Potential Energy & Conservation Explained | Chapter 7 of University Physics

Chapter 7 introduces potential energy as stored energy due to position or configuration and develops the principle of conservation of mechanical energy—an indispensable tool for solving physics problems with elegance and efficiency.

Potential Energy

Potential energy (U) represents energy stored within a system. Two key forms appear in this chapter:

Gravitational Potential Energy

For an object of mass m at height y above a reference level:

U_grav = m g y

As the object moves under gravity, its potential energy converts to kinetic energy. The work done by gravity relates to the change in potential:

W_grav = –ΔU_grav

When only gravity does work, K + U remains constant.

Elastic Potential Energy

Springs store energy when displaced by x, following Hooke’s Law (F = k x):

U_el = ½ k x²

Work done by or against a spring between positions x₁ and x₂ is:

W = ½ k x₂² – ½ k x₁²

In pure spring systems, mechanical energy is conserved.

Conservation of Mechanical Energy

When nonconservative forces (e.g., friction) are negligible, the total mechanical energy is constant:

K₁ + U₁ = K₂ + U₂

If nonconservative work (W_other) occurs:

K₁ + U₁ + W_other = K₂ + U₂

Conservative vs. Nonconservative Forces

• Conservative forces (gravity, springs): path-independent work; recoverable; W = –ΔU; total energy conserved.
• Nonconservative forces (friction): path-dependent work; dissipate mechanical energy into heat; mechanical energy not recovered, though overall energy remains conserved.

Force from Potential Energy

A conservative force relates to potential energy gradients:

F_x = –dU/dx (1D)
F_i = –∂U/∂x_i (3D)

Equilibrium points occur where F = 0, i.e., where dU/dx = 0.

Energy Diagrams & Equilibrium

Plotting U(x) vs. position reveals turning points and stability:

• Turning points: where kinetic energy K = 0 and E = U(x).
• Stable equilibrium: local minimum in U(x); small displacements yield restoring forces.
• Unstable equilibrium: local maximum; displacements grow.
• Neutral equilibrium: flat region; no net force.

Conclusion

By reframing work and kinetic energy in terms of potential energy and exploiting energy conservation, you can solve complex mechanics problems more intuitively. Energy diagrams further illuminate system behavior and stability.

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