + block" is the system.
• The gravitational force is conservative, and potential energy can be defined. Earth's acceleration is negligible compared to the block.
• Work done by the gravitational force due to the block on the earth is zero. The force mg on the block does work (-mgh) if the block ascends through height h, increasing potential energy by mgh.
• If a block of mass m ascends a height h (h << radius of earth), the potential energy of the "earth + block" system increases by mgh. If it descends by h, potential energy decreases by mgh.
• It is customary to call the potential energy of the earth-block system as the potential energy of the block only.
• We can choose any position of the block and call the gravitational potential energy to be zero in this position. The potential energy at a height h above this position is mgh.

• Consider a massless spring of natural length l, one end of which is fastened to a wall.